Engineering Physics (EP) 548 Engineering analysis II...University Course Engineering Physics (EP)...
Transcript of Engineering Physics (EP) 548 Engineering analysis II...University Course Engineering Physics (EP)...
University Course
Engineering Physics (EP) 548Engineering analysis II
University of Wisconsin, MadisonSpring 2017
My Class Notes
Nasser M. Abbasi
Spring 2017
Contents
1 Introduction 11.1 links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Resources 52.1 Review of ODEβs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Sturm Liouville . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 HWs 393.1 HW1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 HW2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.3 HW3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.4 HW4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1783.5 HW5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1893.6 HW6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4 Study notes 2094.1 question asked 2/7/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104.2 note on Airy added 1/31/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.3 note p1 added 2/3/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2134.4 note added 1/31/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2204.5 note p3 figure 3.4 in text (page 92) reproduced in color . . . . . . . . . . . . 2254.6 note p4 added 2/15/17 animations of the solutions to ODE from lecture 2/9/17
for small parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.7 note p5 added 2/15/17 animation figure 9.5 in text page 434 . . . . . . . . . . 2314.8 note p7 added 2/15/17, boundary layer problem solved in details . . . . . . . 2374.9 note p9. added 2/24/17, asking question about problem in book . . . . . . . . 2434.10 note p11. added 3/7/17, Showing that scaling for normalization is same for
any π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.11 note p12. added 3/8/17, comparing exact solution to WKB solution using
Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.12 Convert ODE to Liouville form . . . . . . . . . . . . . . . . . . . . . . . . . . . 2524.13 note p13. added 3/15/17, solving π2π¦β³(π₯) = (π + π₯ + π₯2)π¦(π₯) in Maple . . . . . . 254
5 Exams 255
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Contents CONTENTS
5.1 Exam 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2565.2 Exam 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
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Chapter 1
Introduction
Local contents1.1 links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
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1.1. links CHAPTER 1. INTRODUCTION
1.1 links
1. Professor Leslie M. Smith web page
2. piazza class page Needs login
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1.2. syllabus CHAPTER 1. INTRODUCTION
1.2 syllabus
NE 548
Engineering Analysis II
TR 11:00-12:15 in Van Vleck B341
Leslie Smith: [email protected], Office Hours in Van Vleck Tuesday/Thursday 12:30-2:00, http://www.math.wisc.edu/ lsmith.
Textbook 1 Required: Advanced Mathematical Methods for Scientists and Engineers,Bender and Orszag, Springer.
Textbook 2 Recommended: Applied Partial Differential Equations, Haberman, Pear-son/Prentice Hall. This text is recommended because some of you may already own it (fromMath 322). Almost any intermediate-advanced PDEs text would be suitable alternative asreference.
Pre-requisite: NE 547. If you have not taken NE 547, you should have previously takencourses equivalent to Math 319, 320 or 340, 321, 322.
Assessment: Your grade for the course will be based on two take-home midterm exams(35% each) and selected homework solutions (30%). The homework and midterms examsfor undergraduate and graduate students will be different in scope; please see the separatelearning outcomes below. Graduate students will be expected to synthesize/analyze materialat a deeper level.
Undergraduate Learning Outcomes:
β’ Students will demonstrate knowledge of asymptotic methods to analyze ordinary differ-ential equations, including (but not limited to) boundary layer theory, WKB analysisand multiple-scale analysis.
β’ Students will demonstrate knowledge of commonly used methods to analyze partialdifferential equations, including (but not limited to) Fourier analysis, Greenβs functionsolutions, similarity solutions, and method of characteristics.
β’ Students will apply methods to idealized problems motivated by applications. Such ap-plications include heat conduction (the heat equation), quantum mechanics (Schrodingerβsequation) and plasma turbulence (dispersive wave equations).
Graduate Learning Outcomes:
β’ Students will demonstrate knowledge of asymptotic methods to analyze ordinary differ-ential equations, including (but not limited to) boundary layer theory, WKB analysisand multiple-scale analysis.
β’ Students will demonstrate knowledge of commonly used methods to analyze partialdifferential equations, including (but not limited to) Fourier analysis, Greenβs functionsolutions, similarity solutions, and method of characteristics.
β’ Students will apply methods to idealized problems motivated by applications. Such ap-plications include heat conduction (the heat equation), quantum mechanics (Schrodingerβsequation) and plasma turbulence (dispersive wave equations).
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1.2. syllabus CHAPTER 1. INTRODUCTION
β’ Students will apply methods to solve problems in realistic physical settings.
β’ Students will synthesize multiple techniques to solve equations arising from applica-tions.
Grading Scale for Final Grade: 92-100 A, 89-91 AB, 82-88 B, 79-81 BC, 70-78 C, 60-69D, 59 and below F
Midterm 1: Given out Thursday March 2, 2015 and due Thursday March 9, 2017.
Midterm 2: Given out Thursday April 27, 2017 and due Thursday May 4, 2015.
Homework: Homework problems will be assigned regularly, either each week or every otherweek, paced for 6 hours out-of-class work every week. Homework groups are encouraged,but each student should separately submit solutions reflecting individual understanding ofthe material.
Piazza: There will be a Piazza course page where all course materials will be posted. Piazzais also a forum to facilitate peer-group discussions. Please take advantage of this resource tokeep up to date on class notes, homework and discussions.
Piazza Sign-Up Page: piazza.com/wisc/spring2017/neema548
Piazza Course Page: piazza.com/wisc/spring2017/neema548/home
Course Outline
Part I: Intermediate-Advanced Topics in ODEs from Bender and Orszag.
1. Review of local analysis of ODEs near ordinary points, regular singular points and irregularsingular points (BO Chapter 3, 1.5 weeks)
2. Global analysis using boundary layer theory (BO Chapter 9, 1.5 weeks).
3. Global analysis using WKB theory (BO Chapter 10, 1.5 weeks).
4. Greenβs function solutions (1 week)
5. Multiple-scale analysis (BO Chapter 11, 1.5 weeks).
Part II: Intermediate-Advanced Topics in PDEs
1. Review of Sturm-Liouville theory and eigenfunction expansions (1.5 weeks)
2. Non-homogeneous problems and Greenβs function solutions (1.5 weeks)
3. Infinite domain problems and Fourier transforms (1.5 weeks)
4. Quasilinear PDEs (1.5 weeks)
5. Dispersive wave systems (time remaining)
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Chapter 2
Resources
Local contents2.1 Review of ODEβs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Sturm Liouville . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
2.1 Review of ODEβs
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.1. Review of ODEβs CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
2.2 Sturm Liouville
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.2. Sturm Liouville CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
2.3 Wave equation
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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2.3. Wave equation CHAPTER 2. RESOURCES
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Chapter 3
HWs
Local contents3.1 HW1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 HW2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.3 HW3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.4 HW4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1783.5 HW5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1893.6 HW6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
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3.1 HW1
3.1.1 problem 3.3 (page 138)
3.1.1.1 Part c
problem Classify all the singular points (finite and infinite) of the following
π₯ (1 β π₯) π¦β²β² + (π β (π + π + 1) π₯) π¦β² β πππ¦ = 0
Answer
Writing the DE in standard form
π¦β²β² + π (π₯) π¦β² + π (π₯) π¦ = 0
π¦β²β² +π β (π + π + 1) π₯
π₯ (1 β π₯)π¦β² β
πππ₯ (1 β π₯)
π¦ = 0
π¦β²β² +
π(π₯)
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
ππ₯ (1 β π₯)
β(π + π + 1)(1 β π₯) οΏ½π¦β² β
π(π₯)
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½πππ₯ (1 β π₯)
π¦ = 0
π₯ = 0, 1 are singular points for π (π₯) as well as for π (π₯). Now we classify what type of singularityeach point is. For π (π₯)
limπ₯β0
π₯π (π₯) = limπ₯β0
π₯ οΏ½π
π₯ (1 β π₯)β(π + π + 1)(1 β π₯) οΏ½
= limπ₯β0
οΏ½π
(1 β π₯)βπ₯ (π + π + 1)(1 β π₯) οΏ½
= π
Hence π₯π (π₯) is analytic at π₯ = 0. Therefore π₯ = 0 is regular singularity point. Now we checkfor π (π₯)
limπ₯β0
π₯2π (π₯) = limπ₯β0
π₯2βππ
π₯ (1 β π₯)
= limπ₯β0
βπ₯ππ(1 β π₯)
= 0
Hence π₯2π (π₯) is also analytic at π₯ = 0. Therefore π₯ = 0 is regular singularity point. Now we
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3.1. HW1 CHAPTER 3. HWS
look at π₯ = 1 and classify it. For π (π₯)
limπ₯β1
(π₯ β 1) π (π₯) = limπ₯β1
(π₯ β 1) οΏ½π
π₯ (1 β π₯)β(π + π + 1)(1 β π₯) οΏ½
= limπ₯β1
(π₯ β 1) οΏ½βπ
π₯ (π₯ β 1)+(π + π + 1)(π₯ β 1) οΏ½
= limπ₯β1
οΏ½βππ₯+ (π + π + 1)οΏ½
= βπ + (π + π + 1)
Hence (π₯ β 1) π (π₯) is analytic at π₯ = 1. Therefore π₯ = 1 is regular singularity point. Now wecheck for π (π₯)
limπ₯β1
(π₯ β 1)2 π (π₯) = limπ₯β1
(π₯ β 1)2 οΏ½βππ
π₯ (1 β π₯)οΏ½
= limπ₯β1
(π₯ β 1)2 οΏ½ππ
π₯ (π₯ β 1)οΏ½
= limπ₯β1
(π₯ β 1) οΏ½πππ₯ οΏ½
= 0
Hence (π₯ β 1)2 π (π₯) is analytic also at π₯ = 1. Therefore π₯ = 1 is regular singularity point.Therefore π₯ = 0, 1 are regular singular points for the ODE. Now we check for π₯ at β. Tocheck the type of singularity, if any, at π₯ = β, the DE is first transformed using
π₯ =1π‘
(1)
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3.1. HW1 CHAPTER 3. HWS
This transformation will always results in 1 new ODE in π‘ of this form
π2π¦ππ‘2
+οΏ½βπ (π‘) + 2π‘οΏ½
π‘2+π (π‘)π‘4π¦ = 0 (2)
But
π (π‘) = π (π₯)οΏ½π₯= 1
π‘= οΏ½
π β (π + π + 1) π₯π₯ (1 β π₯) οΏ½
π₯= 1π‘
=
ββββββββββ
π β (π + π + 1) 1π‘1π‘οΏ½1 β 1
π‘οΏ½
ββββββββββ
=ππ‘2 β π‘ (π + π + 1)
(π‘ β 1)(3)
And
π (π‘) = π (π₯)οΏ½π₯= 1
π‘= οΏ½β
πππ₯ (1 β π₯)οΏ½
π₯= 1π‘
= βππ
1π‘οΏ½1 β 1
π‘οΏ½
= βπππ‘2
(π‘ β 1)(4)
1Let π₯ = 1π‘, then
πππ₯
=πππ‘
ππ‘ππ₯
= βπ‘2πππ‘
And
π2
ππ₯2=
πππ₯ οΏ½
πππ₯οΏ½
= οΏ½βπ‘2πππ‘ οΏ½ οΏ½
βπ‘2πππ‘ οΏ½
= βπ‘2πππ‘ οΏ½
βπ‘2πππ‘ οΏ½
= βπ‘2 οΏ½β2π‘πππ‘
β π‘2π2
ππ‘2 οΏ½
= 2π‘3πππ‘
+ π‘4π2
ππ‘2
The original ODE becomes
οΏ½2π‘3πππ‘
+ π‘4π2
ππ‘2 οΏ½π¦ + π (π₯)οΏ½
π₯= 12οΏ½βπ‘2
πππ‘ οΏ½
π¦ + π (π₯)οΏ½π₯= 1
π‘
π¦ = 0
οΏ½2π‘3π¦β² + π‘4π¦β²β²οΏ½ β π‘2π (π‘) π¦β² + π (π‘) π¦ = 0
π‘4π¦β²β² + οΏ½βπ‘2π (π‘) + 2π‘3οΏ½ π¦β² + π (π‘) π¦ = 0
π¦β²β² +οΏ½βπ (π‘) + 2π‘οΏ½
π‘2π¦β² +
π (π‘)π‘4
π¦ = 0
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3.1. HW1 CHAPTER 3. HWS
Substituting equations (3,4) into (2) gives
π¦β²β² +οΏ½β οΏ½ ππ‘
2βπ‘(π+π+1)(π‘β1)
οΏ½ + 2π‘οΏ½
π‘2+οΏ½β πππ‘2
(π‘β1)οΏ½
π‘4π¦ = 0
π¦β²β² +οΏ½2π‘ (π‘ β 1) β π‘2π + (π + π + 1) π‘οΏ½
π‘2 (π‘ β 1)π¦β² β
πππ‘2 (π‘ β 1)
π¦ = 0
Expanding
π¦β²β² +
π(π‘)
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½2π‘ β 1 β π‘π + π + π
π‘ (π‘ β 1) οΏ½π¦β² β
π(π‘)
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½πππ‘2 (π‘ β 1)
π¦ = 0
We see that π‘ = 0 (this means π₯ = β) is singular point for both π (π₯) , π (π₯). Now we checkwhat type it is. For π (π‘)
limπ‘β0
π‘π (π‘) = limπ‘β0
π‘ οΏ½2π‘β
π(π‘ β 1)
+(π + π + 1)π‘ (π‘ β 1) οΏ½
= limπ‘β0
οΏ½2 βπ‘π
(π‘ β 1)+(π + π + 1)(π‘ β 1) οΏ½
= 1 β π β π
π‘π (π‘) is therefore analytic at π‘ = 0. Hence π‘ = 0 is regular singular point. Now we checkfor π (π‘)
limπ‘β0
π‘2π (π‘) = limπ‘β0
π‘2 οΏ½βππ
π‘2 (π‘ β 1)οΏ½
= limπ‘β0
οΏ½βππ(π‘ β 1)οΏ½
= ππ
π‘2π (π‘) is therefore analytic at π‘ = 0. Hence π‘ = 0 is regular singular point for π (π‘). Thereforeπ‘ = 0 is regular singular point which mean that π₯ β β is a regular singular point for theODE.
Summary
Singular points are π₯ = 0, 1. Both are regular singular points. Also π₯ = β is regular singularpoint.
3.1.1.2 Part (d)
Problem Classify all the singular points (finite and infinite) of the following
π₯π¦β²β² + (π β π₯) π¦β² β ππ¦ = 0
solution
Writing the ODE in standard for
π¦β²β² +(π β π₯)π₯
π¦β² βππ₯π¦ = 0
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3.1. HW1 CHAPTER 3. HWS
We see that π₯ = 0 is singularity point for both π (π₯) and π (π₯). Now we check its type. For π (π₯)
limπ₯β0
π₯π (π₯) = limπ₯β0
π₯(π β π₯)π₯
= π
Hence π₯π (π₯) is analytic at π₯ = 0. Therefore π₯ = 0 is regular singularity point for π (π₯). For π (π₯)
limπ₯β0
π₯2π (π₯) = limπ₯β0
π₯2 οΏ½βππ₯οΏ½
= limπ₯β0
(βππ₯)
= 0
Hence π₯2π (π₯) is analytic at π₯ = 0. Therefore π₯ = 0 is regular singularity point for π (π₯).
Now we check for π₯ at β. To check the type of singularity, if any, at π₯ = β, the DE is firsttransformed using
π₯ =1π‘
(1)
This results in (as was done in above part)
π2π¦ππ‘2
+οΏ½βπ (π‘) + 2π‘οΏ½
π‘2+π (π‘)π‘4π¦ = 0
Where
π (π‘) =(π β π₯)π₯
οΏ½π₯= 1
π‘
=οΏ½π β 1
π‘οΏ½
1π‘
= (ππ‘ β 1)
And π (π‘) = β ππ₯ = βππ‘ Hence the new ODE is
π2π¦ππ‘2
+(β (ππ‘ β 1) + 2π‘)
π‘2βππ‘π‘4π¦ = 0
π2π¦ππ‘2
+βππ‘ + 1 + 2π‘
π‘2βππ‘3π¦ = 0
Therefore π‘ = 0 (or π₯ = β) is singular point. Now we will find the singularity type
limπ‘β0
π‘π (π‘) = limπ‘β0
π‘ οΏ½βππ‘ + 1 + 2π‘
π‘2 οΏ½
= limπ‘β0
οΏ½βππ‘ + 1 + 2π‘
π‘ οΏ½
= β
Hence π‘π (π‘) is not analytic, since the limit do not exist, which means π‘ = 0 is an irregular
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3.1. HW1 CHAPTER 3. HWS
singular point for π (π‘) . We can stop here, but will also check for π (π‘)
limπ‘β0
π‘2π (π‘) = limπ‘β0
π‘2 οΏ½βππ‘3οΏ½
= limπ‘β0
οΏ½βππ‘οΏ½
= β
Therefore, π‘ = 0 is irregular singular point, which means π₯ = β is an irregular singular point.
Summary
π₯ = 0 is regular singular point. π₯ = β is an irregular singular point.
3.1.2 problem 3.4
3.1.2.1 part d
problem Classify π₯ = 0 and π₯ = β of the following
π₯2π¦β²β² = π¦π1π₯
Answer
In standard form
π¦β²β² βπ1π₯
π₯2π¦ = 0
Hence π (π₯) = 0, π (π₯) = β π1π₯
π₯2 . The singularity is π₯ = 0. We need to check on π (π₯) only.
limπ₯β0
π₯2π (π₯) = limπ₯β0
π₯2ββββββββπ1π₯
π₯2
βββββββ
= limπ₯β0
οΏ½βπ1π₯ οΏ½
The above is not analytic, since limπ₯β0+ οΏ½βπ1π₯ οΏ½ = β while limπ₯β0β οΏ½βπ
1π₯ οΏ½ = 0. This means π
1π₯ is
not diοΏ½erentiable at π₯ = 0. Here is plot π1π₯ near π₯ = 0
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3.1. HW1 CHAPTER 3. HWS
-0.4 -0.2 0.0 0.2 0.4
0
200
400
600
x
f(x)
Plot of exp(1/x)
Therefore π₯ = 0 is an irregular singular point. We now convert the ODE using π₯ = 1π‘ in order
to check what happens at π₯ = β. This results in (as was done in above part)
π2π¦ππ‘2
+π (π‘)π‘4π¦ = 0
But
π (π‘) = π (π₯)οΏ½π₯= 1
π‘
=
ββββββββπ1π₯
π₯2
βββββββ π₯= 1
π‘
= βπ‘2ππ‘
Hence the ODE becomes
π¦β²β² ββπ‘2ππ‘
π‘4π¦ = 0
π¦β²β² +ππ‘
π‘2π¦ = 0
We now check π (π‘).
limπ‘β0
π‘2π (π‘) = limπ‘β0
π‘2ππ‘
π‘2= lim
π‘β0ππ‘
= 1
This is analytic. Hence π‘ = 0 is regular singular point, which means π₯ = β is regular singularpoint.
Summary π₯ = 0 is irregular singular point, π₯ = β is regular singular point.
46
3.1. HW1 CHAPTER 3. HWS
3.1.2.2 Part e
problem
Classify π₯ = 0 and π₯ = β of the following
(tan π₯) π¦β² = π¦Answer
π¦β² β1
tan π₯π¦ = 0
The function tan π₯ looks like
-Ο - Ο
20 Ο
2Ο
-6
-4
-2
0
2
4
6
x
tan(x)
tan(x)
Therefore, tan (π₯) is not analytic at π₯ = οΏ½π β 12οΏ½ π for π β β€. Hence the function 1
tan(π₯) is not
analytic at π₯ = ππ as seen in this plot
-Ο - Ο
20 Ο
2Ο
-6
-4
-2
0
2
4
6
x
1/tan(x)
1/tan(x)
47
3.1. HW1 CHAPTER 3. HWS
Hence singular points are π₯ = {β― , β2π, βπ, 0, π, 2π,β―}. Looking at π₯ = 0
limπ₯β0
π₯π (π₯) = limπ₯β0
οΏ½π₯1
tan (π₯)οΏ½
= limπ₯β0
βββββββ
ππ₯ππ₯
π tan(π₯)ππ₯
βββββββ
= limπ₯β0
1sec2 π₯
= limπ₯β0
cos2 π₯
= 1
Therefore the point π₯ = 0 is regular singular point. To classify π₯ = β, we use π₯ = 1π‘ substitu-
tion. πππ₯ =
πππ‘
ππ‘ππ₯ = βπ‘
2 πππ‘ and the ODE becomes
οΏ½βπ‘2πππ‘οΏ½
π¦ β1
tan οΏ½1π‘ οΏ½π¦ = 0
βπ‘2π¦β² β1
tan οΏ½1π‘ οΏ½π¦ = 0
π¦β² +1
π‘2 tan οΏ½1π‘ οΏ½π¦ = 0
Hence
π (π‘) =1
π‘2 tan οΏ½1π‘ οΏ½
This function has singularity at π‘ = 0 and at π‘ = 1ππ for π β β€. We just need to consider π‘ = 0
48
3.1. HW1 CHAPTER 3. HWS
since this maps to π₯ = β. Hence
limπ‘β0
π‘π (π‘) = limπ‘β0
ββββββββββπ‘
1
π‘2 tan οΏ½1π‘ οΏ½
ββββββββββ
= limπ‘β0
ββββββββββ
1
π‘ tan οΏ½1π‘ οΏ½
ββββββββββ
= limπ‘β0
ββββββββββ
1π‘
tan οΏ½1π‘ οΏ½
ββββββββββ
βLβhopitals
limπ‘β0
ββββββββββββ
β 1π‘2
βsec2οΏ½ 1π‘ οΏ½
π‘2
ββββββββββββ
= limπ‘β0
ββββββββββ
1
sec2 οΏ½1π‘ οΏ½
ββββββββββ
= limπ‘β0
οΏ½cos2 οΏ½1π‘ οΏ½οΏ½
The following is a plot of cos2 οΏ½1π‘ οΏ½ as π‘ goes to zero.
- Ο
20 Ο
2
0.0
0.2
0.4
0.6
0.8
1.0
t
f(t)
cos21
t
This is the same as asking for limπ₯ββ cos2 (π₯) which does not exist, since cos (π₯) keepsoscillating, hence it has no limit. Therefore, we conclude that π‘π (π‘) is not analytic at π‘ = 0,hence π‘ is irregular singular point, which means π₯ = β is an irregular singular point.
Summary
π₯ = 0 is regular singular point and π₯ = β is an irregular singular point
49
3.1. HW1 CHAPTER 3. HWS
3.1.3 Problem 3.6
3.1.3.1 part b
Problem Find the Taylor series expansion about π₯ = 0 of the solution to the initial valueproblem
π¦β²β² β 2π₯π¦β² + 8π¦ = 0π¦ (0) = 0π¦β² (0) = 4
solution
Since π₯ = 0 is ordinary point, then we can use power series solution
π¦ (π₯) =βοΏ½π=0
πππ₯π
Hence
π¦β² (π₯) =βοΏ½π=0
ππππ₯πβ1 =βοΏ½π=1
ππππ₯πβ1 =βοΏ½π=0
(π + 1) ππ+1π₯π
π¦β²β² (π₯) =βοΏ½π=0
π (π + 1) ππ+1π₯πβ1 =βοΏ½π=1
π (π + 1) ππ+1π₯πβ1 =βοΏ½π=0
(π + 1) (π + 2) ππ+2π₯π
Therefore the ODE becomesβοΏ½π=0
(π + 1) (π + 2) ππ+2π₯π β 2π₯βοΏ½π=0
(π + 1) ππ+1π₯π + 8βοΏ½π=0
πππ₯π = 0
βοΏ½π=0
(π + 1) (π + 2) ππ+2π₯π ββοΏ½π=0
2 (π + 1) ππ+1π₯π+1 +βοΏ½π=0
8πππ₯π = 0
βοΏ½π=0
(π + 1) (π + 2) ππ+2π₯π ββοΏ½π=1
2ππππ₯π +βοΏ½π=0
8πππ₯π = 0
Hence, for π = 0 we obtain(π + 1) (π + 2) ππ+2π₯π + 8πππ₯π = 0
2π2 + 8π0 = 0π2 = β4π0
For π β₯ 1(π + 1) (π + 2) ππ+2 β 2πππ + 8ππ = 0
ππ+2 =2πππ β 8ππ(π + 1) (π + 2)
=ππ (2π β 8)
(π + 1) (π + 2)Hence for π = 1
π3 =π1 (2 β 8)
(1 + 1) (1 + 2)= βπ1
50
3.1. HW1 CHAPTER 3. HWS
For π = 2
π4 =π2 (2 (2) β 8)(2 + 1) (2 + 2)
= β13π2 = β
13(β4π0) =
43π0
For π = 3
π5 =π3 (2 (3) β 8)(3 + 1) (3 + 2)
= β110π3 = β
110(βπ1) =
110π1
For π = 4
π6 =π4 (2 (4) β 8)(4 + 1) (4 + 2)
= 0
For π = 5
π7 =π5 (2 (5) β 8)(5 + 1) (5 + 2)
=121π5 =
121 οΏ½
110π1οΏ½ =
1210
π1
For π = 6
π8 =π6 (2 (6) β 8)(6 + 1) (6 + 2)
=114π6 = 0
For π = 7
π9 =π7 (2 (7) β 8)(7 + 1) (7 + 2)
=112π7 =
112 οΏ½
1210
π1οΏ½ =1
2520π1
Writing now few terms
π¦ (π₯) =βοΏ½π=0
πππ₯π
= π0π₯0 + π1π₯1 + π2π₯2 + π3π₯3 + π4π₯4 + π5π₯5 + π6π₯6 + π7π₯7 + π8π₯8 + π9π₯9 +β―
= π0 + π1π₯ + (β4π0) π₯2 + (βπ1) π₯3 +43π0π₯4 +
110π1π₯5 + 0 +
1210
π1π₯7 + 0 +1
2520π1π₯9 +β―
= π0 + π1π₯ β 4π0π₯2 β π1π₯3 +43π0π₯4 +
110π1π₯5 +
1210
π1π₯7 +1
2520π1π₯9 +β―
= π0 οΏ½1 β 4π₯2 +43π₯4οΏ½ + π1 οΏ½π₯ β π₯3 +
110π₯5 +
1210
π₯7 +1
2520π₯9 +β―οΏ½ (1)
We notice that π0 terms terminates at 43π₯
4 but the π1 terms do not terminate. Now we needto find π0, π1 from initial conditions. At π₯ = 0, π¦ (0) = 0. Hence from (1)
0 = π0Hence the solution becomes
π¦ (π₯) = π1 οΏ½π₯ β π₯3 +110π₯5 +
1210
π₯7 +1
2520π₯9 +β―οΏ½ (2)
Taking derivative of (2), term by term
π¦β² (π₯) = π1 οΏ½1 β 3π₯2 +510π₯4 +
7210
π₯6 +9
2520π₯8 +β―οΏ½
Using π¦β² (0) = 4 the above becomes
4 = π1
51
3.1. HW1 CHAPTER 3. HWS
Hence the solution is
π¦ (π₯) = 4 οΏ½π₯ β π₯3 +110π₯5 +
1210
π₯7 +1
2520π₯9 +β―οΏ½
Or
π¦ (π₯) = 4π₯ β 4π₯3 + 25π₯
5 + 2105π₯
7 + 1630π₯
9 +β―
The above is the Taylor series of the solution expanded around π₯ = 0.
3.1.4 Problem 3.7
Problem: Estimate the number of terms in the Taylor series (3.2.1) and (3.2.2) at page 68 ofthe text, that are necessary to compute
Ai(π₯) and Bi(π₯) correct to three decimal places at π₯ = Β±1, Β±100, Β±10000
Answer:
Ai (π₯) = 3β23
βοΏ½π=0
π₯3π
9ππ!Ξ οΏ½π + 23οΏ½β 3
β43
βοΏ½π=0
π₯3π+1
9ππ!Ξ οΏ½π + 43οΏ½
(3.2.1)
Bi (π₯) = 3β16
βοΏ½π=0
π₯3π
9ππ!Ξ οΏ½π + 23οΏ½β 3
β56
βοΏ½π=0
π₯3π+1
9ππ!Ξ οΏ½π + 43οΏ½
(3.2.2)
The radius of convergence of these series extends from π₯ = 0 to Β±β so we know this willconverge to the correct value of Ai (π₯) ,Bi (π₯) for all π₯, even though we might need largenumber of terms to achieve this, as will be shown below. The following is a plot of Ai (π₯)and Bi (π₯)
-10 -5 0 5 10
-0.4
-0.2
0.0
0.2
0.4
x
AiryAi(x)
0.35503
-10 -5 0
-0.5
0.0
0.5
1.0
1.5
2.0
x
AiryBi(x)
0.6149
52
3.1. HW1 CHAPTER 3. HWS
3.1.4.1 AiryAI series
For Ai (π₯) looking at the (π+1)π‘β
ππ‘β term
Ξ =
3β23
π₯3(π+1)
9π+1(π+1)!ΞοΏ½π+1+ 23 οΏ½β 3
β43
π₯3π+2
9π+1(π+1)!ΞοΏ½π+1+ 43 οΏ½
3β23
π₯3π
9ππ!ΞοΏ½π+ 23 οΏ½β 3
β43
π₯3π+1
9ππ!ΞοΏ½π+ 43 οΏ½
This can be simplified using Ξ (π + 1) = πΞ (π) giving
Ξ οΏ½οΏ½π +23οΏ½+ 1οΏ½ = οΏ½π +
23οΏ½Ξ οΏ½π +
23οΏ½
Ξ οΏ½οΏ½π +43οΏ½+ 1οΏ½ = οΏ½π +
43οΏ½Ξ οΏ½π +
43οΏ½
Hence Ξ becomes
Ξ =
3β23 π₯3(π+1)
9π+1(π+1)!οΏ½π+ 23 οΏ½ΞοΏ½π+ 2
3 οΏ½β 3
β43 π₯3π+2
9π+1(π+1)!οΏ½π+ 43 οΏ½ΞοΏ½π+ 4
3 οΏ½
3β23 π₯3π
9ππ!ΞοΏ½π+ 23 οΏ½β 3
β43 π₯3π+1
9ππ!ΞοΏ½π+ 43 οΏ½
Or
Ξ =
3β23 π₯3(π+1)
9(π+1)οΏ½π+ 23 οΏ½ΞοΏ½π+ 2
3 οΏ½β 3
β43 π₯3π+2
9(π+1)οΏ½π+ 43 οΏ½ΞοΏ½π+ 4
3 οΏ½
3β23 π₯3π
ΞοΏ½π+ 23 οΏ½β 3
β43 π₯3π+1
ΞοΏ½π+ 43 οΏ½
Or
Ξ =
3β23 π₯3(π+1)οΏ½π+ 4
3 οΏ½ΞοΏ½π+ 43 οΏ½β3
β43 π₯3π+2οΏ½π+ 2
3 οΏ½ΞοΏ½π+ 23 οΏ½
9(π+1)οΏ½π+ 23 οΏ½ΞοΏ½π+ 2
3 οΏ½οΏ½π+ 43 οΏ½ΞοΏ½π+ 4
3 οΏ½
3β23 π₯3πΞοΏ½π+ 4
3 οΏ½β3β43 π₯3π+1ΞοΏ½π+ 2
3 οΏ½
ΞοΏ½π+ 23 οΏ½ΞοΏ½π+ 4
3 οΏ½
Or
Ξ =
3β23 π₯3(π+1)οΏ½π+ 4
3 οΏ½ΞοΏ½π+ 43 οΏ½β3
β43 π₯3π+2οΏ½π+ 2
3 οΏ½ΞοΏ½π+ 23 οΏ½
9(π+1)οΏ½π+ 23 οΏ½οΏ½π+ 4
3 οΏ½
3β23 π₯3πΞ οΏ½π + 4
3οΏ½ β 3
β43 π₯3π+1Ξ οΏ½π + 2
3οΏ½
53
3.1. HW1 CHAPTER 3. HWS
For largeπ, we can approximate οΏ½π + 23οΏ½ , οΏ½π + 4
3οΏ½ , (π + 1) to just π+1 and the above becomes
Ξ =
3β23 π₯3(π+1)ΞοΏ½π+ 4
3 οΏ½β3β43 π₯3π+2ΞοΏ½π+ 2
3 οΏ½
9(π+1)2
3β23 π₯3πΞ οΏ½π + 4
3οΏ½ β 3
β43 π₯3π+1Ξ οΏ½π + 2
3οΏ½
Or
Ξ =3β23 π₯3ππ₯3Ξ οΏ½π + 4
3οΏ½ β 3
β43 π₯3ππ₯2Ξ οΏ½π + 2
3οΏ½
9 (π + 1)2 οΏ½3β23 π₯3πΞ οΏ½π + 4
3οΏ½ β 3
β43 π₯3ππ₯Ξ οΏ½π + 2
3οΏ½οΏ½
Or
Ξ =0.488π₯3ππ₯2 οΏ½π₯Ξ οΏ½π + 4
3οΏ½ β 0.488Ξ οΏ½π + 2
3οΏ½οΏ½
(0.488) 9 (π + 1)2 π₯3π οΏ½Ξ οΏ½π + 43οΏ½ β 0.488π₯Ξ οΏ½π + 2
3οΏ½οΏ½
=π₯2
9 (π + 1)2
ββββββββββ
π₯Ξ οΏ½π + 43οΏ½ β 0.488Ξ οΏ½π + 2
3οΏ½
Ξ οΏ½π + 43οΏ½ β 0.488π₯Ξ οΏ½π + 2
3οΏ½
ββββββββββ
We want to solve for π s.t. οΏ½π₯2
9(π+1)2
ββββββπ₯ΞοΏ½π+ 4
3 οΏ½β0.488ΞοΏ½π+ 23 οΏ½
ΞοΏ½π+ 43 οΏ½β0.488π₯ΞοΏ½π+ 2
3 οΏ½
ββββββ οΏ½ < 0.001.
For π₯ = 1
19 (π + 1)2
ββββββββββ
Ξ οΏ½π + 43οΏ½ β 0.488Ξ οΏ½π + 2
3οΏ½
Ξ οΏ½π + 43οΏ½ β 0.488Ξ οΏ½π + 2
3οΏ½
ββββββββββ < 0.001
19 (π + 1)2
< 0.001
9 (π + 1)2 > 1000
(π + 1)2 >10009
π + 1 >οΏ½10009
π > 9.541π = 10
For π₯ = 100
οΏ½οΏ½
1002
9 (π + 1)2
ββββββββββ
100Ξ οΏ½π + 43οΏ½ β 0.488Ξ οΏ½π + 2
3οΏ½
Ξ οΏ½π + 43οΏ½ β 0.488 (100) Ξ οΏ½π + 2
3οΏ½
ββββββββββ
οΏ½οΏ½< 0.001
I could not simplify away the Gamma terms above any more. Is there a way? So wrote small
54
3.1. HW1 CHAPTER 3. HWS
function (in Mathematica, which can compute this) which increments π and evaluate theabove, until the value became smaller than 0.001. At π = 11, 500 this was achieved.
For π₯ = 10000
οΏ½οΏ½100002
9 (π + 1)2
ββββββββββ
100Ξ οΏ½π + 43οΏ½ β 0.488Ξ οΏ½π + 2
3οΏ½
Ξ οΏ½π + 43οΏ½ β 0.488 (10000) Ξ οΏ½π + 2
3οΏ½
ββββββββββ
οΏ½οΏ½< 0.001
Using the same program, found that π = 11, 500, 000 was needed to obtain the result below0.001.
For = β1 also π = 10. For π₯ = β100, π = 10, 030, which is little less than π₯ = +100. Forπ₯ = β10000, π = 10, 030, 000
Summary table for AiryAI(x)
π₯ π1 10β1 10100 11, 500β100 10, 03010000 11, 500, 000β10000 10, 030, 000
The Mathematica function which did the estimate is the following
estimateAiryAi[x_, n_] := (x^2/(9*(n + 1)^2))*((x*Gamma[n + 4/3] -0.488*Gamma[n + 2/3])/(Gamma[n + 4/3] - 0.488*x*Gamma[n + 2/3]))estimateAiryAi[-10000, 10030000] // N-0.0009995904estimateAiryAi[100, 11500] // N0.000928983646707407estimateAiryAi[10000, 11500000] // N0.000929156800155198
55
3.1. HW1 CHAPTER 3. HWS
3.1.4.2 AiryBI series
For Bi (π₯) the diοΏ½erence is the coeοΏ½cients. Hence using the result from above, and justreplace the coeοΏ½cients
Ξ =3β16 π₯3ππ₯3Ξ οΏ½π + 4
3οΏ½ β 3
β58 π₯3ππ₯2Ξ οΏ½π + 2
3οΏ½
9 (π + 1)2 οΏ½3β16 π₯3πΞ οΏ½π + 4
3οΏ½ β 3
β58 π₯3ππ₯Ξ οΏ½π + 2
3οΏ½οΏ½
=π₯2 οΏ½π₯Ξ οΏ½π + 4
3οΏ½ β 0.604 39Ξ οΏ½π + 2
3οΏ½οΏ½
9 (π + 1)2 οΏ½Ξ οΏ½π + 43οΏ½ β 0.604 39π₯Ξ οΏ½π + 2
3οΏ½οΏ½
Hence for π₯ = 1, using the above reduces to1
9 (π + 1)2< 0.001
Which is the same as AiryAI, therefore π = 10. For π₯ = 100, using the same small functionin Mathematica to calculate the above, here are the result.
Summary table for AiryBI(x)
π₯ π1 10β1 10100 11, 290β100 9, 90010000 10, 900, 000β10000 9, 950, 000
The result between AiryAi and AiryBi are similar. AiryBi needs a slightly less number ofterms in the series to obtain same accuracy.
The Mathematica function which did the estimate for the larger π value for the above tableis the following
estimateAiryBI[x_, n_] := Abs[(x^2/(9*(n + 1)^2))*((x*Gamma[n + 4/3] -0.60439*Gamma[n + 2/3])/(Gamma[n + 4/3] - 0.6439*x*Gamma[n + 2/3]))]
3.1.5 Problem 3.8
Problem How many terms in the Taylor series solution to π¦β²β²β² = π₯3π¦ with π¦ (0) = 1, π¦β² (0) =π¦β²β² (0) = 0 are needed to evaluate β«
1
0π¦ (π₯) ππ₯ correct to three decimal places?
Answer
π¦β²β²β² β π₯3π¦ = 056
3.1. HW1 CHAPTER 3. HWS
Since π₯ is an ordinary point, we use
π¦ =βοΏ½π=0
πππ₯π
π¦β² =βοΏ½π=0
ππππ₯πβ1 =βοΏ½π=1
ππππ₯πβ1 =βοΏ½π=0
(π + 1) ππ+1π₯π
π¦β²β² =βοΏ½π=0
π (π + 1) ππ+1π₯πβ1 =βοΏ½π=1
π (π + 1) ππ+1π₯πβ1 =βοΏ½π=0
(π + 1) (π + 2) ππ+2π₯π
π¦β²β²β² =βοΏ½π=0
π (π + 1) (π + 2) ππ+2π₯πβ1 =βοΏ½π=1
π (π + 1) (π + 2) ππ+2π₯πβ1
=βοΏ½π=0
(π + 1) (π + 2) (π + 3) ππ+3π₯π
Hence the ODE becomesβοΏ½π=0
(π + 1) (π + 2) (π + 3) ππ+3π₯π β π₯3βοΏ½π=0
πππ₯π = 0
βοΏ½π=0
(π + 1) (π + 2) (π + 3) ππ+3π₯π ββοΏ½π=0
πππ₯π+3 = 0
βοΏ½π=0
(π + 1) (π + 2) (π + 3) ππ+3π₯π ββοΏ½π=3
ππβ3π₯π = 0
For π = 0(1) (2) (3) π3 = 0
π3 = 0
For π = 1(2) (3) (4) π4 = 0
π4 = 0
For π = 2
π5 = 0
For π β₯ 3, recursive equation is used
(π + 1) (π + 2) (π + 3) ππ+3 β ππβ3 = 0
ππ+3 =ππβ3
(π + 1) (π + 2) (π + 3)Or, for π = 3
π6 =π0
(4) (5) (6)For π = 4
π7 =π1
(4 + 1) (4 + 2) (4 + 3)=
π1(5) (6) (7)
For π = 5
π8 =π2
(5 + 1) (5 + 2) (5 + 3)=
π2(6) (7) (8)
57
3.1. HW1 CHAPTER 3. HWS
For π = 6
π9 =π3
(6 + 1) (6 + 2) (6 + 3)= 0
For π = 7
π10 =π4
(π + 1) (π + 2) (π + 3)= 0
For π = 8
π11 =π5
(8 + 1) (8 + 2) (8 + 3)= 0
For π = 9
π12 =π6
(9 + 1) (9 + 2) (9 + 3)=
π6(10) (11) (12)
=π0
(4) (5) (6) (10) (11) (12)For π = 10
π13 =π7
(10 + 1) (10 + 2) (10 + 3)=
π7(11) (12) (13)
=π1
(5) (6) (7) (11) (12) (13)For π = 11
π14 =π8
(11 + 1) (11 + 2) (11 + 3)=
π8(12) (13) (14)
=π2
(6) (7) (8) (12) (13) (14)And so on. Hence the series is
π¦ =βοΏ½π=0
πππ₯π
= π0 + π1π₯ + π2π₯2 + 0π₯3 + 0π₯4 + 0π₯5 +π0
(4) (5) (6)π₯6 +
π1(5) (6) (7)
π₯7 +π2
(6) (7) (8)π₯8
+ 0 + 0 + 0 +π0
(4) (5) (6) (10) (11) (12)π₯12 +
π1(5) (6) (7) (11) (12) (13)
π₯13
+π2
(6) (7) (8) (12) (13) (14)π₯14 + 0 + 0 + 0 +β―
Or
π¦ (π₯) = π0 + π1π₯ + π2π₯2 +π0
(4) (5) (6)π₯6 +
π1(5) (6) (7)
π₯7 +π2
(6) (7) (8)π₯8+
π0(4) (5) (6) (10) (11) (12)
π₯12 +π1
(5) (6) (7) (11) (12) (13)π₯13 +
π2(6) (7) (8) (12) (13) (14)
π₯14 +β―
Or
π¦ (π₯) = π0 οΏ½1 +π₯6
(4) (5) (6)+
π₯12
(4) (5) (6) (10) (11) (12)+β―οΏ½
+ π1 οΏ½π₯ +π₯7
(5) (6) (7)+
π₯13
(5) (6) (7) (11) (12) (13)+β―οΏ½
+ π2 οΏ½π₯2 +π₯8
(6) (7) (8)+
π₯14
(6) (7) (8) (12) (13) (14)+β―οΏ½
58
3.1. HW1 CHAPTER 3. HWS
π¦ (π₯) = π0 οΏ½1 +1120
π₯6 +1
158400π₯12 +β―οΏ½
+ π1 οΏ½π₯ +1210
π₯7 +1
360360π₯13 +β―οΏ½
+ π2 οΏ½π₯2 +1336
π₯8 +1
733 824π₯14 +β―οΏ½
We now apply initial conditions π¦ (0) = 1, π¦β² (0) = π¦β²β² (0) = 0 . When π¦ (0) = 1
1 = π0Hence solution becomes
π¦ (π₯) = οΏ½1 +1120
π₯6 +1
158400π₯12 +β―οΏ½
+ π1 οΏ½π₯ +1210
π₯7 +1
360360π₯13 +β―οΏ½
+ π2 οΏ½π₯2 +1336
π₯8 +1
733 824π₯14 +β―οΏ½
Taking derivative
π¦β² (π₯) = οΏ½6120
π₯5 +12
158400π₯11 +β―οΏ½
+ π1 οΏ½1 +7210
π₯6 +13
360360π₯12 +β―οΏ½
+ π2 οΏ½2π₯ +8336
π₯7 +12
733 824π₯13 +β―οΏ½
Applying π¦β² (0) = 0 gives
0 = π1And similarly, Applying π¦β²β² (0) = 0 gives π2 = 0. Hence the solution is
π¦ (π₯) = π0 οΏ½1 +π₯6
(4) (5) (6)+
π₯12
(4) (5) (6) (10) (11) (12)+β―οΏ½
= 1 +π₯6
(4) (5) (6)+
π₯12
(4) (5) (6) (10) (11) (12)+
π₯18
(4) (5) (6) (10) (11) (12) (16) (17) (18)+β―
= 1 +π₯6
120+
π₯12
158 400+
π₯18
775 526 400+β―
We are now ready to answer the question. We will do the integration by increasing thenumber of terms by one each time. When the absolute diοΏ½erence between each incrementbecomes less than 0.001 we stop. When using one term For
οΏ½1
0π¦ (π₯) ππ₯ = οΏ½
1
0ππ₯
= 1
59
3.1. HW1 CHAPTER 3. HWS
When using two terms
οΏ½1
01 +
π₯6
120ππ₯ = οΏ½π₯ +
π₯7
120 (7)οΏ½1
0
= 1 +1840
=841840
= 1.001190476
DiοΏ½erence between one term and two terms is 0.001190476. When using three terms
οΏ½1
01 +
π₯6
120+
π₯12
158 400ππ₯ = οΏ½π₯ +
π₯7
840+
π₯13
2059 200οΏ½1
0
= 1 +1840
+1
2059 200
=14 431 56714 414 400
= 1.001190962
Comparing the above result, with the result using two terms, we see that only two terms areneeded since the change in accuracy did not aοΏ½ect the first three decimal points. Hence weneed only this solution with two terms only
π¦ (π₯) = 1 +1120
π₯6
3.1.6 Problem 3.24
3.1.6.1 part e
Problem Find series expansion of all the solutions to the following diοΏ½erential equationabout π₯ = 0. Try to sum in closed form any infinite series that appear.
2π₯π¦β²β² β π¦β² + π₯2π¦ = 0
Solution
π¦β²β² β12π₯π¦β² +
π₯2π¦ = 0
The only singularity is in π (π₯) is π₯ = 0. We will now check if it is removable. (i.e. regular)
limπ₯β0
π₯π (π₯) = limπ₯β0
π₯12π₯
=12
60
3.1. HW1 CHAPTER 3. HWS
Therefore π₯ = 0 is regular singular point. Hence we try Frobenius series
π¦ (π₯) =βοΏ½π=0
πππ₯π+π
π¦β² (π₯) =βοΏ½π=0
(π + π) πππ₯π+πβ1
π¦β²β² (π₯) =βοΏ½π=0
(π + π) (π + π β 1) πππ₯π+πβ2
Substituting the above in 2π₯2π¦β²β² β π₯π¦β² + π₯3π¦ = 0 results in
2π₯2βοΏ½π=0
(π + π) (π + π β 1) πππ₯π+πβ2 β π₯βοΏ½π=0
(π + π) πππ₯π+πβ1 + π₯3βοΏ½π=0
πππ₯π+π = 0
βοΏ½π=0
2 (π + π) (π + π β 1) πππ₯π+π ββοΏ½π=0
(π + π) πππ₯π+π +βοΏ½π=0
πππ₯π+π+3 = 0
βοΏ½π=0
2 (π + π) (π + π β 1) πππ₯π+π ββοΏ½π=0
(π + π) πππ₯π+π +βοΏ½π=3
ππβ3π₯π+π = 0 (1)
The first step is to obtain the indicial equation. As the nature of the roots will tell us howto proceed. The indicial equation is obtained from setting π = 0 in (1) with the assumptionthat π0 β 0. Setting π = 0 in (1) gives
2 (π + π) (π + π β 1) ππ β (π + π) ππ = 02 (π) (π β 1) π0 β ππ0 = 0
Since π0 β 0 then we obtain the indicial equation (quadratic in π)
2 (π) (π β 1) β π = 0π (2 (π β 1) β 1) = 0
π (2π β 3) = 0
Hence roots are
π1 = 0
π2 =32
Since π1 β π2 is not an integer, then we know we can now construct two linearly independentsolutions
π¦1 (π₯) = π₯π1οΏ½π=0πππ₯π
π¦2 (π₯) = π₯π2οΏ½π=0πππ₯π
Or
π¦1 (π₯) = οΏ½π=0πππ₯π
π¦2 (π₯) = οΏ½π=0πππ₯
π+ 32
61
3.1. HW1 CHAPTER 3. HWS
Notice that the coeοΏ½cients are not the same. Since we now know π1, π2, we will use the aboveseries solution to obtain π¦1 (π₯) and π¦2 (π₯).
For π¦1 (π₯) where π1 = 0
π¦1 (π₯) = οΏ½π=0πππ₯π
π¦β² (π₯) = οΏ½π=0ππππ₯πβ1 = οΏ½
π=1ππππ₯πβ1 = οΏ½
π=0(π + 1) ππ+1π₯π
π¦β²β² (π₯) = οΏ½π=0π (π + 1) ππ+1π₯πβ1 = οΏ½
π=1π (π + 1) ππ+1π₯πβ1 = οΏ½
π=0(π + 1) (π + 2) ππ+2π₯π
Substituting the above in 2π₯2π¦β²β² β π₯π¦β² + π₯3π¦ = 0 results in
2π₯2οΏ½π=0
(π + 1) (π + 2) ππ+2π₯π β π₯οΏ½π=0
(π + 1) ππ+1π₯π + π₯3οΏ½π=0πππ₯π = 0
οΏ½π=02 (π + 1) (π + 2) ππ+2π₯π+2 βοΏ½
π=0(π + 1) ππ+1π₯π+1 +οΏ½
π=0πππ₯π+3 = 0
οΏ½π=22 (π β 1) (π) πππ₯π βοΏ½
π=1ππππ₯π +οΏ½
π=3ππβ3π₯π = 0
For π = 1 (index starts at π = 1).
βππππ₯π = 0βπ1 = 0π1 = 0
For π = 2
2 (π β 1) (π) πππ₯π β ππππ₯π = 02 (2 β 1) (2) π2 β 2π2 = 0
2π2 = 0π2 = 0
For π β₯ 3 we have recursive formula
2 (π β 1) (π) ππ β πππ + ππβ3 = 0
ππ =βππβ3
π (2π β 3)Hence, for π = 3
π3 =βπ0
(3) (6 β 3)=βπ09
For π = 4
π4 =βπ1
π (2π β 3)= 0
For π = 5
π5 =βπ2
π (2π β 3)= 0
For π = 6
π6 =βπ3
π (2π β 3)=
βπ3(6) (12 β 3)
=βπ354
=π0
(54) (9)=
1486
π0
62
3.1. HW1 CHAPTER 3. HWS
And for π = 7, 8 we also obtain π7 = 0, π8 = 0, but for π9
π9 =βπ6
9 (2 (9) β 3)=βπ6135
=βπ0
(135) (486)= β
165 610
π0
And so on. Hence from οΏ½π=0πππ₯π we obtain
π¦1 (π₯) = π0 ββπ09π₯3 +
1486
π0π₯6 β1
65 610π0π₯9 +β―
= π0 οΏ½1 β19π₯3 +
1486
π₯6 β1
65 610π₯9 +β―οΏ½
Now that we found π¦1 (π₯).
For π¦2 (π₯) with π2 =32 .
π¦ (π₯) = οΏ½π=0πππ₯
π+ 32
π¦β² (π₯) = οΏ½π=0
οΏ½π +32οΏ½πππ₯
π+ 12
π¦β²β² (π₯) = οΏ½π=0
οΏ½π +12οΏ½ οΏ½
π +32οΏ½πππ₯
πβ 12
Substituting this into 2π₯2π¦β²β² β π₯π¦β² + π₯3π¦ = 0 gives
2π₯2οΏ½π=0
οΏ½π +12οΏ½ οΏ½
π +32οΏ½πππ₯
πβ 12 β π₯οΏ½
π=0οΏ½π +
32οΏ½πππ₯
π+ 12 + π₯3οΏ½
π=0πππ₯
π+ 32 = 0
οΏ½π=02 οΏ½π +
12οΏ½ οΏ½
π +32οΏ½πππ₯
πβ 12+2 βοΏ½
π=0οΏ½π +
32οΏ½πππ₯
π+ 12+1 +οΏ½
π=0πππ₯
π+ 32+3 = 0
οΏ½π=02 οΏ½π +
12οΏ½ οΏ½
π +32οΏ½πππ₯
π+ 32 βοΏ½
π=0οΏ½π +
32οΏ½πππ₯
π+ 32 +οΏ½
π=0πππ₯
π+ 92 = 0
οΏ½π=02 οΏ½π +
12οΏ½ οΏ½
π +32οΏ½πππ₯
π+ 32 βοΏ½
π=0οΏ½π +
32οΏ½πππ₯
π+ 32 +οΏ½
π=3ππβ3π₯
π+ 92β3 = 0
οΏ½π=02 οΏ½π +
12οΏ½ οΏ½
π +32οΏ½πππ₯
π+ 32 βοΏ½
π=0οΏ½π +
32οΏ½πππ₯
π+ 32 +οΏ½
π=3ππβ3π₯
π+ 32 = 0
Now that all the π₯ terms have the same exponents, we can continue.
For π = 0
2 οΏ½π +12οΏ½ οΏ½
π +32οΏ½π0π₯
32 β οΏ½π +
32οΏ½π0π₯
32 = 0
2 οΏ½12οΏ½ οΏ½
32οΏ½π0π₯
32 β οΏ½
32οΏ½π0π₯
32 = 0
32π0π₯
32 β οΏ½
32οΏ½π0π₯
32 = 0
0π0 = 0
63
3.1. HW1 CHAPTER 3. HWS
Hence π0 is arbitrary.
For π = 1
2 οΏ½π +12οΏ½ οΏ½
π +32οΏ½πππ₯
π+ 32 β οΏ½π +
32οΏ½πππ₯
π+ 32 = 0
2 οΏ½1 +12οΏ½ οΏ½
1 +32οΏ½π1 β οΏ½1 +
32οΏ½π1 = 0
5π1 = 0π1 = 0
For π = 2
2 οΏ½π +12οΏ½ οΏ½
π +32οΏ½πππ₯
π+ 32 β οΏ½π +
32οΏ½πππ₯
π+ 32 = 0
2 οΏ½2 +12οΏ½ οΏ½
2 +32οΏ½π2 β οΏ½2 +
32οΏ½π2 = 0
14π2 = 0π2 = 0
For π β₯ 3 we use the recursive formula
2 οΏ½π +12οΏ½ οΏ½
π +32οΏ½ππ β οΏ½π +
32οΏ½ππ + ππβ3 = 0
ππ =βππβ3
π (2π + 3)Hence for π = 3
π3 =βπ03 (9)
=βπ027
For π = 4, π = 5 we will get π4 = 0 and π5 = 0 since π1 = 0 and π2 = 0.
For π = 6
π6 =βπ3
6 (12 + 3)=βπ390
=π0
27 (90)=
π02430
For π = 7, π = 8 we will get π7 = 0 and π8 = 0 since π4 = 0 and π5 = 0
For π = 9
π9 =βπ6
π (2π + 3)=
βπ69 (18 + 3)
=βπ6189
=βπ0
2430 (189)=
βπ0459270
And so on. Hence, from π¦2 (π₯) = οΏ½π=0πππ₯
π+ 32 the series is
π¦2 (π₯) = π0π₯32 β
π027π₯3+
32 +
π02430
π₯6+32 β
π0459 270
π₯9+32 +β―
= π0π₯32 οΏ½1 β
π₯3
27+
π₯6
2430β
π₯9
459270+β―οΏ½
64
3.1. HW1 CHAPTER 3. HWS
The final solution is
π¦ (π₯) = π¦1 (π₯) + π¦2 (π₯)
Or
π¦ (π₯) = π0 οΏ½1 β19π₯
3 + 1486π₯
6 β 165 610π₯
9 +β―οΏ½ + π0π₯32 οΏ½1 β π₯3
27 +π₯6
2430 βπ₯9
459270 +β―οΏ½ (2)
Now comes the hard part. Finding closed form solution.
The Taylor series of cos οΏ½13π₯32β2οΏ½ is (using CAS)
cos οΏ½13π₯32β2οΏ½ β 1 β
19π₯3 +
1486
π₯6 β1
65610π₯9 +β― (3)
And the Taylor series for sin οΏ½13π₯32β2οΏ½ is (Using CAS)
sin οΏ½13π₯32β2οΏ½ β
13π₯32β2 β
181π₯92β2 +
17290
π₯152 β2 ββ―
= π₯32 οΏ½13β
2 β181π₯3β2 +
17290
π₯6β2 ββ―οΏ½
=13β
2π₯32 οΏ½1 β
127π₯3 +
12430
π₯6 ββ―οΏ½ (4)
Comparing (3,4) with (2) we see that (2) can now be written as
π¦ (π₯) = π0 cos οΏ½13π₯32β2οΏ½ +
π013β2
sin οΏ½13π₯32β2οΏ½
Or letting π = π013β2
π¦ (π₯) = π0 cos οΏ½13π₯32β2οΏ½ + π0 sin οΏ½13π₯
32β2οΏ½
This is the closed form solution. The constants π0, π0 can be found from initial conditions.
3.1.6.2 part f
Problem Find series expansion of all the solutions to the following diοΏ½erential equationabout π₯ = 0. Try to sum in closed form any infinite series that appear.
(sin π₯) π¦β²β² β 2 (cos π₯) π¦β² β (sin π₯) π¦ = 0Solution
In standard form
π¦β²β² β 2 οΏ½cos π₯sin π₯
οΏ½ π¦β² β π¦ = 0
65
3.1. HW1 CHAPTER 3. HWS
Hence the singularities are in π (π₯) only and they occur when sin π₯ = 0 or π₯ = 0, Β±π, Β±2π,β―but we just need to consider π₯ = 0. Let us check if the singularity is removable.
limπ₯β0
π₯π (π₯) = 2 limπ₯β0
π₯cos π₯sin π₯ = 2 lim
π₯β0π₯1 β π₯2
2 +π₯4
4! ββ―
π₯ β π₯3
3! +π₯5
5! ββ―= 2 lim
π₯β0
1 β π₯2
2 +π₯4
4! ββ―
1 β π₯2
3! +π₯4
5! ββ―= 2
Hence the singularity is regular. So we can use Frobenius series
π¦ (π₯) =βοΏ½π=0
πππ₯π+π
π¦β² (π₯) =βοΏ½π=0
(π + π) πππ₯π+πβ1
π¦β²β² (π₯) =βοΏ½π=0
(π + π) (π + π β 1) πππ₯π+πβ2
Substituting the above in sin (π₯) π¦β²β² β 2 cos (π₯) π¦β² β sin (π₯) π¦ = 0 results in
sin (π₯)βοΏ½π=0
(π + π) (π + π β 1) πππ₯π+πβ2 β 2 cos (π₯)βοΏ½π=0
(π + π) πππ₯π+πβ1 β sin (π₯)βοΏ½π=0
πππ₯π+π = 0
Now using Taylor series for sin π₯, cos π₯ expanded around 0, the above becomesβοΏ½π=0
(β1)ππ₯2π+1
(2π + 1)!
βοΏ½π=0
(π + π) (π + π β 1) πππ₯π+πβ2
β2βοΏ½π=0
(β1)ππ₯2π
(2π)!
βοΏ½π=0
(π + π) πππ₯π+πβ1
ββοΏ½π=0
(β1)ππ₯2π+1
(2π + 1)!
βοΏ½π=0
πππ₯π+π = 0 (1)
We need now to evaluate products of power series. Using what is called Cauchy product rule,where
π (π₯) π (π₯) = οΏ½βοΏ½π=0
πππ₯ποΏ½ οΏ½βοΏ½π=0
πππ₯ποΏ½ =βοΏ½π=0
βοΏ½π=0
πππππ₯π+π (2)
Applying (2) to first term in (1) givesβοΏ½π=0
(β1)ππ₯2π+1
(2π + 1)!
βοΏ½π=0
(π + π) (π + π β 1) πππ₯π+πβ2 =βοΏ½π=0
βοΏ½π=0
(β1)π (π + π) (π + π β 1)(2π + 1)!
πππ₯2π+π+πβ1
(3)
Applying (2) to second term in (1) givesβοΏ½π=0
(β1)ππ₯2π
(2π)!
βοΏ½π=0
(π + π) πππ₯π+πβ1 =βοΏ½π=0
βοΏ½π=0
(β1)π (π + π)(2π)!
πππ₯2π+π+πβ1 (4)
Applying (2) to the last term in (1) givesβοΏ½π=0
(β1)ππ₯2π+1
(2π + 1)!
βοΏ½π=0
πππ₯π+π =βοΏ½π=0
βοΏ½π=0
(β1)π ππ(2π + 1)!
π₯2π+π+π+1 (5)
66
3.1. HW1 CHAPTER 3. HWS
Substituting (3,4,5) back into (1) givesβοΏ½π=0
βοΏ½π=0
(β1)π (π + π) (π + π β 1)(2π + 1)!
πππ₯2π+π+πβ1
β2βοΏ½π=0
βοΏ½π=0
(β1)π (π + π)(2π)!
πππ₯2π+π+πβ1
ββοΏ½π=0
βοΏ½π=0
(β1)π ππ(2π + 1)!
π₯2π+π+π+1 = 0
We now need to make all π₯ exponents the same. This givesβοΏ½π=0
βοΏ½π=0
(β1)π (π + π) (π + π β 1)(2π + 1)!
πππ₯2π+π+πβ1
β2βοΏ½π=0
βοΏ½π=0
(β1)π (π + π)(2π)!
πππ₯2π+π+πβ1
ββοΏ½π=0
βοΏ½π=2
(β1)π ππβ2(2π + 1)!
π₯2π+π+πβ1 = 0 (6)
The first step is to obtain the indicial equation. As the nature of the roots will tell us how toproceed. The indicial equation is obtained from setting π = π = 0 in (6) with the assumptionthat π0 β 0. This results in
(β1)π (π + π) (π + π β 1)(2π + 1)!
πππ₯2π+π+πβ1 β 2(β1)π (π + π)
(2π)!πππ₯2π+π+πβ1 = 0
(π) (π β 1) π0 β 2ππ0 = 0
π0 οΏ½π2 β π β 2ποΏ½ = 0
Since π0 β 0 then we obtain the indicial equation (quadratic in π)
π2 β 3π = 0π (π β 3) = 0
Hence roots are
π1 = 3π2 = 0
(it is always easier to make π1 > π2). Since π1 β π2 = 3 is now an integer, then this is case IIpart (b) in textbook, page 72. In this case, the two linearly independent solutions are
π¦1 (π₯) = π₯3οΏ½π=0πππ₯π = οΏ½
π=0πππ₯π+3
π¦2 (π₯) = ππ¦1 (π₯) ln (π₯) + π₯π2οΏ½π=0πππ₯π = ππ¦1 (π₯) ln (π₯) +οΏ½
π=0πππ₯π
67
3.1. HW1 CHAPTER 3. HWS
Where π is some constant. Now we will find π¦1. From (6), where now we set π = 3βοΏ½π=0
βοΏ½π=0
(β1)π (π + 3) (π + 2)(2π + 1)!
πππ₯2π+π++2
β2βοΏ½π=0
βοΏ½π=0
(β1)π (π + 3)(2π)!
πππ₯2π+π+2
ββοΏ½π=0
βοΏ½π=2
(β1)π ππβ2(2π + 1)!
π₯2π+π+2 = 0
For π = 0, π = 1(β1)π (π + 3) (π + 2)
(2π + 1)!ππ β 2 οΏ½
(β1)π (π + 3)(2π)!
πποΏ½ = 0
(4) (3) π1 β 2 (4π1) = 0π1 = 0
For π = 0, π β₯ 2 we obtain recursive equation
(π + 3) (π + 2) ππ β 2 (π + 3) ππ β ππβ2 = 0
ππ =βππβ2
(π + 3) (π + 2) β 2 (π + 3)=
βππβ2π (π + 3)
Hence, for π = 0, π = 2
π2 =βπ010
For π = 0, π = 3
π3 =βπ1
π (π + 3)= 0
For π = 0, π = 4
π4 =βπ24 (7)
=π0280
For π = 0, π = 5
π5 = 0
For π = 0, π = 6
π6 =βπ4
6 (6 + 3)=
βπ06 (6 + 3) (280)
=βπ015120
For π = 0, π = 7, π7 = 0 and for π = 0, π = 8
π8 =βπ6
8 (8 + 3)=
11330560
π0
68
3.1. HW1 CHAPTER 3. HWS
And so on. Since π¦1 (π₯) = οΏ½π=0πππ₯π+3, then the first solution is now found. It is
π¦1 (π₯) = π0π₯3 + π1π₯4 + π2π₯5 + π3π₯6 + π4π₯7 + π5π₯8 + π6π₯9 + π7π₯10 + π8π₯11 +β―
= π0π₯3 + 0 βπ010π₯5 + 0 +
π0280
π₯7 + 0 βπ0
15 120π₯9 + 0 +
11330560
π0π₯11 +β―
= π0π₯3 οΏ½1 βπ₯2
10+π₯5
280β
π₯6
15120+
π₯8
1330560ββ―οΏ½ (7)
The second solution can now be found from
π¦2 = ππ¦1 (π₯) ln (π₯) +οΏ½π=0πππ₯π
I could not find a way to convert the complete solution to closed form solution, or even findclosed form for π¦1 (π₯). The computer claims that the closed form final solution is
π¦ (π₯) = π¦1 (π₯) + π¦2 (π₯)
= π0 cos (π₯) + π0 οΏ½ββcos2 π₯ β 1 + cos π₯ ln οΏ½cos (π₯) + βcos2 π₯ β 1οΏ½οΏ½
Which appears to imply that (7) is cos (π₯) series. But it is not. Converting series solution toclosed form solution is hard. Is this something we are supposed to know how to do? Otherby inspection, is there a formal process to do it?
69
3.1. HW1 CHAPTER 3. HWS
3.1.7 key solution of selected problems
3.1.7.1 section 3 problem 8
Problem Statement: How many terms are needed in the Taylor series solution to yβ²β²β² = x3y,
y(0) = 1, yβ²(0) = 0, yβ²β²(0) = 0 are needed to evaluateββ«0
y(x)dx correct to 3 decimal places?
Solution: Let
y =ββ
n=0
anxn
yβ² =ββ
n=0
nanxnβ1
yβ²β² =ββ
n=0
nan(nβ 1)xnβ1
yβ²β²β² =
ββn=0
n(nβ 1)(nβ 2)anxnβ3 = x3y = x3
ββn=0
anxn =
ββn=0
anxn+3.
Shift indices by letting n β (nβ 6) in the RHS:
ββn=0
anxn+3 β
ββn=6
anβ6xnβ3
ββn=0
n(nβ 1)(nβ 2)anxnβ3 = 0 Β· xβ3 + 0 Β· xβ2 + 0 Β· xβ1 + 3 Β· 2 Β· 1 Β· a3x
0 + 4 Β· 3 Β· 2 Β· a4x1
+ 5 Β· 4 Β· 3 Β· a5x2 +
ββn=6
n(nβ 1)(nβ 2)anxnβ3
From here,a3 = 0, a4 = 0, a5 = 0
xnβ3{n(nβ 1)(nβ 2)an β anβ6} = 0 for n = 6, 7, 8, ...
The recurrence relation becomes:
an =1
n(nβ 1)(nβ 2)anβ6 n β₯ 6
Solution to D.E. becomes:
y =
ββn=0
anxn = a0 + a1 Β· x+ a2 Β· x
2 +
ββn=6
[1
n(nβ 1)(nβ 2)anβ6x
n]
y(0) = a0 = 1 ==> a0 = 1
1
70
3.1. HW1 CHAPTER 3. HWS
yβ²(0) = a1 = 0 ==> a1 = 0
yβ²β²(0) = 2 Β· a2 = 0 ==> a2 = 0
y(x) = 1 +
ββn=6
1
n(nβ 1)(nβ 2)anβ6x
n
β« 1
0
y(x)dx = [x+ββ
n=6
1
(n+ 1)n(nβ 1)(nβ 2)anβ6x
n+1]10
= 1 +
ββn=6
1
(n+ 1)(n)(nβ 1)(nβ 2)anβ6
= 1 +1
7 Β· 6 Β· 5 Β· 4(1) +
1
13 Β· 12 Β· 11 Β· 10(
1
6 Β· Β·4(1)) + ...
= 1 +1
840+
1
2, 059, 200+ ...
β« 1
0
y(x)dx β 1 + 0.00119 + 4.856Γ 10β7
So the 2nd non-zero term is required. All other terms are below the specified tolerance of0.001.
2
71
3.1. HW1 CHAPTER 3. HWS
3.1.7.2 section 3 problem 6
72
3.1. HW1 CHAPTER 3. HWS
73
3.1. HW1 CHAPTER 3. HWS
3.1.7.3 section 3 problem 4
74
3.1. HW1 CHAPTER 3. HWS
75
3.1. HW1 CHAPTER 3. HWS
76
3.1. HW1 CHAPTER 3. HWS
3.1.7.4 section 3 problem 24
1 Closed form of Problem 3.24e
In problem 3.24e, we find Frobenius series solutions
yΞ±(x) =ββ
n=0
anxn+Ξ± (1)
for Ξ± = 0, 3
2, and recurrence relation on the coefficients:
an+3 =βan
(n+ Ξ± + 3)(2n+ 2Ξ±+ 3).
For each Ξ±, a1 = a2 = 0, so only the a3k survive. Rewriting the recurrence relation:
a3k+3 =βa3k
(3k + 3 + Ξ±)(6k + 3 + 2Ξ±)=
βa3k9(k + 1 + Ξ±/3)(2k + 1 + 2Ξ±/3)
.
To identify a closed-form sum, we need to find the general form of the coefficients a3k. Towards thisend, consider the case Ξ± = 0. Now
a3k =β1/9
k(2k β 1)a3kβ3 =
(β1/9)2
k(k β 1) Β· (2k β 1)(2k β 3)a3kβ6 = Β· Β· Β· =
(β1/9)k
k! Β· (2k β 1)!!a0.
Now note that
(2k β 1)!! =(2k)!
(2k)!!=
(2k)!
2k Β· k!=β k! Β· (2k β 1)!! = 2βk(2k)!,
so
a3k =(β1/9)k
2βk(2k)!a0 =
(β2/9)k
(2k)!a0.
Therefore,
y0(x) = a0
ββ
k=0
(β2/9)k
(2k)!x3k = a0
ββ
k=0
(β1)k
(2k)!(x3/2
β2/3)2k =: a0
ββ
k=0
βz2k
(2k)!,
where z := x3/2β2/3. We recognize the remaining series as the Taylor series for cos z, so
y0(x) = a0 cos z(x) = a0 cos
(β2
3x3/2
)
.
For Ξ± = 3
2,
a3k+3 =βa3k
9(k + 3/2)(2k + 2)=
βa3k9(k + 1)(2k + 3)
.
Following the same argument as before,
a3k =(β1/9)k
k! Β· (2k + 1)!!a0.
This is the same as before, but divided by 2k + 1, so
y3/2(x) = a0
ββ
k=0
(β2/9)k
(2k + 1)!x3k = a0 sin
(β2
3x3/2
)
.
Therefore, the general solution is
y(x) = y0(x) + y3/2(x) = c1 cos
(β2
3x3/2
)
+ c2 sin
(β2
3x3/2
)
.
1
77
3.1. HW1 CHAPTER 3. HWS
2 Closed form of Problem 3.24f
Problem 3.24f asks us to solve (sin x)yβ²β² β 2(cosx)yβ² β (sin x)y = 0. Dividing by sin x,
yβ²β² β 2(cotx)yβ² β y = 0,
so x = 0 is a regular singular point (cotx is not analytic at x = 0, but x cot x is). Thus, we can look fora Frobenius series solution.
Expanding sin x and cos x:
0 =
(
ββ
m=0
(β1)m
(2m+ 1)!x2m+1
)
ββ
n=0
(n+ Ξ±)(n+ Ξ±β 1)anxn+Ξ±β2 β 2
(
ββ
m=0
(β1)m
(2m)!x2m
)
ββ
n=0
(n + Ξ±)anxn+Ξ±β1
β
(
ββ
m=0
(β1)m
(2m+ 1)!x2m+1
)
ββ
n=0
anxn+Ξ±.
The lowest order is xΞ±β1:0 = Ξ±(Ξ±β 1)a0 β 2Ξ±a0 = Ξ±(Ξ±β 3)a0,
so Ξ± = 0, 3. We are in case II(b) on page 72.Let Ξ± = 0. Then
0 =
(
ββ
m=0
(β1)m
(2m+ 1)!x2m+1
)
ββ
n=0
n(nβ 1)anxnβ2 β 2
(
ββ
m=0
(β1)m
(2m)!x2m
)
ββ
n=0
nanxnβ1
β
(
ββ
m=0
(β1)m
(2m+ 1)!x2m+1
)
ββ
n=0
anxn.
We solve order-by-order:
x0 : 0 = β2a1 =β a1 = 0,
x1 : 0 = 2a2 β 2 Β· 2a2 β a0 =β a2 = βa02,
x2k : 0 =
kβ
β=0
(β1)kββ 2β(2β+ 1)
(2k β 2β+ 1)!a2β+1 β 2
kβ
β=0
(β1)kββ 2β+ 1
(2k β 2β)!a2β+1 β
kβ1β
β=0
(β1)kβββ1
(2k β 2ββ 1)!a2β+1,
x2k+1 : 0 =
k+1β
β=0
(β1)kββ+12β(2ββ 1)
(2k β 2β+ 3)!a2β β 2
k+1β
β=0
(β1)kββ+12β
(2k β 2β+ 2)!a2β β
kβ
β=0
(β1)kββ
(2k β 2β+ 1)!a2β,
where k β₯ 1. From the x2k terms, we see that a2n+1 depends only on a1, a3, ..., a2nβ1. As a1 = 0, it is asimple (strong) induction to show that each a2n+1 = 0.
We use the x2k+1 terms for k = 1 to see that
0 =
(
0a0 β2
6a2 +
12
1a4
)
β 2
(
0a0 β2
2a2 +
4
1a4
)
β(
β1
6a0 +
1
1a2
)
=β a4 = β1
4
(
1
6a0 β
2
3a2
)
.
As a2 = β1
2a0, we see that a4 = 1
24a0. We notice a pattern that these coefficients are starting to look
like those in the Taylor series expansion of y0(x) = a0 cos x, which we now verify by plugging into theODE:
(sin x)yβ²β²0(x)β 2(cosx)yβ²0(x)β (sin x)y0(x) = βa0 sin x cos x+ 2a0 cos x sin xβ a0 sin x cosx = 0.
2
78
3.1. HW1 CHAPTER 3. HWS
To find the other solution, we use reduction of order: look for a solution of the form y3(x) = v(x) cosx(we keep the Ξ± = 3 subscript to remind us that we expect the Taylor series of our solution to start atthe x3-term). Plugging into the ODE:
(sin x)(vβ²β²(x) cosxβ 2vβ²(x) sin xβ v(x) cosx)β (2 cosx)(vβ²(x) cos xβ v(x) sin x)β (sin x)(v(x) cosx) = 0.
Simplifying,
(sin x cosx)vβ²β²(x)β 2vβ²(x) = 0 =β vβ²(x) = c1 tan2 x =β v(x) = c1(tanxβ x) + c0.
We set c0 = 0, since this corresponds to y0(x). Then y3(x) = c1(tan xβ x) cosx = c1(sin xβ x cosx).The full solution is therefore
y(x) = c0 cosx+ c1(sin xβ x cosx).
3
79
3.2. HW2 CHAPTER 3. HWS
3.2 HW2
3.2.1 problem 3.27 (page 138)
Problem derive 3.4.28. Below is a screen shot from the book giving 3.4.28 at page 88, andthe context it is used in before solving the problem
Solution
For ππ‘β order ODE, π0 (π₯) is given by
π0 (π₯) βΌ ποΏ½π₯π (π‘)
1π ππ‘
And (page 497, textbook)
π1 (π₯) βΌ1 β π2π
ln (π (π₯)) + π (10.2.11)
80
3.2. HW2 CHAPTER 3. HWS
Therefore
π¦ (π₯) βΌ exp (π0 + π1)
βΌ exp οΏ½ποΏ½π₯π (π‘)
1π ππ‘ +
1 β π2π
ln (π (π₯)) + ποΏ½
βΌ π [π (π₯)]1βπ2π exp οΏ½ποΏ½
π₯π (π‘)
1π ππ‘οΏ½
Note: I have tried other methods to proof this, such as a proof by induction. But was notable to after many hours trying. The above method uses a given formula which the bookdid not indicate how it was obtained. (see key solution)
3.2.2 Problem 3.33(b) (page 140)
Problem Find leading behavior as π₯ β 0+ for π₯4π¦β²β²β² β 3π₯2π¦β² + 2π¦ = 0
Solution Let
π¦ (π₯) = ππ(π₯)
π¦β² = πβ²ππ
π¦β²β² = πβ²β²ππ + (πβ²)2 ππ
π¦β²β²β² = πβ²β²β²ππ + πβ²β²πβ²ππ + 2πβ²πβ²β²ππ + (πβ²)3 ππ
Hence the ODE becomes
π₯4 οΏ½πβ²β²β² + 3πβ²πβ²β² + (πβ²)3οΏ½ β 3π₯2πβ² = β2 (1)
Now, we define π (π₯) as sum of a number of leading terms, which we try to find
π (π₯) = π0 (π₯) + π1 (π₯) + π2 (π₯) +β―
Therefore (1) becomes (using only two terms for now π = π0 + π1)
{π0 + π1}β²β²β² + 3 οΏ½(π0 + π1) (π0 + π1)
β²β²οΏ½ + οΏ½(π0 + π1)β²οΏ½3β3π₯2{π0 + π1}
β² = β2π₯4
οΏ½πβ²β²β²0 + πβ²β²β²1 οΏ½ + 3 οΏ½(π0 + π1) οΏ½πβ²β²0 + πβ²β²1 οΏ½οΏ½ + οΏ½πβ²0 + πβ²1οΏ½3β3π₯2οΏ½πβ²0 + πβ²1οΏ½ = β
2π₯4
οΏ½πβ²β²β²0 + πβ²β²β²1 οΏ½ + 3 οΏ½πβ²β²0 πβ²0 + πβ²β²0 πβ²1 + πβ²β²1 πβ²0οΏ½ + οΏ½οΏ½πβ²0οΏ½3+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0 οΏ½πβ²1οΏ½
2οΏ½ β
3π₯2οΏ½πβ²0 + πβ²1οΏ½ = β
2π₯4
(2)
Assuming that πβ²0 β πβ²1, πβ²β²β²0 β πβ²β²β²1 , οΏ½πβ²0οΏ½3β 3οΏ½πβ²0οΏ½
2πβ²1 then equation (2) simplifies to
πβ²β²β²0 + 3πβ²β²0 πβ²0 + οΏ½πβ²0οΏ½3β3π₯2πβ²0 βΌ β
2π₯4
Assuming οΏ½πβ²0οΏ½3β πβ²β²β²0 , οΏ½πβ²0οΏ½
3β 3πβ²β²0 πβ²0, οΏ½πβ²0οΏ½
3β 3
π₯2πβ²0 (which we need to verify later), then
the above becomes
οΏ½πβ²0οΏ½3βΌ β
2π₯4
Verification2
2When carrying out verification, all constant multipliers and signs are automticaly simplified and removed
81
3.2. HW2 CHAPTER 3. HWS
Since πβ²0 βΌ οΏ½β2π₯4οΏ½13= 1
π₯43then πβ²β²0 βΌ 1
π₯73and πβ²β²β²0 βΌ 1
π₯103. Now we need to verify the three
assumptions made above, which we used to obtain πβ²0.
οΏ½πβ²0οΏ½3β πβ²β²β²0
1π₯4β
1
π₯103
Yes.
οΏ½πβ²0οΏ½3β 3πβ²β²0 πβ²0
1π₯4β
ββββββ1
π₯73
ββββββ
ββββββ1
π₯43
ββββββ
1π₯4β
ββββββ1
π₯113
ββββββ
Yes.
οΏ½πβ²0οΏ½3β
3π₯2πβ²0
1π₯4β οΏ½
1π₯2 οΏ½
1
π₯43
1π₯4β
1
π₯103
Yes. Assumed balance ise verified. Therefore
οΏ½πβ²0οΏ½3βΌ β
2π₯4
πβ²0 βΌ ππ₯β43
Where π3 = β2. Integrating
π0 βΌ ποΏ½π₯β43 ππ₯
βΌ ποΏ½π₯β43ππ₯
βΌ β3ππ₯β13
Where we ignored the constant of integration since subdominant. To find leading behavior,we go back to equation (2) and now solve for π1.
οΏ½πβ²β²β²0 + πβ²β²β²1 οΏ½ + 3 οΏ½πβ²β²0 πβ²0 + πβ²β²0 πβ²1 + πβ²β²1 πβ²0οΏ½ + οΏ½οΏ½πβ²0οΏ½3+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0 οΏ½πβ²1οΏ½
2οΏ½ β
3π₯2οΏ½πβ²0 + πβ²1οΏ½ = β
2π₯4
Moving all known quantities (those which are made of π0 and its derivatives) to the RHS
going from one step to the next, as they do not aοΏ½ect the final result.
82
3.2. HW2 CHAPTER 3. HWS
and simplifying, gives
οΏ½πβ²β²β²1 οΏ½ + 3 οΏ½πβ²β²0 πβ²1 + πβ²β²1 πβ²0οΏ½ + οΏ½3 οΏ½πβ²0οΏ½2πβ²1 + 3πβ²0 οΏ½πβ²1οΏ½
2οΏ½ β
3πβ²1π₯2
βΌ βπβ²β²β²0 β 3πβ²β²0 πβ²0 +3π₯2πβ²0
Now we assume the following (then will verify later)
3 οΏ½πβ²0οΏ½2πβ²1 β 3πβ²0 οΏ½πβ²1οΏ½
2
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²β²1
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²1 πβ²0
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²0 πβ²1
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²1 πβ²0
Hence
3 οΏ½πβ²0οΏ½2πβ²1 β
3πβ²1π₯2
βΌ βπβ²β²β²0 β 3πβ²β²0 πβ²0 +3πβ²0π₯2
(3)
But
πβ²0 βΌ ππ₯β43
οΏ½πβ²0οΏ½2βΌ π2π₯
β83
πβ²β²0 βΌ β43ππ₯
β73
πβ²β²β²0 βΌ289ππ₯
β103
Hence (3) becomes
3 οΏ½π2π₯β83 οΏ½ πβ²1 β
3πβ²1π₯2
βΌ289ππ₯
β103 + 3 οΏ½
43π2π₯
β73 π₯
β43 οΏ½ +
3ππ₯β43
π₯2
3π2π₯β83 πβ²1 β 3π₯β2πβ²1 βΌ
289ππ₯
β103 + 4π2π₯
β113 + 3ππ₯
β103
For small π₯, π₯β83 πβ²1 β π₯β2πβ²1 and π₯
β113 β π₯
β103 , then the above simplifies to
3π2π₯β83 πβ²1 βΌ 4π2π₯
β113
πβ²1 βΌ43π₯β1
π1 βΌ43
ln π₯
Where constant of integration was dropped, since subdominant.
83
3.2. HW2 CHAPTER 3. HWS
Verification Using πβ²0 β½ π₯β43 , οΏ½πβ²0οΏ½
2β½ π₯
β83 , πβ²β²0 β½ π₯
β73 , πβ²1 βΌ
1π₯ , π
β²β²1 βΌ 1
π₯2 , πβ²β²β²1 βΌ 1
π₯3
3 οΏ½πβ²0οΏ½2πβ²1 β 3πβ²0 οΏ½πβ²1οΏ½
2
π₯β831π₯β π₯
β431π₯2
π₯β83 β π₯
β73
Yes.
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²β²1
π₯β831π₯β
1π₯3
1
π₯83
β1π₯2
Yes.
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²1 πβ²0
π₯β831π₯β
1π₯2π₯β43
π₯β83 β π₯
β73
Yes.
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²0 πβ²1
π₯β831π₯β π₯
β731π₯
π₯β83 β π₯
β73
Yes
3 οΏ½πβ²0οΏ½2πβ²1 β πβ²β²1 πβ²0
π₯β831π₯β
1π₯2π₯β43
π₯β83 β π₯
β73
Yes. All verified. Leading behavior is
π¦ (π₯) β½ ππ0(π₯)+π1(π₯)
= exp οΏ½πππ₯β13 +
43
ln π₯οΏ½
= π₯43 ππππ₯
β13
I now wanted to see how Maple solution to this problem compare with the leading behaviornear π₯ = 0. To obtain a solution from Maple, one have to give initial conditions a little bitremoved from π₯ = 0 else no solution could be generated. So using arbitrary initial conditionsat π₯ = 1
100 a solution was obtained and compared to the above leading behavior. Another
84
3.2. HW2 CHAPTER 3. HWS
problem is how to select π in the above leading solution. By trial and error a constant wasselected. Here is screen shot of the result. The exact solution generated by Maple is verycomplicated, in terms of hypergeom special functions.
ode:=x^4*diff(y(x),x$3)-3*x^2*diff(y(x),x)+2*y(x);pt:=1/100:ic:=y(pt)=500,D(y)(pt)=0,(D@@2)(y)(pt)=0:sol:=dsolve({ode,ic},y(x)):leading:=(x,c)->x^(4/3)*exp(c*x^(-1/3));plot([leading(x,.1),rhs(sol)],x=pt..10,y=0..10,color=[blue,red],legend=["leading behavior","exact solution"],legendstyle=[location=top]);
3.2.3 problem 3.33(c) (page 140)
Problem Find leading behavior as π₯ β 0+ for π¦β²β² = βπ₯π¦
Solution
Let π¦ (π₯) = ππ(π₯). Hence
π¦ (π₯) = ππ0(π₯)
π¦β² (π₯) = πβ²0ππ0
π¦β²β² = πβ²β²0 ππ0 + οΏ½πβ²0οΏ½2ππ0
= οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ0
Substituting in the ODE gives
πβ²β²0 + οΏ½πβ²0οΏ½2= βπ₯ (1)
85
3.2. HW2 CHAPTER 3. HWS
Assuming πβ²β²0 βΌ οΏ½πβ²0οΏ½2then (1) becomes
πβ²β²0 βΌ β οΏ½πβ²0οΏ½2
Let πβ²0 = π§ then the above becomes π§β² = βπ§2. Hence ππ§ππ₯
1π§2 = β1 or ππ§
π§2 = βππ₯. Integratingβ1π§ = βπ₯ + π or π§ =
1π₯+π1
. Hence πβ²0 =1
π₯+π1. Integrating again gives
π0 (π₯) βΌ ln |π₯ + π1| + π2
Verification
πβ²0 =1
π₯+π1, οΏ½πβ²0οΏ½
2= 1
(π₯+π1)2 , πβ²β²0 =
β1(π₯+π1)
2
πβ²β²0 β π₯12
1(π₯ + π1)
2 β π₯12
Yes for π₯ β 0+.
οΏ½πβ²0οΏ½2β π₯
12
1(π₯ + π1)
2 β π₯12
Yes for π₯ β 0+. Verified. Controlling factor is
π¦ (π₯) βΌ ππ0(π₯)
βΌ πln|π₯+π1|+π2
βΌ π΄π₯ + π΅
3.2.4 problem 3.35
Problem: Obtain the full asymptotic behavior for small π₯ of solutions to the equation
π₯2π¦β²β² + (2π₯ + 1) π¦β² β π₯2 οΏ½π2π₯ + 1οΏ½ π¦ = 0
Solution
86
3.2. HW2 CHAPTER 3. HWS
Let π¦ (π₯) = ππ(π₯). Hence
π¦ (π₯) = ππ0(π₯)
π¦β² (π₯) = πβ²0ππ0
π¦β²β² = πβ²β²0 ππ0 + οΏ½πβ²0οΏ½2ππ0
= οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ0
Substituting in the ODE gives
π₯2 οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ0 + (2π₯ + 1) πβ²0ππ0 β π₯2 οΏ½π
2π₯ + 1οΏ½ ππ0(π₯) = 0
π₯2 οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ + (2π₯ + 1) πβ²0 β π₯2 οΏ½π
2π₯ + 1οΏ½ = 0
π₯2 οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ + (2π₯ + 1) πβ²0 = π₯2 οΏ½π
2π₯ + 1οΏ½
πβ²β²0 + οΏ½πβ²0οΏ½2+(2π₯ + 1)π₯2
πβ²0 = π2π₯ + 1
Assuming balance
οΏ½πβ²0οΏ½2βΌ οΏ½π
2π₯ + 1οΏ½
οΏ½πβ²0οΏ½2βΌ π
2π₯
πβ²0 βΌ Β±π1π₯
Where 1 was dropped since subdominant to π1π₯ for small π₯.
Verification Since οΏ½πβ²0οΏ½2βΌ π
2π₯ then πβ²0 βΌ π
1π₯ and πβ²β²0 βΌ β 1
π₯2 π1π₯ , hence
οΏ½πβ²0οΏ½2β πβ²β²0
π2π₯ β
1π₯2π1π₯
π1π₯ β
1π₯2
Yes, As π₯ β 0+
οΏ½πβ²0οΏ½2β
(2π₯ + 1) πβ²0π₯2
π2π₯ β
(2π₯ + 1)π₯2
π1π₯
π1π₯ β
(2π₯ + 1)π₯2
π1π₯ β
2π₯+1π₯2
87
3.2. HW2 CHAPTER 3. HWS
Yes as π₯ β 0+. Verified. Hence both assumptions used were verified OK. Hence
πβ²0 βΌ Β±π2π₯
π0 βΌ Β±οΏ½π1π₯ππ₯
Since the integral do not have closed form, we will do asymptotic expansion on the integral.
Rewriting β« π1π₯ππ₯ as β« π
1π₯
οΏ½βπ₯2οΏ½οΏ½βπ₯2οΏ½ ππ₯. Using β«π’ππ£ = π’π£ β β«π£ππ’, where π’ = βπ₯2, ππ£ = βπ
1π₯
π₯2 , gives
ππ’ = β2π₯ and π£ = π1π₯ , hence
οΏ½π1π₯ππ₯ = π’π£ βοΏ½π£ππ’
= βπ₯2π1π₯ βοΏ½β2π₯π
1π₯ππ₯
= βπ₯2π1π₯ + 2οΏ½π₯π
1π₯ππ₯ (1)
Now we apply integration by parts on β«π₯π1π₯ππ₯ = β«π₯ π
1π₯
βπ₯3οΏ½βπ₯3οΏ½ ππ’ = β« π
1π₯
βπ₯2οΏ½βπ₯3οΏ½ ππ’, where
π’ = βπ₯3, ππ£ = π1π₯
βπ₯2 , hence ππ’ = β3π₯2, π£ = π
1π₯ , hence we have
οΏ½π₯π1π₯ππ₯ = π’π£ βοΏ½π£ππ’
= βπ₯3π1π₯ +οΏ½3π₯2π
1π₯ππ₯
Substituting this into (1) gives
οΏ½π1π₯ππ₯ = βπ₯2π
1π₯ + 2 οΏ½βπ₯3π
1π₯ +οΏ½3π₯2π
1π₯ππ₯οΏ½
= βπ₯2π1π₯ β 2π₯3π
1π₯ + 6οΏ½3π₯2π
1π₯ππ₯
And so on. The series will become
οΏ½π1π₯ππ₯ = βπ₯2π
1π₯ β 2π₯3π
1π₯ + 6π₯4π
1π₯ + 24π₯5π
1π₯ +β―+ π!π₯π+1π
1π₯ +β―
= βπ1π₯ οΏ½π₯2 + 2π₯3 + 6π₯4 +β―+ π!π₯π+1 +β―οΏ½
Now as π₯ β 0+, we can decide how many terms to keep in the RHS, If we keep one term,then we can say
π0 βΌ Β±οΏ½π1π₯ππ₯
βΌ Β±π₯2π1π₯
For two terms
π0 βΌ Β±οΏ½π1π₯ππ₯ βΌ Β±π
1π₯ οΏ½π₯2 + 2π₯3οΏ½
And so on. Let us use one term for now for the rest of the solution.
π0 βΌ Β±π₯2π1π₯
88
3.2. HW2 CHAPTER 3. HWS
To find leading behavior, let
π (π₯) = π0 (π₯) + π1 (π₯)
Then π¦ (π₯) = ππ0(π₯)+π1(π₯) and hence now
π¦β² (π₯) = (π0 (π₯) + π1 (π₯))β² ππ0+π1
π¦β²β² (π₯) = οΏ½(π0 + π1)β²οΏ½2ππ0+π1 + (π0 + π1)
β²β² ππ0+π1
Substituting into the given ODE gives
π₯2 οΏ½οΏ½(π0 + π1)β²οΏ½2+ (π0 + π1)
β²β²οΏ½ + (2π₯ + 1) (π0 (π₯) + π1 (π₯))β² β π₯2 οΏ½π
2π₯ + 1οΏ½ = 0
π₯2 οΏ½οΏ½πβ²0 + πβ²1οΏ½2+ (π0 + π1)
β²β²οΏ½ + (2π₯ + 1) οΏ½πβ²0 (π₯) + πβ²1 (π₯)οΏ½ β π₯2 οΏ½π2π₯ + 1οΏ½ = 0
π₯2 οΏ½οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1 + οΏ½πβ²β²0 + πβ²β²1 οΏ½οΏ½ + (2π₯ + 1) πβ²0 (π₯) + (2π₯ + 1) πβ²1 (π₯) β π₯2 οΏ½π
2π₯ + 1οΏ½ = 0
π₯2 οΏ½πβ²0οΏ½2+ π₯2 οΏ½πβ²1οΏ½
2+ 2π₯2πβ²0πβ²1 + π₯2πβ²β²0 + π₯2πβ²β²1 + (2π₯ + 1) πβ²0 (π₯) + (2π₯ + 1) πβ²1 (π₯) = π₯2 οΏ½π
2π₯ + 1οΏ½
But π₯2 οΏ½πβ²0οΏ½2βΌ π₯2 οΏ½π
2π₯ + 1οΏ½ since we found that πβ²0 βΌ π
1π₯ . Hence the above simplifies to
π₯2 οΏ½πβ²1οΏ½2+ 2π₯2πβ²0πβ²1 + π₯2πβ²β²0 + π₯2πβ²β²1 + (2π₯ + 1) πβ²0 (π₯) + (2π₯ + 1) πβ²1 (π₯) = 0
οΏ½πβ²1οΏ½2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 +
(2π₯ + 1)π₯2
πβ²0 (π₯) +(2π₯ + 1)π₯2
πβ²1 (π₯) = 0 (2)
Now looking at πβ²β²0 +(2π₯+1)π₯2 πβ²0 (π₯) terms in the above. We can simplify this since we know
πβ²0 = π1π₯ ,πβ²β²0 = β
1π₯2 π
1π₯ This terms becomes
β1π₯2π1π₯ +
(2π₯ + 1)π₯2
π1π₯ =
βπ1π₯ + 2π₯π
1π₯ + π
1π₯
π₯2=2π₯π
1π₯
π₯2=2π
1π₯
π₯Therefore (2) becomes
οΏ½πβ²1οΏ½2+ 2πβ²0πβ²1 + πβ²β²1 +
(2π₯ + 1)π₯2
πβ²1 (π₯) βΌβ2π
1π₯
π₯οΏ½πβ²1οΏ½
2
πβ²0+ 2πβ²1 +
πβ²β²1πβ²0
+(2π₯ + 1)π₯2πβ²0
πβ²1 (π₯) βΌβ2π
1π₯
π₯πβ²0οΏ½πβ²1οΏ½
2
π1π₯
+ 2πβ²1 +πβ²β²1π1π₯
+(2π₯ + 1)
π₯2π1π₯
πβ²1 (π₯) βΌβ2π₯
Assuming the balance is
πβ²1 βΌβ1π₯
Hence
π1 (π₯) βΌ β ln (π₯) + π
89
3.2. HW2 CHAPTER 3. HWS
Since π subdominant as π₯ β 0+ then
π1 (π₯) βΌ β ln (π₯)
Verification
πβ²1 βοΏ½πβ²1οΏ½
2
π1π₯
1π₯β
1π₯2
1
π1π₯
π1π₯ β
1π₯
Yes, for π₯ β 0+
πβ²1 βπβ²β²1π1π₯
1π₯β
1π₯2
1
π1π₯
Yes.
πβ²1 β(2π₯ + 1)
π₯2π1π₯
πβ²1 (π₯)
1π₯β
(2π₯ + 1)
π₯2π1π₯
1π₯
1π₯β
1
π₯3π1π₯
Yes. All assumptions verified. Hence leading behavior is
π¦ (π₯) βΌ exp (π0 (π₯) + π1 (π₯))
βΌ exp οΏ½Β±π₯2π1π₯ β ln (π₯)οΏ½
βΌ1π₯οΏ½exp οΏ½π₯2π
1π₯ οΏ½ + exp οΏ½βπ₯2π
1π₯ οΏ½οΏ½
For small π₯, then we ignore exp οΏ½βπ₯2π1π₯ οΏ½ since much smaller than exp οΏ½π₯2π
1π₯ οΏ½. Therefore
π¦ (π₯) βΌ1π₯
exp οΏ½π₯2π1π₯ οΏ½
3.2.5 problem 3.39(h)
problem Find leading asymptotic behavior as π₯ β β for π¦β²β² = πβ3π₯π¦
90
3.2. HW2 CHAPTER 3. HWS
solution Let π¦ (π₯) = ππ(π₯). Hence
π¦ (π₯) = ππ0(π₯)
π¦β² (π₯) = πβ²0ππ0
π¦β²β² = πβ²β²0 ππ0 + οΏ½πβ²0οΏ½2ππ0
= οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ0
Substituting in the ODE gives
οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ0 = πβ
3π₯ ππ0
πβ²β²0 + οΏ½πβ²0οΏ½2= πβ
3π₯
Assuming οΏ½πβ²0οΏ½2β πβ²β²0 the above becomes
οΏ½πβ²0οΏ½2βΌ πβ
3π₯
πβ²0 βΌ Β±πβ32π₯
Hence
π0 βΌ Β±οΏ½πβ32π₯ππ₯
Integration by parts. Since πππ₯π
β32π₯ = 3
2π₯2 πβ 32π₯ , then we rewrite the integral above as
οΏ½πβ32π₯ππ₯ = οΏ½
32π₯2
πβ32π₯ οΏ½
2π₯2
3 οΏ½ ππ₯
And now apply integration by parts. Let ππ£ = 32π₯2 π
β 32π₯ β π£ = πβ
32π₯ , π’ = 2π₯2
3 β ππ’ = 43π₯, hence
οΏ½πβ32π₯ππ₯ = [π’π£] βοΏ½π£ππ’
=2π₯2
3πβ
32π₯ βοΏ½
43π₯πβ
32π₯ππ₯
Ignoring higher terms, then we use
π0 βΌ Β±2π₯2
3πβ
32π₯
Verification
οΏ½πβ²0οΏ½2β πβ²β²0
οΏ½πβ32π₯ οΏ½
2β
32π₯2
πβ32π₯
πβ3π₯ β
32π₯2
πβ32π₯
Yes, as π₯ β β. To find leading behavior, let
π (π₯) = π0 (π₯) + π1 (π₯)
91
3.2. HW2 CHAPTER 3. HWS
Then π¦ (π₯) = ππ0(π₯)+π1(π₯) and hence now
π¦β² (π₯) = (π0 (π₯) + π1 (π₯))β² ππ0+π1
π¦β²β² (π₯) = οΏ½(π0 + π1)β²οΏ½2ππ0+π1 + (π0 + π1)
β²β² ππ0+π1
Using the above, the ODE π¦β²β² = πβ3π₯π¦ now becomes
οΏ½(π0 + π1)β²οΏ½2+ (π0 + π1)
β²β² = πβ3π₯
οΏ½πβ²0 + πβ²1οΏ½2+ πβ²β²0 + πβ²β²1 = π
β 3π₯
οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 = π
β 3π₯
But οΏ½πβ²0οΏ½2βΌ πβ
3π₯ hence the above simplifies to
οΏ½πβ²1οΏ½2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 = 0
Assuming οΏ½2πβ²0πβ²1οΏ½ β πβ²β²1 the above becomes
οΏ½πβ²1οΏ½2+ 2πβ²0πβ²1 + πβ²β²0 = 0
Assuming 2πβ²0πβ²1 β οΏ½πβ²1οΏ½2
2πβ²0πβ²1 + πβ²β²0 = 0
πβ²1 βΌ βπβ²β²02πβ²0
π1 βΌ β12
ln οΏ½πβ²0οΏ½
But πβ²0 βΌ πβ32π₯ , hence the above becomes
π1 βΌ β12
ln οΏ½πβ32π₯ οΏ½ + π
βΌ34π₯+ π
Verification
οΏ½2πβ²0πβ²1οΏ½ β πβ²β²1
οΏ½2πβ32π₯β34π₯2 οΏ½
β32π₯3
32πβ
32π₯
π₯2β
32π₯3
For large π₯ the above simplifies to1π₯2β
1π₯3
92
3.2. HW2 CHAPTER 3. HWS
Yes.
οΏ½2πβ²0πβ²1οΏ½ β οΏ½πβ²1οΏ½2
οΏ½2πβ32π₯β34π₯2 οΏ½
β οΏ½β34π₯2 οΏ½
2
32πβ
32π₯
π₯2β
916π₯4
For large π₯ the above simplifies to1π₯2β
1π₯4
Yes. All verified. Therefore, the leading behavior is
π¦ (π₯) βΌ exp (π0 (π₯) + π1 (π₯))
βΌ exp οΏ½Β±2π₯2
3πβ
32π₯ β
12
ln οΏ½πβ32π₯ οΏ½ + ποΏ½
βΌ ππ34π₯ exp οΏ½Β±
2π₯2
3πβ
32π₯ οΏ½ (1)
Check if we can use 3.4.28 to verify:
limπ₯ββ
|π₯ππ (π₯)| = limπ₯ββ
οΏ½π₯2πβ3π₯ οΏ½ β β
We can use it. Lets verify using 3.4.28
π¦ (π₯) βΌ π [π (π₯)]1βπ2π exp οΏ½ποΏ½
π₯π [π‘]
1π ππ‘οΏ½
Where π2 = 1. For π = 2,π (π₯) = πβ3π₯ , the above gives
π¦ (π₯) βΌ π οΏ½πβ3π₯ οΏ½
1β24
exp
βββββββποΏ½
π₯οΏ½πβ
3π‘ οΏ½
12ππ‘
βββββββ
βΌ π οΏ½πβ3π₯ οΏ½
β14
exp οΏ½ποΏ½π₯πβ
32π‘ππ‘οΏ½
βΌ ππ34π₯ exp οΏ½ποΏ½
π₯πβ
32π‘ππ‘οΏ½ (2)
We see that (1,2) are the same. Verified OK. Notice that in (1), we use the approximation for
the πβ32π₯ππ₯ β 2π₯2
3 πβ 32π₯ we found earlier. This was done, since there is no closed form solution
for the integral.
QED.
3.2.6 problem 3.42(a)
Problem: Extend investigation of example 1 of section 3.5 (a) Obtain the next few correctionsto the leading behavior (3.5.5) then see how including these terms improves the numerical
93
3.2. HW2 CHAPTER 3. HWS
approximation of π¦ (π₯) in 3.5.1.
Solution Example 1 at page 90 is π₯π¦β²β² + π¦β² = π¦. The leading behavior is given by 3.5.5 as(π₯ β β)
π¦ (π₯) βΌ ππ₯β14 π2π₯
12 (3.5.5)
Where the book gives π = 12π
β12 on page 91. And 3.5.1 is
π¦ (π₯) =βοΏ½π=0
π₯π
(π!)2(3.5.1)
To see the improvement, the book method is followed. This is described at end of page 91.This is done by plotting the leading behavior as ratio to π¦ (π₯) as given in 3.5.1. Hence forthe above leading behavior, we need to plot
12π
β12 π₯
β14 π2π₯
12
π¦ (π₯)We are given π0 (π₯) , π1 (π₯) in the problem. They are
π0 (π₯) = 2π₯12
π1 (π₯) = β14
ln π₯ + π
Hence
πβ²0 (π₯) = π₯β12
πβ²β²0 =β12π₯β
32
πβ²1 (π₯) = β141π₯
πβ²β²1 (π₯) =14π₯2
(1)
We need to find π2 (π₯) , π3 (π₯) ,β― to see that this will improve the solution π¦ (π₯) βΌ exp (π0 + π1 + π2 +β―)as π₯ β π₯0 compared to just using leading behavior π¦ (π₯) βΌ exp (π0 + π1). So now we need tofind π2 (π₯)
Let π¦ (π₯) = ππ, then the ODE becomes
π₯ οΏ½πβ²β² + (πβ²)2οΏ½ + πβ² = 1
Replacing π by π0 (π₯) + π1 (π₯) + π2 (π₯) in the above gives
(π0 + π1 + π2)β²β² + οΏ½(π0 + π1 + π2)
β²οΏ½2+1π₯(π0 + π1 + π2)
β² βΌ1π₯
οΏ½πβ²β²0 + πβ²β²1 + πβ²β²2 οΏ½ + οΏ½οΏ½πβ²0 + πβ²1 + πβ²2οΏ½οΏ½2+1π₯οΏ½πβ²0 + πβ²1 + πβ²2οΏ½ βΌ
1π₯
οΏ½πβ²β²0 + πβ²β²1 + πβ²β²2 οΏ½ + οΏ½οΏ½πβ²0οΏ½2+ 2πβ²0πβ²1 + 2πβ²0πβ²2 + οΏ½πβ²1οΏ½
2+ 2πβ²1πβ²2 + οΏ½πβ²2οΏ½
2οΏ½ +
1π₯οΏ½πβ²0 + πβ²1 + πβ²2οΏ½ βΌ
1π₯
Moving all known quantities to the RHS, these are πβ²β²0 , πβ²β²1 , οΏ½πβ²0οΏ½2, 2πβ²0πβ²1, πβ²0, πβ²1, οΏ½πβ²1οΏ½
2then the
94
3.2. HW2 CHAPTER 3. HWS
above reduces to
οΏ½πβ²β²2 οΏ½ + οΏ½+2πβ²0πβ²2 + 2πβ²1πβ²2 + οΏ½πβ²2οΏ½2οΏ½ +
1π₯οΏ½πβ²2οΏ½ βΌ
1π₯β πβ²β²0 β πβ²β²1 β οΏ½πβ²0οΏ½
2β 2πβ²0πβ²1 β
1π₯πβ²0 β
1π₯πβ²1 β οΏ½πβ²1οΏ½
2
Replacing known terms, by using (1) into the above gives
οΏ½πβ²β²2 οΏ½ + οΏ½2πβ²0πβ²2 + 2πβ²1πβ²2 + οΏ½πβ²2οΏ½2οΏ½ +
1π₯οΏ½πβ²2οΏ½ βΌ
1π₯+12π₯β
32 β
14π₯2
β οΏ½π₯β12 οΏ½
2β 2 οΏ½π₯
β12 οΏ½ οΏ½β
141π₯οΏ½β1π₯οΏ½π₯
β12 οΏ½ β
1π₯ οΏ½β141π₯οΏ½β οΏ½β
141π₯οΏ½
2
Simplifying gives
οΏ½πβ²β²2 οΏ½ + οΏ½2πβ²0πβ²2 + 2πβ²1πβ²2 + οΏ½πβ²2οΏ½2οΏ½ +
1π₯οΏ½πβ²2οΏ½ βΌ
1π₯+12π₯β
32 β
14π₯2
β π₯β1 +12π₯β32 β π₯
β32 +
141π₯2β116
1π₯2
Hence
οΏ½πβ²β²2 οΏ½ + οΏ½2πβ²0πβ²2 + 2πβ²1πβ²2 + οΏ½πβ²2οΏ½2οΏ½ +
1π₯οΏ½πβ²2οΏ½ βΌ β
116
1π₯2
Lets assume now that
2πβ²0πβ²2 βΌ β116
1π₯2
(2)
Therefore
πβ²2 βΌ β132
1πβ²0π₯2
βΌ β132
1
οΏ½π₯β12 οΏ½ π₯2
βΌ β132π₯β32
We can now verify this before solving the ODE. We need to check that (as π₯ β β)
2πβ²0πβ²2 β πβ²β²22πβ²0πβ²2 β 2πβ²1πβ²2
2πβ²0πβ²2 β οΏ½πβ²2οΏ½2
2πβ²0πβ²2 β1π₯πβ²2
Where πβ²β²2 βΌ π₯β52 , Hence
2πβ²0πβ²2 β πβ²β²2
π₯β12 οΏ½π₯
β32 οΏ½ β π₯
β52
π₯β2 β π₯β52
95
3.2. HW2 CHAPTER 3. HWS
Yes.
2πβ²0πβ²2 β 2πβ²1πβ²2
π₯β2 β οΏ½1π₯οΏ½οΏ½π₯
β32 οΏ½
π₯β2 β π₯β52
Yes
2πβ²0πβ²2 β οΏ½πβ²2οΏ½2
π₯β2 β οΏ½π₯β32 οΏ½
2
π₯β2 β π₯β3
Yes
2πβ²0πβ²2 β1π₯πβ²2
π₯β2 β1π₯π₯β32
π₯β2 β π₯β52
Yes. All assumptions are verified. Therefore we can g ahead and solve for π2 using (2)
2πβ²0πβ²2 βΌ β116
1π₯2
πβ²2 βΌ β132
1π₯2
1πβ²0
βΌ β132
1π₯2
1
π₯β12
βΌ β132
1
π₯32
Hence
π2 βΌ116
1
βπ₯
The leading behavior now is
π¦ (π₯) βΌ exp (π0 + π1 + π2)
βΌ exp οΏ½2π₯12 β
14
ln π₯ + π + 116
1
βπ₯οΏ½
Now we will find π3. From
π₯ οΏ½πβ²β² + (πβ²)2οΏ½ + πβ² = 1
96
3.2. HW2 CHAPTER 3. HWS
Replacing π by π0 + π1 + π2 + π3 in the above gives
(π0 + π1 + π2 + π3)β²β² + οΏ½(π0 + π1 + π2 + π3)
β²οΏ½2+1π₯(π0 + π1 + π2 + π3)
β² βΌ1π₯
οΏ½πβ²β²0 + πβ²β²1 + πβ²β²2 + πβ²β²3 οΏ½ + οΏ½οΏ½πβ²0 + πβ²1 + πβ²2 + πβ²3οΏ½οΏ½2+1π₯οΏ½πβ²0 + πβ²1 + πβ²2 + πβ²3οΏ½ βΌ
1π₯
Hence
οΏ½πβ²β²0 + πβ²β²1 + πβ²β²2 + πβ²β²3 οΏ½ +
οΏ½οΏ½πβ²0οΏ½2+ 2πβ²0πβ²1 + 2πβ²0πβ²2 + 2πβ²1πβ²2 + οΏ½πβ²1οΏ½
2+ οΏ½πβ²2οΏ½
2+ 2πβ²0πβ²3 + 2πβ²1πβ²3 + 2πβ²2πβ²3 + οΏ½πβ²3οΏ½
2οΏ½
+1π₯οΏ½πβ²0 + πβ²1 + πβ²2 + πβ²3οΏ½ βΌ
1π₯
Moving all known quantities to the RHS gives
οΏ½πβ²β²3 οΏ½ + οΏ½2πβ²0πβ²3 + 2πβ²1πβ²3 + 2πβ²2πβ²3 + οΏ½πβ²3οΏ½2οΏ½ +
1π₯οΏ½πβ²3οΏ½
βΌ1π₯β πβ²β²0 β πβ²β²1 β πβ²β²2 β οΏ½πβ²0οΏ½
2β 2πβ²0πβ²1 β 2πβ²0πβ²2 β 2πβ²1πβ²2 β οΏ½πβ²1οΏ½
2β οΏ½πβ²2οΏ½
2β1π₯πβ²0 β
1π₯πβ²1 β
1π₯πβ²2 (3)
Now we will simplify the RHS, since it is all known. Using
πβ²0 (π₯) = π₯β12
οΏ½πβ²0οΏ½2= π₯β1
πβ²β²0 =β12π₯β
32
πβ²1 (π₯) = β141π₯
οΏ½πβ²1 (π₯)οΏ½2=116
1π₯2
πβ²β²1 (π₯) =14π₯2
πβ²2 (π₯) = β132
1
π₯32
οΏ½πβ²2 (π₯)οΏ½2=
11024π₯3
πβ²β²2 (π₯) =364
1
π₯52
2πβ²0πβ²1 = 2 οΏ½π₯β12 οΏ½ οΏ½β
141π₯οΏ½= β
121
π₯32
2πβ²0πβ²2 = 2 οΏ½π₯β12 οΏ½βββββββ
132
1
π₯32
ββββββ = β
116π₯2
2πβ²1πβ²2 = 2 οΏ½β141π₯οΏ½
βββββββ
132
1
π₯32
ββββββ =
1
64π₯52
97
3.2. HW2 CHAPTER 3. HWS
Hence (3) becomes
οΏ½πβ²β²3 οΏ½ + οΏ½2πβ²0πβ²3 + 2πβ²1πβ²3 + 2πβ²2πβ²3 + οΏ½πβ²3οΏ½2οΏ½ +
1π₯οΏ½πβ²3οΏ½ βΌ
1π₯+
1
2π₯32
β14π₯2
β364
1
π₯52
β1π₯+121
π₯32
+1
16π₯2β
1
64π₯52
β116
1π₯2β
11024π₯3
β1
π₯32
+141π₯2+132
1
π₯52
Simplifying gives
οΏ½πβ²β²3 οΏ½ + οΏ½2πβ²0πβ²3 + 2πβ²1πβ²3 + 2πβ²2πβ²3 + οΏ½πβ²3οΏ½2οΏ½ +
1π₯οΏ½πβ²3οΏ½ βΌ β
ββββββ
32
1024π₯52
+1
1024π₯3
ββββββ
Let us now assume that
πβ²0πβ²3 β πβ²1πβ²3πβ²0πβ²3 β πβ²2πβ²3
πβ²0πβ²3 β οΏ½πβ²3οΏ½2
πβ²0πβ²3 β1π₯οΏ½πβ²3οΏ½
πβ²0πβ²3 β πβ²β²3Therefore, we end up with the balance
2πβ²0πβ²3 βΌ βββββββ
32
1024π₯52
+1
1024π₯3
ββββββ
πβ²3 βΌ β
βββββββ
32
1024π₯52πβ²0
+1
1024π₯3πβ²0
βββββββ
βΌ β
ββββββββββ
32
1024π₯52 οΏ½π₯
β12 οΏ½
+1
1024π₯3 οΏ½π₯β12 οΏ½
ββββββββββ
βΌ βββββββ1
32π₯2+
1
1024π₯52
ββββββ
Hence
π3 βΌ1
1024οΏ½ 232π₯2 +
32π₯οΏ½
Where constant of integration was ignored. Let us now verify the assumptions made
πβ²0πβ²3 β πβ²1πβ²3πβ²0 β πβ²1
Yes.
πβ²0πβ²3 β πβ²2πβ²3πβ²0 β πβ²2
98
3.2. HW2 CHAPTER 3. HWS
Yes
πβ²0πβ²3 β οΏ½πβ²3οΏ½2
πβ²0 β πβ²3Yes.
πβ²0πβ²3 β1π₯οΏ½πβ²3οΏ½
πβ²0 β1π₯
π₯β12 β
1π₯
Yes, as π₯ β β, and finally
πβ²0πβ²3 β πβ²β²3
π₯β12
ββββββ1
32π₯2+
1
1024π₯52
ββββββ β
ββββββ
5
2048π₯72
+1
16π₯3
ββββββ
οΏ½32βπ₯ + 1οΏ½1024π₯3
β128βπ₯ + 5
2048π₯72
Yes, as π₯ β β. All assumptions verified. The leading behavior now is
π¦ (π₯) βΌ exp (π0 + π1 + π2 + π3)
βΌ exp οΏ½2π₯12 β
14
ln π₯ + π + 116
1
βπ₯+
11024 οΏ½
232π₯2
+32π₯ οΏ½οΏ½
βΌ ππ₯β14 exp οΏ½2π₯
12 +
116
1
βπ₯+
11024 οΏ½
232π₯2
+32π₯ οΏ½οΏ½
Now we will show how adding more terms to leading behavior improved the π¦ (π₯) solutionfor large π₯. When plotting the solutions, we see that exp(π0+π1+π2+π3)
π¦(π₯) approached the ratio
1 sooner than exp(π0+π1+π2)π¦(π₯) and this in turn approached the ratio 1 sooner than just using
exp(π0+π1)π¦(π₯) . So the eοΏ½ect of adding more terms, is that the solution becomes more accurate
for larger range of π₯ values. Below is the code used and the plot generated.
99
3.2. HW2 CHAPTER 3. HWS
ClearAll[y, x];
s0[x_] :=
1
2 Pi12
Exp2 x12
y[x, 300];
s0s1[x_] :=
1
2 Pi12
Exp2 x12 -
1
4Log[x]
y[x, 300];
s0s1s2[x_] :=
1
2 Pi12
Exp2 x12 -
1
4Log[x] +
1
16
1
Sqrt[x]
y[x, 300];
s0s1s2s3[x_] :=
1
2 Pi12
Exp2 x12 -
1
4Log[x] +
1
16
1
Sqrt[x]+
1
1024
2
32 x2+
32
x
y[x, 300];
y[x_, max_] := Sum[ x^n / (Factorial[n]^2), {n, 0, max}];
LogLinearPlot[Evaluate[{s0s1[x], s0s1s2[x], s0s1s2s3[x]}], {x, 1, 30},
PlotRange β All, Frame β True, GridLines β Automatic, GridLinesStyle β LightGray,
PlotLegends β {"exp(S0+S1)", "exp(S0+S1+S2)", "exp(S0+S1+S2+S3)"},
FrameLabel β {{None, None}, {"x", "Showing improvement as more terms are added"}},
PlotStyle β {Red, Blue, Black}, BaseStyle β 14]
Out[59]=
1 2 5 10 20
0.92
0.94
0.96
0.98
1.00
x
Showing improvement as more terms are added
exp(S0+S1)
exp(S0+S1+S2)
exp(S0+S1+S2+S3)
100
3.2. HW2 CHAPTER 3. HWS
3.2.7 problem 3.49(c)
Problem Find the leading behavior as π₯ β β of the general solution of π¦β²β² + π₯π¦ = π₯5
Solution This is non-homogenous ODE. We solve this by first finding the homogenoussolution (asymptotic solution) and then finding particular solution. Hence we start with
π¦β²β²β + π₯π¦β = 0
π₯ = β is ISP point. Therefore, we assume π¦β (π₯) = ππ(π₯) and obtain
πβ²β² + (πβ²)2 + π₯ = 0 (1)
Let
π (π₯) = π0 + π1 +β―
Therefore (1) becomes
οΏ½πβ²β²0 + πβ²β²1 +β―οΏ½ + οΏ½πβ²0 + πβ²1 +β―οΏ½2= βπ₯
οΏ½πβ²β²0 + πβ²β²1 +β―οΏ½ + οΏ½οΏ½πβ²0οΏ½2+ 2πβ²0πβ²1 + οΏ½πβ²1οΏ½
2+β―οΏ½ = βπ₯ (2)
Assuming οΏ½πβ²0οΏ½2β πβ²β²0 we obtain
οΏ½πβ²0οΏ½2βΌ βπ₯
πβ²0 βΌ πβπ₯
Where π = Β±π
Verification
οΏ½πβ²0οΏ½2β πβ²β²0
π₯ β121
βπ₯Yes, as π₯ β β. Hence
π0 βΌ32ππ₯
32
Now we will find π1. From (2), and moving all known terms to RHS
οΏ½πβ²β²1 +β―οΏ½ + οΏ½2πβ²0πβ²1 + οΏ½πβ²1οΏ½2+β―οΏ½ βΌ βπ₯ β πβ²β²0 β οΏ½πβ²0οΏ½
2(3)
Assuming
2πβ²0πβ²1 β πβ²β²1
2πβ²0πβ²1 β οΏ½πβ²1οΏ½2
101
3.2. HW2 CHAPTER 3. HWS
Then (3) becomes (where πβ²0 βΌ πβπ₯, οΏ½πβ²0οΏ½2βΌ βπ₯, πβ²β²0 βΌ 1
2π1
βπ₯)
2πβ²0πβ²1 βΌ βπ₯ β πβ²β²0 β οΏ½πβ²0οΏ½2
πβ²1 βΌβπ₯ β πβ²β²0 β οΏ½πβ²0οΏ½
2
2πβ²0
βΌβπ₯ β 1
2π1
βπ₯β (βπ₯)
2πβπ₯
βΌ β14π₯
Verification (where πβ²β²1 βΌ 141π₯2 )
2πβ²0πβ²1 β πβ²β²1
βπ₯ οΏ½14π₯οΏ½
β141π₯2
1
π₯12
β1π₯2
Yes, as π₯ β β
2πβ²0πβ²1 β οΏ½πβ²1οΏ½2
1
π₯12
β οΏ½14π₯οΏ½
2
1
π₯12
β1
16π₯2
Yes, as π₯ β β. All validated. We solve for π1
πβ²1 βΌ β14π₯
π1 βΌ β14
ln π₯ + π
π¦β is found. It is given by
π¦β (π₯) βΌ exp (π0 (π₯) + π1 (π₯))
βΌ exp οΏ½32ππ₯
32 β
14
ln π₯ + ποΏ½
βΌ ππ₯β14 exp οΏ½
32ππ₯
32 οΏ½
Now that we have found π¦β, we go back and look at
π¦β²β² + π₯π¦ = π₯5
And consider two cases (a) π¦β²β² βΌ π₯5 (b) π₯π¦ βΌ π₯5. The case of π¦β²β² βΌ π₯π¦ was covered above.This is what we did to find π¦β (π₯).
102
3.2. HW2 CHAPTER 3. HWS
case (a)
π¦β²β²π βΌ π₯5
π¦β²π βΌ15π₯4
π¦π βΌ120π₯3
Where constants of integration are ignored since subdominant for π₯ β β. Now we check ifthis case is valid.
π₯π¦π β π₯5
π₯120π₯3 β π₯5
π₯4 β π₯5
No. Therefore case (a) did not work out. We try case (b) now
π₯π¦π βΌ π₯5
π¦π βΌ π₯4
Now we check is this case is valid.
π¦β²β²π β π₯5
12π₯2 β π₯5
Yes. Therefore, we found
π¦π βΌ π₯4
Hence the complete asymptotic solution is
π¦ (π₯) βΌ π¦β (π₯) + π¦π (π₯)
βΌ ππ₯β14 exp οΏ½
32ππ₯
32 οΏ½ + π₯4
103
3.2. HW2 CHAPTER 3. HWS
3.2.8 key solution of selected problems
3.2.8.1 section 3 problem 27
104
3.2. HW2 CHAPTER 3. HWS
105
3.2. HW2 CHAPTER 3. HWS
106
3.2. HW2 CHAPTER 3. HWS
107
3.2. HW2 CHAPTER 3. HWS
3.2.8.2 section 3 problem 33 b
NEEP 548: Engineering Analysis II 2/6/2011
Problem 3.33: Bender & Orszag
Instructor: Leslie Smith
1 Problem Statement
Find the leading behavoir of the following equation as x β 0+:
x4yβ²β²β² β 3x2yβ² + 2y = 0 (1)
2 Solution
Need to make the substitution: y = eS(x)
yβ² = Sβ²eS(x)
yβ²β² = Sβ²β²eS(x) + (Sβ²)2eS(x)
yβ²β²β² = Sβ²β²β²eS(x) + 3Sβ²β²Sβ²eS(x) + (Sβ²)3eS(x)
After substituting and diving through by y, one obtains,
Sβ²β²β² + 3Sβ²β²Sβ² + (Sβ²)3 β3
x2Sβ² +
2
x4βΌ 0 (2)
Typically, Sβ²β² << (Sβ²)2 =β Sβ²β²β² << (Sβ²)3. Similarly, assume Sβ²β²β² << (Sβ²)3
With these assumptions, 2 becomes:
(Sβ²)3 β3
x2Sβ²
βΌ β2
x4
which is still a difficult problem to solve. Therefore, also want to assume (Sβ²)3 >> 3x2S
β² as x β 0+.Then,
(Sβ²)3 βΌβ2
x4x β 0+
Sβ²βΌ wxβ4/3 x β 0+
w = (β2)1/3
So(x) βΌ β3wxβ1/3 x β 0+
Sβ²β²βΌ β
4w
3xβ7/3
Sβ²β²β²βΌ
28w
9xβ10/3
1
108
3.2. HW2 CHAPTER 3. HWS
Now, check our assumptions,
(Sβ²)3 βΌ β2xβ12/3 >> β3Sβ²xβ2βΌ β3wxβ10/3 x β 0+ X
(Sβ²)3 βΌ β2xβ12/3 >> Sβ²β²β²βΌ
28w
9xβ10/3 x β 0+ X
(Sβ²)3 βΌ β2xβ12/3 >> Sβ²β²Sβ²βΌ
β4w2
3xβ11/3 x β 0+ X
Now estimate the inegrating function C(x) by letting,
S(x) = So(x) + C(x) (3)
and substitute this into 2.
Sβ²β²β²
o + C β²β²β² + 3 (Sβ²β²
o + C β²β²)(Sβ²
o + C β²)οΈΈ οΈ·οΈ· οΈΈ
term 1
+(Sβ²
o + C β²)3οΈΈ οΈ·οΈ· οΈΈ
term 2
β3
x2(Sβ²
o + C β²) +2
x4= 0 (4)
term 1: (Sβ²β²
oSβ²
o +C β²β²Sβ²
o + Sβ²β²
oCβ² + C β²β²C β²)
term 2: (Sβ²
o)3 + (C β²)3 + 3(Sβ²)2C β² + 3(Sβ²)(C β²)2
From here, notice that we have already balanced the terms (Sβ²
o)3 and β2
x4 , so they are removedat this point. Now we make the following assumptions:
Sβ²
o >> C β², Sβ²β²
o >> C β²β², Sβ²β²β²
o >> C β²β²β² as x β 0+
These assumptions result in,
Sβ²β²β²
o + 3Sβ²β²
oSβ²
o + 3(Sβ²
o)2C β²
βΌ3Sβ²
o
x2+
3C β²
x2x β 0+
Insert the value of So found in the previous step,
28w
9xβ10/3
β 4w2xβ11/3 + 3w2xβ8/3C β²βΌ 3wxβ10/3 + 3C β²xβ6/3
In keeping with the dominant balance idea, it is clear that the middle two terms dominate, thusleading to the following simplified relation,
3w2xβ8/3C β²βΌ 4w2xβ11/3 x β 0+
C β²βΌ
4
3xβ1 x β 0+
C βΌ4
3ln(x) x β 0+
Then C β²β²βΌ
β43 xβ2, C β²β²β²
βΌ83x
β3 and once again, check assumptions made,
2
109
3.2. HW2 CHAPTER 3. HWS
Sβ²
o βΌ wxβ4/3 >> C β²βΌ
4
3xβ1 x β 0+ X
Sβ²β²
o βΌβ4
3xβ7/3 >> C β²β²
βΌβ4
3xβ2 x β 0+ X
Sβ²β²β²
o βΌ28w
9xβ10/3 >> C β²β²β²
βΌ8
3xβ3 x β 0+ X
With the appearance of the ln(x) term, we have likely found the leading behavior already. However,lets find D, the third term, just to check.
Substitute y = e(So+Co+D), then divide by y,
yβ² = [Sβ²
o + C β²
o +Dβ²]e(So+Co+D)
yβ²β² = [Sβ²β²
o + C β²β²
o +Dβ²β²]e(So+Co+D) + [Sβ²
o + C β²
o +Dβ²]2e(So+Co+D)
yβ²β²β² = [Sβ²β²β²
o + C β²β²β²
o +Dβ²β²β²]e(So+Co+D) + 3[Sβ²β²
o + C β²β²
o +Dβ²β²][Sβ²
o + C β²
o +Dβ²]e(So+Co+D) + [Sβ²
o + C β²
o +Dβ²]3e(So+Co+D)
And...here we go...
Sβ²β²β²
o +οΏ½οΏ½C β²β²β²
o +οΏ½οΏ½Dβ²β²β²+3[οΏ½οΏ½οΏ½Sβ²β²
oSβ²
o + Sβ²β²
oCβ²
o +οΏ½οΏ½οΏ½Sβ²β²
oDβ² + C β²β²
oSβ²
o +οΏ½οΏ½οΏ½C β²β²
oCβ²
o +οΏ½οΏ½οΏ½C β²β²
oDβ² +οΏ½οΏ½οΏ½
Dβ²β²Sβ²
o +οΏ½οΏ½οΏ½Dβ²β²C β²
o +οΏ½οΏ½οΏ½Dβ²β²Dβ²]
+οΏ½οΏ½οΏ½οΏ½οΏ½6Sβ²
oCβ²
oDβ² + 3[οΏ½οΏ½οΏ½οΏ½
(Sβ²
o)2C β²
o + (Sβ²
o)2Dβ² + (C β²
o)2Sβ²
oοΈΈ οΈ·οΈ· οΈΈ
Unknown balance
+οΏ½οΏ½οΏ½οΏ½(C β²
o)2Dβ² +οΏ½οΏ½οΏ½οΏ½
(Dβ²)2Sβ²
o +οΏ½οΏ½οΏ½οΏ½οΏ½(Dβ²)2C β²
o] +οΏ½οΏ½οΏ½(Sβ²
o)3
+οΏ½οΏ½οΏ½(C β²
o)3 +οΏ½οΏ½οΏ½(Dβ²)3 β
3
x2[Sβ²
o + C β²
o +Dβ²] +οΏ½οΏ½οΏ½2
x4= 0
Assumptions: Sβ²
o >> C β²
o >> Dβ², Sβ²β²
o >> C β²β²
o >> Dβ²β², Sβ²β²β²
o >> C β²β²β²
o >> Dβ²β²β² all as x β 0+ and removepreviously balanced terms,
Sβ²β²β²
o + 3Sβ²β²
oCβ²
o + 3C β²β²
oSβ²
o + 3(Sβ²
o)2Dβ²
βΌ β3(C β²
o)Sβ²
o x β 0+
Subbing in the previously found values for So, Co,
β3wxβ10/3 +28w
9xβ10/3
β
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½16w
3xβ10/3
β 4wxβ10/3 + 3w2xβ8/3Dβ²βΌοΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β16w
3xβ10/3
3
110
3.2. HW2 CHAPTER 3. HWS
β3w2xβ8/3Dβ²βΌ
35w
9xβ10/3
Dβ²βΌ
35
27wxβ2/3
D βΌ35
27w(1
3)x1/3 + d
So since x1/3 β 0 as x β 0+, the leading order behavior is
y βΌ exp[βw
3xβ1/3 +
4
3ln(x) + d] x β 0+
4
111
3.2. HW2 CHAPTER 3. HWS
3.2.8.3 section 3 problem 33 c
112
3.2. HW2 CHAPTER 3. HWS
113
3.2. HW2 CHAPTER 3. HWS
114
3.2. HW2 CHAPTER 3. HWS
115
3.2. HW2 CHAPTER 3. HWS
116
3.2. HW2 CHAPTER 3. HWS
3.2.8.4 section 3 problem 35
Some notes on BO 3.35 (to get the leading behavior):
β’ a consistent balance at lowest order seems to be
(S β²
o)2 βΌ exp(2/x), x β 0
β’ taking the square root and integrating leads to
So βΌ Β±
β« x
1
exp(1/s)ds
β’ then change variables s = 1/t to find
So βΌ Β±
β« (1/x)
1
exp(t)
(
β1
t2
)
dt
β’ integration by parts gives
So βΌ βx2 exp(1/x) +Β±ao +β
β« (1/x)
1
2
t3exp(t)dt
βΌ βx2 exp(1/x)
and subdominant terms have been dropped. [Why are these terms subdominant andwhy is this the only consistent integration by parts?]
117
3.2. HW2 CHAPTER 3. HWS
3.2.8.5 section 3 problem 42a
118
3.2. HW2 CHAPTER 3. HWS
119
3.2. HW2 CHAPTER 3. HWS
120
3.2. HW2 CHAPTER 3. HWS
121
3.2. HW2 CHAPTER 3. HWS
122
3.2. HW2 CHAPTER 3. HWS
123
3.3. HW3 CHAPTER 3. HWS
3.3 HW3
3.3.1 problem 9.3 (page 479)
problem (a) show that if π (π₯) < 0 for 0 β€ π₯ β€ 1 then the solution to 9.1.7 has boundarylayer at π₯ = 1. (b) Find a uniform approximation with error π (π) to the solution 9.1.7 whenπ (π₯) < 0 for 0 β€ π₯ β€ 1 (c) Show that if π (π₯) > 0 it is impossible to match to a boundary layerat π₯ = 1
solution
3.3.1.1 Part a
Equation 9.1.7 at page 422 is
ππ¦β²β² + π (π₯) π¦β² + π (π₯) π¦ (π₯) = 0 (9.1.7)
π¦ (0) = π΄π¦ (1) = π΅
For 0 β€ π₯ β€ 1. Now we solve for π¦ππ (π₯), but first we introduce inner variable π. Weassume boundary layer is at π₯ = 0, then show that this leads to inconsistency. Let π = π₯
ππbe the inner variable. We express the original ODE using this new variable. We also needto determine π. Since ππ¦
ππ₯ =ππ¦ππ
ππππ₯ then ππ¦
ππ₯ =ππ¦πππ
βπ. Hence πππ₯ β‘ π
βπ πππ
π2
ππ₯2=πππ₯
πππ₯
= οΏ½πβπππποΏ½ οΏ½
πβπππποΏ½
= πβ2ππ2
ππ2
Therefore π2π¦ππ₯2 = π
β2π π2π¦ππ2 and (9.1.7) becomes
ππβ2ππ2π¦ππ2
+ π (π₯) πβπππ¦ππ
+ π¦ = 0
π1β2ππ2π¦ππ2
+ π (π₯) πβπππ¦ππ
+ π¦ = 0
The largest terms are οΏ½π1β2π, πβποΏ½, therefore balance gives 1 β 2π = βπ or π = 1. The ODE nowbecomes
πβ1π2π¦ππ2
+ π (π₯) πβ1ππ¦ππ
+ π¦ = 0 (1)
Assuming that
π¦ππ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
124
3.3. HW3 CHAPTER 3. HWS
And substituting the above into (1) gives
πβ1 οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + π (π₯) πβ1 οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0 (1A)
Collecting powers of ποΏ½πβ1οΏ½ terms, gives the ODE to solve for π¦ππ0 as
π¦β²β²0 βΌ βπ (π₯) π¦β²0In the rapidly changing region, because the boundary layer is very thin, we can approximateπ (π₯) by π (0). The above becomes
π¦β²β²0 βΌ βπ (0) π¦β²0But we are told that π (π₯) < 0, so π (0) < 0, hence βπ (0) is positive. Let βπ (0) = π2, to make itmore clear this is positive, then the ODE to solve is
π¦β²β²0 βΌ π2π¦β²0The solution to this ODE is
π¦0 (π) βΌπΆ1π2ππ2π + πΆ2
Using π¦ (0) = π΄, then the above gives π΄ = πΆ1π2 + πΆ2 or πΆ2 = π΄ β
πΆ1π2 and the ODE becomes
π¦0 (π) βΌπΆ1π2ππ2π + οΏ½π΄ β
πΆ1π2 οΏ½
βΌπΆ1π2οΏ½ππ2π β 1οΏ½ + π΄
We see from the above solution for the inner layer, that as π increases (meaning we aremoving away from π₯ = 0), then the solution π¦0 (π) and its derivative is increasing and notdecreasing since π¦β²0 (π) = πΆ1ππ
2π and π¦β²β²0 (π) = πΆ1π2ππ2π.
But this contradicts what we assumed that the boundary layer is at π₯ = 0 since we expectthe solution to change less rapidly as we move away from π₯ = 0. Hence we conclude that ifπ (π₯) < 0, then the boundary layer can not be at π₯ = 0.
Let us now see what happens by taking the boundary layer to be at π₯ = 1. We repeat thesame process as above, but now the inner variable as defined as
π =1 β π₯ππ
We express the original ODE using this new variable and determine π. Since ππ¦ππ₯ =
ππ¦ππ
ππππ₯ then
ππ¦ππ₯ =
ππ¦ππ(βπβπ). Hence π
ππ₯ β‘ (βπβπ) π
ππ
π2
ππ₯2=πππ₯
πππ₯
= οΏ½(βπβπ)ππποΏ½ οΏ½
(βπβπ)ππποΏ½
= πβ2ππ2
ππ2
125
3.3. HW3 CHAPTER 3. HWS
Therefore π2π¦ππ₯2 = π
β2π π2π¦ππ2 and equation (9.1.7) becomes
ππβ2ππ2π¦ππ2
β π (π₯) πβπππ¦ππ
+ π¦ = 0
π1β2ππ2π¦ππ2
β π (π₯) πβπππ¦ππ
+ π¦ = 0
The largest terms are οΏ½π1β2π, πβποΏ½, therefore matching them gives 1 β 2π = βπ or
π = 1
The ODE now becomes
πβ1π2π¦ππ2
β π (π₯) πβ1ππ¦ππ
+ π¦ = 0 (2)
Assuming that
π¦ππ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
And substituting the above into (2) gives
πβ1 οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ β π (π₯) πβ1 οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0 (2A)
Collecting ποΏ½πβ1οΏ½ terms, gives the ODE to solve for π¦ππ0 as
π¦β²β²0 βΌ π (π₯) π¦β²0In the rapidly changing region, πΌ = π (1), because the boundary layer is very thin, weapproximated π (π₯) by π (1). The above becomes
π¦β²β²0 βΌ πΌπ¦β²0But we are told that π (π₯) < 0, so πΌ < 0, and the above becomes
π¦β²β²0 βΌ πΌπ¦β²0The solution to this ODE is
π¦0 (π) βΌπΆ1πΌππΌπ + πΆ2 (3)
Using
π¦ (π₯ = 1) = π¦ (π = 0)= π΅
Then (3) gives π΅ = πΆ1πΌ + πΆ2 or πΆ2 = π΅ β
πΆ1πΌ and (3) becomes
π¦0 (π) βΌπΆ1πΌππΌπ + οΏ½π΅ β
πΆ1πΌ οΏ½
βΌπΆ1πΌοΏ½ππΌπ β 1οΏ½ + π΅ (4)
From the above, π¦β²0 (π) = βπΆ1ππΌπ and π¦β²β²0 (π) = πΆ1πΌππΌπ. We now see that as that as π increases(meaning we are moving away from π₯ = 1 towards the left), then the solution π¦0 (π) is actuallychanging less rapidly. This is because πΌ < 0. The solution is changing less rapidly as we
126
3.3. HW3 CHAPTER 3. HWS
move away from the boundary layer as what we expect. Therefore, we conclude that ifπ (π₯) < 0 then the boundary layer can not be at π₯ = 0 and has to be at π₯ = 1.
3.3.1.2 Part b
To find uniform approximation, we need now to find π¦ππ’π‘ (π₯) and then do the matching. Sincefrom part(a) we concluded that π¦ππ is near π₯ = 1, then we assume now that π¦ππ’π‘ (π₯) is nearπ₯ = 0. Let
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Substituting this into (9.1.7) gives
π οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + π (π₯) οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ + π (π₯) οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0
Collecting terms of π (1) gives the ODE
π (π₯) π¦β²0 + π (π₯) π¦0 = 0
The solution to this ODE is
π¦0 (π₯) = πΆ2πββ«
π₯0
π(π )π(π )ππ
Applying π¦ (0) = π΄ gives
π΄ = πΆ2πββ«
10
π(π )π(π )ππ
= πΆ2πΈ
Where πΈ is constant, which is the value of the definite integral πΈ = πββ«10
π(π )π(π )ππ . Hence the
solution π¦ππ’π‘ (π₯) can now be written as
π¦ππ’π‘0 (π₯) =π΄πΈπββ«
π₯0
π(π )π(π )ππ
We are now ready to do the matching.
limπββ
π¦ππ (π) βΌ limπ₯β0
π¦ππ’π‘ (π₯)
limπββ
πΆ1πΌοΏ½ππΌπ β 1οΏ½ + π΅ βΌ lim
π₯β0
π΄πΈπββ«
π₯0
π(π )π(π )ππ
But since πΌ = π (1) < 0 then the above simplifies to
βπΆ1π (1)
+ π΅ =π΄πΈ
πΆ1 = βπ (1) οΏ½π΄πΈβ π΅οΏ½
127
3.3. HW3 CHAPTER 3. HWS
Hence inner solution becomes
π¦ππ0 (π) βΌβπ (1) οΏ½π΄πΈ β π΅οΏ½
π (1)οΏ½ππ(1)π β 1οΏ½ + π΅
βΌ οΏ½π΅ βπ΄πΈ οΏ½
οΏ½ππ(1)π β 1οΏ½ + π΅
βΌ π΅ οΏ½ππ(1)π β 1οΏ½ βπ΄πΈοΏ½ππ(1)π β 1οΏ½ + π΅
βΌ οΏ½π΅ βπ΄πΈ οΏ½
οΏ½ππ(1)π β 1οΏ½ + π΅
The uniform solution is
π¦uniform (π₯) βΌ π¦ππ (π) + π¦ππ’π‘ (π₯) β π¦πππ‘πβ
βΌ
π¦ππ
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½π΅ β
π΄πΈ οΏ½
οΏ½ππ(1)π β 1οΏ½ + π΅ +
π¦ππ’π‘
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½π΄πΈπββ«
π₯0
π(π )π(π )ππ β
π΄πΈ
Or in terms of π₯ only
π¦uniform (π₯) βΌ οΏ½π΅ βπ΄πΈ οΏ½
οΏ½ππ(1)1βπ₯π β 1οΏ½ + π΅ +
π΄πΈπββ«
π₯0
π(π )π(π )ππ β
π΄πΈ
3.3.1.3 Part c
We now assume the boundary layer is at π₯ = 1 but π (π₯) > 0. From part (a), we found thatthe solution for π¦ππ0 (π) where boundary layer at π₯ = 1 is
π¦0 (π) βΌπΆ1πΌοΏ½ππΌπ β 1οΏ½ + π΅
But now πΌ = π (1) > 0 and not negative as before. We also found that π¦ππ’π‘0 (π₯) solution was
π¦ππ’π‘0 (π₯) =π΄πΈπββ«
π₯0
π(π )π(π )ππ
Lets now try to do the matching and see what happens
limπββ
π¦ππ (π) βΌ limπ₯β0
π¦ππ’π‘ (π₯)
limπββ
πΆ1πΌοΏ½ππΌπ β 1οΏ½ + π΅ + π (π) βΌ lim
π₯β0
π΄πΈπββ«
π₯0
π(π )π(π )ππ + π (π)
limπββ
πΆ1 οΏ½ππΌπ
πΌβ1πΌοΏ½
βΌπ΄πΈβ π΅
Since now πΌ > 0, then the term on the left blows up, while the term on the right is finite.Not possible to match, unless πΆ1 = 0. But this means the boundary layer solution is just aconstant π΅ and that π΄
πΈ = π΅. So the matching does not work in general for arbitrary conditions.This means if π (π₯) > 0, it is not possible to match boundary layer at π₯ = 1.
128
3.3. HW3 CHAPTER 3. HWS
3.3.2 problem 9.4(b)
Problem Find the leading order uniform asymptotic approximation to the solution of
ππ¦β²β² + οΏ½1 + π₯2οΏ½ π¦β² β π₯3π¦ (π₯) = 0 (1)
π¦ (0) = 1π¦ (1) = 1
For 0 β€ π₯ β€ 1 in the limit as π β 0.
solution
Since π (π₯) = οΏ½1 + π₯2οΏ½ is positive, we expect the boundary layer to be near π₯ = 0. First we findπ¦ππ’π‘ (π₯), which is near π₯ = 1. Assuming
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
And substituting this into (1) gives
π οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + οΏ½1 + π₯2οΏ½ οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ β π₯3 οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0
Collecting terms in π (1) gives the ODE
οΏ½1 + π₯2οΏ½ π¦β²0 βΌ π₯3π¦0
The ODE becomes π¦β²0 βΌπ₯3
οΏ½1+π₯2οΏ½π¦0 with integrating factor π = π
β« βπ₯3
οΏ½1+π₯2οΏ½ππ₯. To evaluate β« βπ₯3
οΏ½1+π₯2οΏ½ππ₯,
let π’ = π₯2, hence ππ’ππ₯ = 2π₯ and the integral becomes
οΏ½βπ₯3
οΏ½1 + π₯2οΏ½ππ₯ = βοΏ½
π’π₯(1 + π’)
ππ’2π₯
= β12 οΏ½
π’(1 + π’)
ππ’
But
οΏ½π’
(1 + π’)ππ’ = οΏ½1 β
1(1 + π’)
ππ’
= π’ β ln (1 + π’)But π’ = π₯2, hence
οΏ½βπ₯3
οΏ½1 + π₯2οΏ½ππ₯ =
β12οΏ½π₯2 β ln οΏ½1 + π₯2οΏ½οΏ½
129
3.3. HW3 CHAPTER 3. HWS
Therefore the integrating factor is π = exp οΏ½β12 π₯2 + 1
2 ln οΏ½1 + π₯2οΏ½οΏ½. The ODE becomes
πππ₯οΏ½ππ¦0οΏ½ = 0
ππ¦0 βΌ π
π¦ππ’π‘ (π₯) βΌ π exp οΏ½12π₯2 β
12
ln οΏ½1 + π₯2οΏ½οΏ½
βΌ ππ12π₯
2πlnοΏ½1+π₯
2οΏ½β12
βΌ πππ₯22
β1 + π₯2To find π, using boundary conditions π¦ (1) = 1 gives
1 = ππ12
β2
π = β2πβ12
Hence
π¦ππ’π‘0 (π₯) βΌ β2ππ₯2β12
β1 + π₯2
Now we find π¦ππ (π₯) near π₯ = 0. Let π = π₯ππ be the inner variable. We express the original ODE
using this new variable and determine π. Since ππ¦ππ₯ =
ππ¦ππ
ππππ₯ then ππ¦
ππ₯ =ππ¦πππ
βπ. Hence πππ₯ β‘ π
βπ πππ
π2
ππ₯2=πππ₯
πππ₯
= οΏ½πβπππποΏ½ οΏ½
πβπππποΏ½
= πβ2ππ2
ππ2
Therefore π2π¦ππ₯2 = π
β2π π2π¦ππ2 and ππ¦
β²β² + οΏ½1 + π₯2οΏ½ π¦β² β π₯3π¦ (π₯) = 0 becomes
ππβ2ππ2π¦ππ2
+ οΏ½1 + (πππ)2οΏ½ πβπππ¦ππ
β (πππ)3 π¦ = 0
π1β2ππ2π¦ππ2
+ οΏ½1 + π2π2ποΏ½ πβπππ¦ππ
β π3π3ππ¦ = 0
The largest terms are οΏ½π1β2π, πβποΏ½, therefore matching them gives 1 β 2π = βπ or π = 1. TheODE now becomes
πβ1π2π¦ππ2
+ οΏ½1 + π2π2οΏ½ πβ1ππ¦ππ
β π3π3π¦ = 0 (2)
130
3.3. HW3 CHAPTER 3. HWS
Assuming that
π¦ππ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
And substituting the above into (2) gives
πβ1 οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + οΏ½1 + π2π2οΏ½ πβ1 οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ β π3π3 οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0 (2A)
Collecting terms in ποΏ½πβ1οΏ½ gives the ODE
π¦β²β²0 (π) βΌ βπ¦β²0 (π)
The solution to this ODE is
π¦ππ0 (π) βΌ π1 + π2πβπ (3)
Applying π¦ππ0 (0) = 1 gives
1 = π1 + π2π1 = 1 β π2
Hence (3) becomes
π¦ππ0 (π) βΌ (1 β π2) + π2πβπ
βΌ 1 + π2 οΏ½πβπ β 1οΏ½ (4)
Now that we found π¦ππ’π‘ and π¦ππ, we apply matching to find π2 in the π¦ππ solution.
limπββ
π¦ππ0 (π) βΌ limπ₯β0+
π¦ππ’π‘0 (π₯)
limπββ
1 + π2 οΏ½πβπ β 1οΏ½ βΌ limπ₯β0+
β2ππ₯2β12
β1 + π₯2
1 β π2 βΌοΏ½2π
limπ₯β0+
ππ₯22
β1 + π₯2
βΌοΏ½2π
limπ₯β0+
ππ₯22
β1 + π₯2
=οΏ½2π
Hence
π2 = 1 βοΏ½2π
131
3.3. HW3 CHAPTER 3. HWS
Therefore the π¦ππ0 (π) becomes
π¦ππ0 (π) βΌ 1 +ββββββ1 βοΏ½
2π
ββββββ οΏ½π
βπ β 1οΏ½
βΌ 1 + οΏ½πβπ β 1οΏ½ βοΏ½2ποΏ½πβπ β 1οΏ½
βΌ πβπ βοΏ½2ποΏ½πβπ β 1οΏ½
βΌ πβπ βοΏ½2ππβπ +
οΏ½2π
βΌ πβπββββββ1 βοΏ½
2π
ββββββ +οΏ½
2π
βΌ 0.858 + 0.142πβπ
Therefore, the uniform solution is
π¦π’ππππππ βΌ π¦ππ (π₯) + π¦ππ’π‘ (π₯) β π¦πππ‘πβ + π (π) (4)
Where π¦πππ‘πβ is π¦ππ (π₯) at the boundary layer matching location. (or π¦ππ’π‘ at same matchinglocation). Hence
π¦πππ‘πβ βΌ 1 β π2
βΌ 1 βββββββ1 βοΏ½
2π
ββββββ
βΌοΏ½2π
Hence (4) becomes
π¦π’ππππππ βΌ
π¦πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½πβπ
ββββββ1 βοΏ½
2π
ββββββ +οΏ½
2π+
π¦ππ’π‘οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
β2ππ₯2β12
β1 + π₯2β
π¦πππ‘πβοΏ½
οΏ½2π
βΌ πβπ₯π
ββββββ1 βοΏ½
2π
ββββββ + β2
ππ₯2β12
β1 + π₯2+ π (π)
This is the leading order uniform asymptotic approximation solution. To verify the result,the numerical solution was plotted against the above solution for π = {0.1, 0.05, 0.01}. We seefrom these plots that as π becomes smaller, the asymptotic solution becomes more accuratewhen compared to the numerical solution. This is because the error, which is π (π) , becomessmaller. The code used to generate these plots is
132
3.3. HW3 CHAPTER 3. HWS
In[180]:= eps = 0.1;
sol = NDSolve[{1 / 100 y''[x] + (1 + x^2) y'[x] - x^3 y[x] β©΅ 0, y[0] β©΅ 1, y[1] β©΅ 1}, y, {x, 0, 1}];
p1 = Plot[Evaluate[y[x] /. sol], {x, 0, 1}, Frame β True,
FrameLabel β {{"y(x)", None}, {"x", Row[{"numberical vs. asymptotic for eps =", eps}]}},
GridLines β Automatic, GridLinesStyle β LightGray];
mysol[x_, eps_] := Exp-x
eps 1 - Sqrt
2
Exp[1] +
Sqrt[2] Exp x2-1
2
Sqrt[1 + x^2];
p2 = Plot[mysol[x, eps], {x, 0, 1}, PlotRange β All, PlotStyle β Red];
Show[Legended[p1, Style["Numerical", Red]], Legended[p2, Style["Asymtotic", Blue]]]
The following are the three plots for each value of π
Out[6]=
0.0 0.2 0.4 0.6 0.8 1.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
x
y(x)
numberical vs. asymptotic for eps =0.1
NumericalAsymptotic
Out[12]=
0.0 0.2 0.4 0.6 0.8 1.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
x
y(x)
numberical vs. asymptotic for eps =0.05
NumericalAsymptotic
133
3.3. HW3 CHAPTER 3. HWS
Out[17]=
0.0 0.2 0.4 0.6 0.8 1.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
x
y(x)
numberical vs. asymptotic for eps =0.01
NumericalAsymptotic
To see the eοΏ½ect on changing π on only the asymptotic approximation, the following plotgives the approximation solution only as π changes. We see how the approximation convergesto the numerical solution as π becomes smaller.
In[236]:= Plot[{mysol[x, .1], mysol[x, .05], mysol[x, .01]}, {x, 0, 1},
PlotLegends β {"β=0.1", "β=0.05", "β=0.01"}, Frame β True,
FrameLabel β {{"y(x)", None}, {"x", "Asymptotic solution as eps changes"}},
BaseStyle β 14]
Out[236]=
0.0 0.2 0.4 0.6 0.8 1.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
x
y(x)
Asymptotic solution as eps changes
β=0.1
β=0.05
β=0.01
3.3.3 problem 9.6
Problem Consider initial value problem
π¦β² = οΏ½1 +π₯β2
100οΏ½π¦2 β 2π¦ + 1
With π¦ (1) = 1 on the interval 0 β€ π₯ β€ 1. (a) Formulate this problem as perturbation problemby introducing a small parameter π. (b) Find outer approximation correct to order π witherrors of order π2. Where does this approximation break down? (c) Introduce inner variableand find the inner solution valid to order 1 (with errors of order π). By matching to the outersolution find a uniform valid solution to π¦ (π₯) on interval 0 β€ π₯ β€ 1. Estimate the accuracy of
134
3.3. HW3 CHAPTER 3. HWS
this approximation. (d) Find inner solution correct to order π (with errors of order π2) andshow that it matches to the outer solution correct to order π.
solution
3.3.3.1 Part a
Since 1100 is relatively small compared to all other coeοΏ½cients, we replace it with π and the
ODE becomes
π¦β² β οΏ½1 +ππ₯2οΏ½ π¦2 + 2π¦ = 1 (1)
3.3.3.2 Part b
Assuming boundary layer is on the left side at π₯ = 0. We now solve for π¦ππ’π‘ (π₯), which is thesolution near π₯ = 1.
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Substituting this into (1) gives
οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ β οΏ½1 +ππ₯2οΏ½ οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½
2+ 2 οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 1
Expanding the above to see more clearly the terms gives
οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ β οΏ½1 +ππ₯2οΏ½ οΏ½π¦20 + π οΏ½2π¦0π¦1οΏ½ + π2 οΏ½2π¦0π¦2 + π¦21οΏ½ +β―οΏ½ + 2 οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 1
(2)
The leading order are those terms of coeοΏ½cient π (1). This gives
π¦β²0 β π¦20 + 2π¦0 βΌ 1
With boundary conditions π¦ (1) = 1.ππ¦0ππ₯
βΌ π¦20 β 2π¦0 + 1
This is separableππ¦0
π¦20 β 2π¦0 + 1βΌ ππ₯
ππ¦0οΏ½π¦0 β 1οΏ½
2 βΌ ππ₯
135
3.3. HW3 CHAPTER 3. HWS
For π¦0 β 1. Integrating
οΏ½ππ¦0
οΏ½π¦0 β 1οΏ½2 βΌοΏ½ππ₯
β1π¦0 β 1
βΌ π₯ + πΆ
οΏ½π¦0 β 1οΏ½ (π₯ + πΆ) βΌ β1
π¦0 βΌβ1π₯ + πΆ
+ 1 (3)
To find πΆ, from π¦ (1) = 1, we find
1 βΌβ11 + πΆ
+ 1
1 βΌπΆ
1 + πΆThis is only possible if πΆ = β. Therefore from (2), we conclude that
π¦0 (π₯) βΌ 1
The above is leading order for the outer solution. Now we repeat everything to find π¦ππ’π‘1 (π₯).From (2) above, we now keep all terms with π (π) which gives
π¦β²1 β 2π¦0π¦1 + 2π¦1 βΌ1π₯2π¦20
But we found π¦0 (π₯) βΌ 1 from above, so the above ODE becomes
π¦β²1 β 2π¦1 + 2π¦1 βΌ1π₯2
π¦β²1 βΌ1π₯2
Integrating gives
π¦1 (π₯) βΌ β1π₯+ πΆ
The boundary condition now becomes π¦1 (1) = 0 (since we used π¦ (1) = 1 earlier with π¦0).This gives
0 = β11+ πΆ
πΆ = 1
Therefore the solution becomes
π¦1 (π₯) βΌ 1 β 1π₯
Therefore, the outer solution is
π¦ππ’π‘ (π₯) βΌ π¦0 + ππ¦1
136
3.3. HW3 CHAPTER 3. HWS
Or
π¦ (π₯) βΌ 1 + π οΏ½1 β 1π₯οΏ½ + π οΏ½π2οΏ½
Since the ODE is π¦β² β οΏ½1 + ππ₯2οΏ½ π¦2 + 2π¦ = 1, the approximation breaks down when π₯ < βπ or
π₯ < 110 . Because when π₯ < βπ, the
ππ₯2 will start to become large. The term π
π₯2 should remainsmall for the approximation to be accurate. The following are plots of the π¦0 and π¦0 + ππ¦1solutions (using π = 1
100 ) showing that with two terms the approximation has improved forthe outer layer, compared to the full solution of the original ODE obtained using CAS. Butthe outer solution breaks down near π₯ = 0.1 and smaller as can be seen in these plots. Hereis the solution of the original ODE obtained using CAS
In[46]:= eps =1
100;
ode = y'[x] == 1 +eps
x2 y[x]^2 - 2 y[x] + 1;
sol = y[x] /. First@DSolve[{ode, y[1] β©΅ 1}, y[x], x]
Out[48]=
10 x -12 - 5 6 - 12 x2 65 + 5 6 x
2 65
- 6 - 120 x - 50 6 x + 6 x2 65 - 120 x1+
2 65 + 50 6 x1+
2 65
In[53]:= Plot[sol, {x, 0, 1}, PlotRange β All, Frame β True, GridLines β Automatic, GridLinesStyle β LightGray,
FrameLabel β {{"y(x)", None}, {"x", "Exact solution to use to compare with"}}, BaseStyle β 14,
PlotStyle β Red, ImageSize β 400]
Out[53]=
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y(x)
Exact solution to use to compare with
In the following plot, the π¦0 and the π¦0 + ππ¦1 solutions are superimposed on same figure, toshow how the outer solution has improved when adding another term. But we also noticethat the outer solution π¦0+ππ¦1 only gives good approximation to the exact solution for aboutπ₯ > 0.1 and it breaks down quickly as π₯ becomes smaller.
137
3.3. HW3 CHAPTER 3. HWS
In[164]:= outer1 = 1;
outer2 = 1 + eps * (1 - 1 / x);
p2 = Plot[Callout[outer1, "y0", Scaled[0.1]], {x, 0, 1}, PlotRange β All, Frame β True,
GridLines β Automatic, GridLinesStyle β LightGray,
FrameLabel β {{"y(x)", None}, {"x", "comparing outer solution y0 with y0+ βy1"}},
BaseStyle β 14, PlotStyle β Red, ImageSize β 400];
p3 = Plot[Callout[outer2, "y0+ βy1", {Scaled[0.5], Below}], {x, 0.01, 1}, AxesOrigin β {0, 0}];
Show[p2, p3, PlotRange β {{0.01, .5}, {0, 1.2}}]
Out[168]=
y0
y0+ βy1
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x
y(x)
comparing outer solution y0 with y0+ βy1
3.3.3.3 Part c
Now we will obtain solution inside the boundary layer π¦ππ (π) = π¦ππ0 (π) + π (π). The first stepis to always introduce new inner variable. Since the boundary layer is on the right side, then
π =π₯ππ
And then to express the original ODE using this new variable. We also need to determineπ in the above expression. Since the original ODE is π¦β² β οΏ½1 + ππ₯β2οΏ½ π¦2 + 2π¦ = 1, then ππ¦
ππ₯ =ππ¦ππ
ππππ₯ =
ππ¦ππ(πβπ), then the ODE now becomes
ππ¦πππβπ β οΏ½1 +
ππππ2 οΏ½
π¦2 + 2π¦ = 1
ππ¦πππβπ β οΏ½1 +
π1β2π
π2 οΏ½ π¦2 + 2π¦ = 1
Where in the above π¦ β‘ π¦ (π). We see that we have οΏ½πβπ, ποΏ½1β2ποΏ½οΏ½ as the two biggest terms to
match. This means βπ = 1 β 2π or
π = 1
Hence the above ODE becomesππ¦πππβ1 β οΏ½1 +
πβ1
π2 οΏ½π¦2 + 2π¦ = 1
138
3.3. HW3 CHAPTER 3. HWS
We are now ready to replace π¦ (π) with ββπ=0 π
ππ¦π which gives
οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ πβ1 β οΏ½1 +πβ1
π2 οΏ½οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½
2+ 2 οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 1
οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ πβ1 β οΏ½1 +πβ1
π2 οΏ½οΏ½π¦20 + π οΏ½2π¦0π¦1οΏ½ +β―οΏ½ + 2 οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 1 (3)
Collecting terms with ποΏ½πβ1οΏ½ gives
π¦β²0 βΌ1π2π¦20
This is separable
οΏ½ππ¦0π¦20
βΌοΏ½1π2ππ
βπ¦β10 βΌ βπβ1 + πΆ1π¦0
βΌ1πβ πΆ
1π¦0
βΌ1 β πΆππ
π¦ππ0 βΌπ
1 β ππΆNow we use matching with π¦ππ’π‘ to find πΆ. We have found before that π¦ππ’π‘0 (π₯) βΌ 1 therefore
limπββ
π¦ππ0 (π) + π (π) = limπ₯β0
π¦ππ’π‘0 (π₯) + π (π)
limπββ
π1 β ππΆ
= 1 + π (π)
limπββ
(βπΆ) + π οΏ½πβ1οΏ½ + π (π) = 1 + π (π)
βπΆ = 1
Therefore
π¦ππ0 (π) βΌπ
1 + π(4)
Therefore,
π¦π’ππππππ = π¦ππ0 + π¦ππ’π‘0 β π¦πππ‘πβ
=
π¦πποΏ½π1+ π
+π¦ππ’π‘β1 β 1
Since π¦πππ‘πβ = 1 (this is what limπββ π¦ππ0 is). Writing everything in π₯, using π = π₯π the above
becomes
π¦uniform =π₯π
1 + π₯π
=π₯
π + π₯
139
3.3. HW3 CHAPTER 3. HWS
The following is a plot of the above, using π = 1100 to compare with the exact solution.,
In[189]:= y[x_, eps_] :=x
x + eps
p1 = Plot[{exactSol, y[x, 1 / 100]}, {x, 0.02, 1}, PlotRange β All,
Frame β True, GridLines β Automatic, GridLinesStyle β LightGray,
FrameLabel β
{{"y(x)", None},
{"x", "Exact solution vs. uniform solution found. 9.6 part(c)"}},
BaseStyle β 14, PlotStyle β {Red, Blue}, ImageSize β 400,
PlotLegends β {"exact", "uniform approximation"}]
Plot[y[x, 1 / 100], {x, 0, 1}, PlotRange β {{.05, 1}, Automatic}]
Out[190]=
0.0 0.2 0.4 0.6 0.8 1.00.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
x
y(x)
Exact solution vs. uniform solution found. 9.6 part(c)
exact
uniform approximation
3.3.3.4 Part (d)
Now we will obtain π¦ππ1 solution inside the boundary layer. Using (3) we found in part (c),reproduced here
οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ πβ1 β οΏ½1 +πβ1
π2 οΏ½οΏ½π¦20 + π οΏ½2π¦0π¦1οΏ½ +β―οΏ½ + 2 οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 1 (3)
But now collecting all terms of order of π (1), this results in
π¦β²1 β π¦20 β2π2π¦0π¦1 + 2π¦0 βΌ 1
Using π¦ππ0 found in part (c) into the above gives
π¦β²1 β2π οΏ½
11 + ποΏ½
π¦1 βΌ 1 β 2 οΏ½π
1 + ποΏ½+ οΏ½
π1 + ποΏ½
2
π¦β²1 β οΏ½2
π (1 + π)οΏ½π¦1 βΌ
1(π + 1)2
140
3.3. HW3 CHAPTER 3. HWS
This can be solved using integrating factor π = πβ« β2
π+π2ππ
using partial fractions gives π =exp (β2 ln π + 2 ln (1 + π)) or π = 1
π2(1 + π)2. Hence we obtain
πππ₯οΏ½ππ¦1οΏ½ βΌ π
1(π + 1)2
πππ₯ οΏ½
1π2(1 + π)2 π¦1οΏ½ βΌ
1π2
Integrating1π2(1 + π)2 π¦1 βΌοΏ½
1π2ππ
1π2(1 + π)2 π¦1 βΌ
β1π+ πΆ2
(1 + π)2 π¦1 βΌ βπ + π2πΆ2
π¦1 βΌβπ + π2πΆ2
(1 + π)2
Therefore, the inner solution becomes
π¦ππ (π) = π¦0 + ππ¦1
=π
1 + ππΆ1+ π
π2πΆ2 β π(1 + π)2
To find πΆ1, πΆ2 we do matching with with π¦ππ’π‘ that we found in part (a) which is π¦ππ’π‘ (π₯) βΌ1 + π οΏ½1 β 1
π₯οΏ½
limπββ
οΏ½π
1 + ππΆ1+ π
π2πΆ2 β π(1 + π)2
οΏ½ βΌ limπ₯β0
1 + π οΏ½1 β1π₯οΏ½
Doing long division π1+ππΆ1
= 1πΆ1β 1
ππΆ21+ 1
π2πΆ31+β― and π2πΆ2βπ
(1+π)2= πΆ2 β
2πΆ2+1π ββ―, hence the above
becomes
limπββ
οΏ½οΏ½1πΆ1
β1ππΆ2
1+
1π2πΆ3
1+β―οΏ½ + οΏ½ππΆ2 β π
2πΆ2 + 1π
+β―οΏ½οΏ½ βΌ limπ₯β0
1 + π οΏ½1 β1π₯οΏ½
1πΆ1
+ ππΆ2 βΌ limπ₯β0
1 + π οΏ½1 β1π₯οΏ½
Using π₯ = ππ on the RHS, the above simplifies to
1πΆ1
+ ππΆ2 βΌ limπββ
1 + π οΏ½1 β1πποΏ½
βΌ limπββ
1 + οΏ½π β1ποΏ½
βΌ 1 + π
141
3.3. HW3 CHAPTER 3. HWS
Therefore, πΆ1 = 1 and πΆ2 = 1. Hence the inner solution is
π¦ππ (π) = π¦0 + ππ¦1
=π
1 + π+ π
π2 β π(1 + π)2
Therefore
π¦uniform = π¦ππ + π¦ππ’π‘ β π¦πππ‘πβ
=
π¦πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½π1 + π
+ ππ2 β π(1 + π)2
+
π¦ππ’π‘οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½1 + π οΏ½1 β
1π₯οΏ½β (1 + π)
Writing everything in π₯, using π = π₯π the above becomes
π¦uniform =π₯π
1 + π₯π
+ ππ₯2
π2 βπ₯π
οΏ½1 + π₯ποΏ½2 + 1 + π οΏ½1 β
1π₯οΏ½β (1 + π)
=π₯
π + π₯+
π₯2 β π₯π
π οΏ½1 + π₯ποΏ½2 + 1 + π β
ππ₯β 1 β π
=π₯
π + π₯+
π₯2 β π₯π
π οΏ½1 + π₯ποΏ½2 β
ππ₯+ π οΏ½π2οΏ½
The following is a plot of the above, using π = 1100 to compare with the exact solution.
In[192]:= y[x_, eps_] :=x
x + eps+
x2 - x eps
eps 1 +x
eps2-eps
x
p1 = Plot[{exactSol, y[x, 1 / 100]}, {x, 0.02, 1}, PlotRange β All, Frame β True,
GridLines β Automatic, GridLinesStyle β LightGray,
FrameLabel β {{"y(x)", None}, {"x", "Exact solution vs. uniform solution found. 9.6 part(d)"}},
BaseStyle β 14, PlotStyle β {Red, Blue}, ImageSize β 400,
PlotLegends β {"exact", "uniform approximation"}]
Plot[y[x, 1 / 100], {x, 0, 1}, PlotRange β {{.05, 1}, Automatic}]
Out[193]=
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
x
y(x)
Exact solution vs. uniform solution found. 9.6 part(d)
exact
uniform approximation
142
3.3. HW3 CHAPTER 3. HWS
Let us check if π¦uniform (π₯) satisfies π¦ (1) = 1 or not.
π¦uniform (1) =1
π + 1+
1 β π
π οΏ½1 + 1ποΏ½2 β π + π οΏ½π
2οΏ½
=1 β π3 β 3π2 + π
(π + 1)2
Taking the limit π β 0 gives 1. Therefore π¦uniform (π₯) satisfies π¦ (1) = 1.
3.3.4 problem 9.9
problem Use boundary layer methods to find an approximate solution to initial value problem
ππ¦β²β² + π (π₯) π¦β² + π (π₯) π¦ = 0 (1)
π¦ (0) = 1π¦β² (0) = 1
And π > 0. Show that leading order uniform approximation satisfies π¦ (0) = 1 but not π¦β² (0) = 1for arbitrary π. Compare leading order uniform approximation with the exact solution tothe problem when π (π₯) , π (π₯) are constants.
Solution
Since π (π₯) > 0 then we expect the boundary layer to be at π₯ = 0. We start by finding π¦ππ’π‘ (π₯).
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Substituting this into (1) gives
π οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + π οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ + π οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0
Collecting terms with π (1) results in
ππ¦β²0 βΌ βππ¦0ππ¦0ππ₯
βΌ βπππ¦0
This is separable
οΏ½ππ¦0π¦0
βΌ βοΏ½π (π₯)π (π₯)
ππ₯
ln οΏ½π¦0οΏ½ βΌ βοΏ½π (π₯)π (π₯)
ππ₯ + πΆ
π¦0 βΌ πΆπββ«π₯1
π(π )π(π )ππ
Now we find π¦ππ. First we introduce interval variable
π =π₯ππ
143
3.3. HW3 CHAPTER 3. HWS
And transform the ODE. Since ππ¦ππ₯ =
ππ¦ππ
ππππ₯ then ππ¦
ππ₯ =ππ¦πππ
βπ. Hence πππ₯ β‘ π
βπ πππ
π2
ππ₯2=πππ₯
πππ₯
= οΏ½πβπππποΏ½ οΏ½
πβπππποΏ½
= πβ2ππ2
ππ2
Therefore π2π¦ππ₯2 = π
β2π π2π¦ππ2 and the ODE becomes
ππβ2ππ2π¦ππ2
+ π (π)ππ¦πππβπ + π (π) π¦ = 0
π1β2ππ¦β²β² + ππβππ¦β² + ππ¦ = 0
Balancing 1 β 2π with βπ shows that
π = 1
Hence
πβ1π¦β²β² + ππβ1π¦β² + ππ¦ = 0
Substituting π¦ππ = ββπ=0 π
ππ¦π = π¦0 + ππ¦1 + π2π¦2 +β― in the above gives
πβ1 οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + ππβ1 οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ + π οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0
Collecting terms with order ποΏ½πβ1οΏ½ gives
π¦β²β²0 βΌ βππ¦β²0Assuming π§ = π¦β²0 then the above becomes π§β² βΌ βππ§ or ππ§
ππ βΌ βππ§. This is separable. The
solution is ππ§π§ βΌ βπππ or
ln |π§| βΌ βοΏ½π
0π (π ) ππ + πΆ1
π§ βΌ πΆ1πββ«
π0
π(π )ππ
Henceππ¦0ππ
βΌ πΆ1πββ«
π0
π(π )ππ
ππ¦0 βΌ οΏ½πΆ1πββ«
π0
π(π )ππ οΏ½ ππ
Integrating again
π¦ππ0 βΌοΏ½π
0οΏ½πΆ1π
ββ«π0
π(π )ππ οΏ½ ππ + πΆ2
Applying initial conditions at π¦ (0) since this is where the π¦ππ exist. Using π¦ππ (0) = 1 then theabove becomes
1 = πΆ2
144
3.3. HW3 CHAPTER 3. HWS
Hence the solution becomes
π¦ππ0 βΌοΏ½π
0οΏ½πΆ1π
ββ«π0
π(π )ππ οΏ½ ππ + 1
To apply the second initial condition, which is π¦β² (0) = 1, we first take derivative of the abovew.r.t. π
π¦β²0 βΌ πΆ1πββ«
π0
π(π )ππ
Applying π¦β²0 (0) = 1 gives
1 = πΆ1
Hence
π¦ππ0 βΌ 1 +οΏ½π
0πββ«
π0
π(π )ππ ππ
Now to find constant of integration for π¦ππ’π‘ from earlier, we need to do matching.
limπββ
π¦ππ0 βΌ limπ₯β0
π¦ππ’π‘0
limπββ
1 +οΏ½π
0πββ«
π0
π(π )ππ ππ βΌ limπ₯β0
πΆπββ«π₯1
π(π )π(π )ππ
On the LHS the integral β«π
0πββ«
π0
π(π )ππ ππ since π > 0 and negative power on the exponential.So as π β β the integral value is zero. So we have now
1 βΌ limπ₯β0
πΆπββ«π₯1
π(π )π(π )ππ
Let limπ₯β0 πββ«
π₯1
π(π )π(π )ππ β πΈ, where πΈ is the value of the definite integral πΆπββ«
01
π(π )π(π )ππ . Another
constant, which if we know π (π₯) , π (π₯) we can evaluate. Hence the above gives the value of πΆas
πΆ =1πΈ
The uniform solution can now be written as
π¦uniform = π¦ππ + π¦ππ’π‘ β π¦match
= 1 +οΏ½π
0πββ«
π0
π(π )ππ ππ +1πΈπββ«
π₯1
π(π )π(π )ππ β 1
= οΏ½π
0πββ«
π0
π(π )ππ ππ +1πΈπββ«
π₯1
π(π )π(π )ππ (2)
Finally, we need to show that π¦uniform (0) = 1 but not π¦β²uniform (0) = 1. From (2), at π₯ = 0 whichalso means π = 0, since boundary layer at left side, equation (2) becomes
π¦uniform (0) = 0 +1πΈ
limπ₯β0
πββ«π₯1
π(π )π(π )ππ
But we said that limπ₯β0 πββ«
π₯1
π(π )π(π )ππ = πΈ, therefore
π¦uniform (0) = 1
145
3.3. HW3 CHAPTER 3. HWS
Now we take derivative of (2) w.r.t. π₯ and obtain
π¦β²uniform (π₯) =πππ₯
βββββοΏ½
π₯π
0πββ«
π0
π(π )ππ ππβββββ +
1πΈπππ₯ οΏ½
πββ«π₯1
π(π )π(π )ππ οΏ½
= πββ«π₯π
0π(π )ππ β
1πΈπ (π₯)π (π₯)
πββ«π₯1
π(π )π(π )ππ
And at π₯ = 0 the above becomes
π¦β²uniform (0) = 1 β1πΈπ (0)π (0)
The above is zero only if π (0) = 0 (since we know π (0) > 0). Therefore, we see thatπ¦β²uniform (0) β 1 for any arbitrary π (π₯). Which is what we are asked to show.
Will now solve the whole problem again, when π, π are constants.
ππ¦β²β² + ππ¦β² + ππ¦ = 0 (1A)
π¦ (0) = 1π¦β² (0) = 1
And π > 0. And compare leading order uniform approximation with the exact solution tothe problem when π (π₯) , π (π₯) are constants. Since π > 0 then boundary layer will occur atπ₯ = 0. We start by finding π¦ππ’π‘ (π₯).
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Substituting this into (1) gives
π οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + π οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ + π οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0
Collecting terms with π (1) results in
ππ¦β²0 βΌ βππ¦0ππ¦0ππ₯
βΌ βπππ¦0
This is separable
οΏ½ππ¦0π¦0
βΌ βππππ₯
ln οΏ½π¦0οΏ½ βΌ βπππ₯ + πΆ
π¦ππ’π‘0 βΌ πΆ1πβ πππ₯
Now we find π¦ππ. First we introduce internal variable π =π₯ππ and transform the ODE as we
did above. This results in
πβ1 οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + ππβ1 οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ + π οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0
Collecting terms with order ποΏ½πβ1οΏ½ gives
π¦β²β²0 βΌ βππ¦β²0
146
3.3. HW3 CHAPTER 3. HWS
Assuming π§ = π¦β²0 then the above becomes π§β² βΌ βππ§ or ππ§ππ βΌ βππ§. This is separable. The
solution is ππ§π§ βΌ βπππ or
ln |π§| βΌ βππ + πΈ1π§ βΌ πΈ1πβππ
Henceππ¦0ππ
βΌ πΈ1πβππ
ππ¦0 βΌ πΈ1πβππππ
Integrating again
π¦ππ0 βΌ πΈ1 οΏ½β1π οΏ½
πβππ + πΈ2
Applying initial conditions at π¦ (0) since this is where the π¦ππ exist. Using π¦ππ (0) = 1 then theabove becomes
1 = πΈ1 οΏ½β1π οΏ½
+ πΈ2
π (πΈ2 β 1) = πΈ1Hence the solution becomes
π¦ππ0 βΌ (1 β πΈ2) πβππ + πΈ2 (1B)
To apply the second initial condition, which is π¦β² (0) = 1, we first take derivative of the abovew.r.t. π
π¦β²0 βΌ βπ (1 β πΈ2) πβππ
Hence π¦β² (0) = 1 gives
1 = βπ (1 β πΈ2)1 = βπ + ππΈ2
πΈ2 =1 + ππ
And the solution π¦ππ in (1B) becomes
π¦ππ0 βΌ οΏ½1 β1 + ππ οΏ½ πβππ +
1 + ππ
βΌ οΏ½β1π οΏ½
πβππ +1 + ππ
βΌ(1 + π) β πβππ
π
147
3.3. HW3 CHAPTER 3. HWS
Now to find constant of integration for π¦ππ’π‘ (π₯) from earlier, we need to do matching.
limπββ
π¦ππ0 βΌ limπ₯β0
π¦ππ’π‘0
limπββ
(1 + π) β πβππ
πβΌ lim
π₯β0πΆ1π
β πππ₯
1 + ππ
βΌ πΆ1
Hence now the uniform solution can be written as
π¦uniform (π₯) βΌ π¦ππ + π¦ππ’π‘ β π¦match
βΌ
π¦πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(1 + π) β πβπ
π₯π
π+
π¦ππ’π‘οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½1 + ππ
πβπππ₯ β
1 + ππ
βΌ(1 + π)π
βπβπ
π₯π
π+1 + ππ
πβπππ₯ β
1 + ππ
βΌ βπβπ
π₯π
π+1 + ππ
πβπππ₯
βΌ1ποΏ½(1 + π) πβ
πππ₯ β πβπ
π₯π οΏ½ (2A)
Now we compare the above, which is the leading order uniform approximation, to the exactsolution. Since now π, π are constants, then the exact solution is
π¦ππ₯πππ‘ (π₯) = π΄ππ1π₯ + π΅ππ2π₯ (3)
Where π1,2 are roots of characteristic equation of ππ¦β²β² + ππ¦β² + ππ¦ = 0. These are π = βπ2π Β±
12πβπ
2 β 4ππ. Hence
π1 =βπ2+12βπ2 β 4ππ
π2 =βπ2β12βπ2 β 4ππ
Applying initial conditions to (3). π¦ (0) = 1 gives
1 = π΄ + π΅π΅ = 1 β π΄
And solution becomes π¦ππ₯πππ‘ (π₯) = π΄ππ1π₯ + (1 β π΄) ππ2π₯. Taking derivatives gives
π¦β²ππ₯πππ‘ (π₯) = π΄π1ππ1π₯ + (1 β π΄) π2ππ2π₯
Using π¦β² (0) = 1 gives
1 = π΄π1 + (1 β π΄) π21 = π΄ (π1 β π2) + π2
π΄ =1 β π2π1 β π2
148
3.3. HW3 CHAPTER 3. HWS
Therefore, π΅ = 1 β 1βπ2π1βπ2
and the exact solution becomes
π¦ππ₯πππ‘ (π₯) =1 β π2π1 β π2
ππ1π₯ + οΏ½1 β1 β π2π1 β π2
οΏ½ ππ2π₯
=1 β π2π1 β π2
ππ1π₯ + οΏ½(π1 β π2) β (1 β π2)
π1 β π2οΏ½ ππ2π₯
=1 β π2π1 β π2
ππ1π₯ + οΏ½π1 β 1π1 β π2
οΏ½ ππ2π₯ (4)
While the uniform solution above was found to be 1ποΏ½(1 + π) πβ
πππ₯ β πβπ
π₯π οΏ½. Here is a plot
of the exact solution above, for π = {1/10, 1/50, 1/100} , and for some values for π, π such asπ = 1, π = 10 in order to compare with the uniform solution. Note that the uniform solutionis π (π). As π becomes smaller, the leading order uniform solution will better approximatethe exact solution. At π = 0.01 the uniform approximation gives very good approximation.This is using only leading term approximation.
In[134]:= ClearAll[x, y]
eps = 1/ 10; a = 1; b = 10;
mySol = 1/ a ((1 + a)* Exp[-b/ a x] - Exp[-a x/ eps]);
sol = y[x] /. First@DSolve[{eps y''[x] + a y'[x] + b y[x] β©΅ 0, y[0] β©΅ 1, y'[0] β©΅ 1}, y[x], x]
Out[137]=15β -5 x
5 Cos5 3 x + 2 3 Sin5 3 x
In[125]:= Plot[{sol, mySol}, {x, 0, 1}, PlotRange β All, PlotStyle β {Blue, Red}, PlotLegends β {"Exact", "approximation"},
Frame β True, FrameLabel β {{"y(x)", None}, {"x", Row[{"Exact vs. approximation for Ο΅ =", eps}]}},
BaseStyle β 14, GridLines β Automatic, GridLinesStyle β LightGray]
Out[125]=
0.0 0.2 0.4 0.6 0.8 1.0-0.2
0.0
0.2
0.4
0.6
0.8
1.0
x
y(x)
Exact vs. approximation for Ο΅ = 1
10
Exact
approximation
149
3.3. HW3 CHAPTER 3. HWS
In[126]:= ClearAll[x, y]
eps = 1/ 50; a = 1; b = 10;
sol = y[x] /. First@DSolve[{eps y''[x] + a y'[x] + b y[x] β©΅ 0, y[0] β©΅ 1, y'[0] β©΅ 1}, y[x], x]
Plot[{sol, mySol}, {x, 0, 1}, PlotRange β All, PlotStyle β {Blue, Red}, PlotLegends β {"Exact", "approximation"},
Frame β True, FrameLabel β {{"y(x)", None}, {"x", Row[{"Exact vs. approximation for Ο΅ =", eps}]}},
BaseStyle β 14, GridLines β Automatic, GridLinesStyle β LightGray]
Out[128]=150
25 β -25-5 5 x- 26 5 β
-25-5 5 x+ 25 β -25+5 5 x
+ 26 5 β -25+5 5 x
Out[129]=
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y(x)
Exact vs. approximation for Ο΅ = 1
50
Exact
approximation
In[130]:= ClearAll[x, y]
eps = 1/ 100; a = 1; b = 10;
sol = y[x] /. First@DSolve[{eps y''[x] + a y'[x] + b y[x] β©΅ 0, y[0] β©΅ 1, y'[0] β©΅ 1}, y[x], x]
Plot[{sol, mySol}, {x, 0, 1}, PlotRange β All, PlotStyle β {Blue, Red}, PlotLegends β {"Exact", "approximation"},
Frame β True, FrameLabel β {{"y(x)", None}, {"x", Row[{"Exact vs. approximation for Ο΅ =", eps}]}},
BaseStyle β 14, GridLines β Automatic, GridLinesStyle β LightGray]
Out[132]=1100
50 β -50-10 15 x- 17 15 β
-50-10 15 x+ 50 β -50+10 15 x
+ 17 15 β -50+10 15 x
Out[133]=
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y(x)
Exact vs. approximation for Ο΅ = 1
100
Exact
approximation
3.3.5 problem 9.15(b)
Problem Find first order uniform approximation valid as π β 0+ for 0 β€ π₯ β€ 1
ππ¦β²β² + οΏ½π₯2 + 1οΏ½ π¦β² β π₯3π¦ = 0 (1)
π¦ (0) = 1π¦ (1) = 1
Solution
Since π (π₯) = οΏ½π₯2 + 1οΏ½ is positive for 0 β€ π₯ β€ 1, therefore we expect the boundary layer to be
150
3.3. HW3 CHAPTER 3. HWS
on the left side at π₯ = 0. Assuming this is the case for now (if it is not, then we expect notto be able to do the matching). We start by finding π¦ππ’π‘ (π₯).
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Substituting this into (1) gives
π οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + οΏ½π₯2 + 1οΏ½ οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ β π₯3 οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0 (2)
Collecting terms with π (1) results inοΏ½π₯2 + 1οΏ½ π¦β²0 βΌ π₯3π¦0
ππ¦0ππ₯
βΌπ₯3
οΏ½π₯2 + 1οΏ½π¦0
This is separable.
οΏ½ππ¦0π¦0
βΌοΏ½π₯3
οΏ½π₯2 + 1οΏ½ππ₯
ln οΏ½π¦0οΏ½ βΌοΏ½π₯ βπ₯
1 + π₯2ππ₯
βΌπ₯2
2β12
ln οΏ½1 + π₯2οΏ½ + πΆ
Hence
π¦0 βΌ ππ₯22 β 1
2 lnοΏ½1+π₯2οΏ½+πΆ
βΌπΆπ
π₯22
β1 + π₯2
Applying π¦ππ’π‘0 (1) = 1 to the above (since this is where the outer solution is), we solve for πΆ
1 βΌπΆπ
12
β2
πΆ βΌ β2πβ12
Therefore
π¦ππ’π‘0 βΌ β2πβ12 π
π₯22
β1 + π₯2
βΌοΏ½2π
ππ₯22
β1 + π₯2
Now we need to find π¦ππ’π‘1 . From (2), but now collecting terms in π (π) gives
π¦β²β²0 + οΏ½π₯2 + 1οΏ½ π¦β²1 β½ π₯3π¦1 (3)
151
3.3. HW3 CHAPTER 3. HWS
In the above π¦β²β²0 is known.
π¦β²0 (π₯) = οΏ½2ππππ₯
βββββββββ
ππ₯22
β1 + π₯2
βββββββββ
=οΏ½2π
π₯3ππ₯22
οΏ½1 + π₯2οΏ½32
And
π¦β²β²0 (π₯) = οΏ½2ππ₯2π
π₯22 οΏ½π₯4 + π₯2 + 3οΏ½
οΏ½1 + π₯2οΏ½52
Hence (3) becomes
οΏ½π₯2 + 1οΏ½ π¦β²1 β½ π₯3π¦1 β π¦β²β²0
οΏ½π₯2 + 1οΏ½ π¦β²1 β½ π₯3π¦1 βοΏ½2ππ₯2π
π₯22 οΏ½π₯4 + π₯2 + 3οΏ½
οΏ½1 + π₯2οΏ½52
π¦β²1 βπ₯3
οΏ½π₯2 + 1οΏ½π¦1 β½ βοΏ½
2ππ₯2π
π₯22 οΏ½π₯4 + π₯2 + 3οΏ½
οΏ½1 + π₯2οΏ½72
Integrating factor is π = πββ« π₯3
οΏ½π₯2+1οΏ½ππ₯= πβ
π₯22 + 1
2 lnοΏ½1+π₯2οΏ½ = οΏ½1 + π₯2οΏ½12 π
βπ₯22 , hence the above becomes
πππ₯ οΏ½
οΏ½1 + π₯2οΏ½12 π
βπ₯22 π¦1οΏ½ β½ βοΏ½
2ποΏ½1 + π₯2οΏ½
12 π
βπ₯22π₯2π
π₯22 οΏ½π₯4 + π₯2 + 3οΏ½
οΏ½1 + π₯2οΏ½72
β½ βοΏ½2ππ₯2 οΏ½π₯4 + π₯2 + 3οΏ½
οΏ½1 + π₯2οΏ½3
Integrating gives (with help from CAS)
οΏ½1 + π₯2οΏ½12 π
βπ₯22 π¦1 (π₯) β½ βοΏ½
2π οΏ½
π₯2 οΏ½π₯4 + π₯2 + 3οΏ½
οΏ½1 + π₯2οΏ½3 ππ₯
β½ βοΏ½2π οΏ½
1 β3
οΏ½1 + π₯2οΏ½3 +
4
οΏ½1 + π₯2οΏ½2 β
21 + π₯2
ππ₯
β½ βοΏ½2π
ββββββββπ₯ β
3π₯
4 οΏ½1 + π₯2οΏ½2 +
7π₯8 οΏ½1 + π₯2οΏ½
β 9arctan (π₯)
8
ββββββββ + πΆ1
152
3.3. HW3 CHAPTER 3. HWS
Hence
π¦ππ’π‘1 (π₯) β½ βοΏ½2π
ππ₯22
οΏ½1 + π₯2οΏ½12
ββββββββπ₯ β
3π₯
4 οΏ½1 + π₯2οΏ½2 +
7π₯8 οΏ½1 + π₯2οΏ½
β 9arctan (π₯)
8
ββββββββ + πΆ1
ππ₯22
οΏ½1 + π₯2οΏ½12
Now we find πΆ1 from boundary conditions π¦1 (1) = 0. (notice the BC now is π¦1 (1) = 0 andnot π¦1 (1) = 1, since we used π¦1 (1) = 1 already).
οΏ½2π
π12
(1 + 1)12οΏ½1 β
34 (1 + 1)2
+7
8 (1 + 1)β 9
arctan (1)8 οΏ½ = πΆ1
π12
(1 + 1)12
οΏ½2ππ12
β2οΏ½1 β
316+716β98
arctan (1)οΏ½ = πΆ1π12
β2Simplifying
1 β316+716β98
arctan (1) = πΆ1π12
β2
πΆ1 = οΏ½2π οΏ½54β98
arctan (1)οΏ½
πΆ1 = οΏ½2π οΏ½54β932ποΏ½
= 0.31431
Hence
π¦ππ’π‘1 βΌ βοΏ½2π
ππ₯22
οΏ½οΏ½1 + π₯2οΏ½
ββββββββπ₯ β
3π₯
4 οΏ½1 + π₯2οΏ½2 +
7π₯8 οΏ½1 + π₯2οΏ½
β98
arctan (π₯)
ββββββββ +οΏ½
2π οΏ½54β932ποΏ½
ππ₯22
οΏ½οΏ½1 + π₯2οΏ½
βΌοΏ½2π
ππ₯22
οΏ½οΏ½1 + π₯2οΏ½
ββββββββοΏ½54β932ποΏ½ β
ββββββββπ₯ β
3π₯
4 οΏ½1 + π₯2οΏ½2 +
7π₯8 οΏ½1 + π₯2οΏ½
β98
arctan (π₯)
ββββββββ
ββββββββ
βΌοΏ½2π
ππ₯22
οΏ½οΏ½1 + π₯2οΏ½
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ
153
3.3. HW3 CHAPTER 3. HWS
Hence
π¦ππ’π‘ (π₯) βΌ π¦ππ’π‘0 + ππ¦ππ’π‘1
βΌοΏ½2π
ππ₯22
β1 + π₯2+ π
οΏ½2π
ππ₯22
οΏ½οΏ½1 + π₯2οΏ½
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ + π οΏ½π
2οΏ½
βΌοΏ½2π
ππ₯22
β1 + π₯2
ββββββββ1 + π
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ
ββββββββ + π οΏ½π
2οΏ½
(3A)
Now that we found π¦ππ’π‘ (π₯), we need to find π¦ππ (π₯) and then do the matching and the finduniform approximation. Since the boundary layer at π₯ = 0, we introduce inner variable π = π₯
ππand then express the original ODE using this new variable. We also need to determine π inthe above expression. Since ππ¦
ππ₯ =ππ¦ππ
ππππ₯ then ππ¦
ππ₯ =ππ¦πππ
βπ. Hence πππ₯ β‘ π
βπ πππ
π2
ππ₯2=πππ₯
πππ₯
= οΏ½πβπππποΏ½ οΏ½
πβπππποΏ½
= πβ2ππ2
ππ2
Therefore π2π¦ππ₯2 = π
β2π π2π¦ππ2 and the ODE ππ¦β²β² + οΏ½π₯2 + 1οΏ½ π¦β² β π₯3π¦ = 0 now becomes
ππβ2ππ2π¦ππ2
+ οΏ½(πππ)2 + 1οΏ½ πβπππ¦ππ
β (πππ)3 π¦ = 0
π1β2ππ2π¦ππ2
+ οΏ½π2ππ + πβποΏ½ππ¦ππ
β π3π3ππ¦ = 0
The largest terms are οΏ½π1β2π, πβποΏ½, therefore matching them gives π = 1. The ODE nowbecomes
πβ1π2π¦ππ2
+ οΏ½π2π + πβ1οΏ½ππ¦ππ
β π3π3π¦ = 0 (4)
Assuming that
π¦ππ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
And substituting the above into (4) gives
πβ1 οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + οΏ½π2π + πβ1οΏ½ οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ β π3π3 οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0 (4A)
Collecting terms in ποΏ½πβ1οΏ½ gives
π¦β²β²0 β½ βπ¦β²0
154
3.3. HW3 CHAPTER 3. HWS
Letting π§ = π¦β²0, the above becomesππ§ππ
β½ βπ§
ππ§π§β½ βππ
ln |π§| β½ βπ + πΆ1
π§ β½ πΆ1πβπ
Henceππ¦0ππ
β½ πΆ1πβπ
π¦0 β½ πΆ1οΏ½πβπππ + πΆ2
β½ βπΆ1πβπ + πΆ2 (5)
Applying boundary conditions π¦ππ0 (0) = 1 gives
1 = βπΆ1 + πΆ2
πΆ2 = 1 + πΆ1
And (5) becomes
π¦ππ0 (π) β½ βπΆ1πβπ + (1 + πΆ1)
β½ 1 + πΆ1 οΏ½1 β πβποΏ½ (6)
We now find π¦ππ1 . Going back to (4) and collecting terms in π (1) gives the ODE
π¦β²β²1 β½ π¦β²1This is the same ODE we solved above. But it will have diοΏ½erent B.C. Hence
π¦ππ1 β½ βπΆ3πβπ + πΆ4
Applying boundary conditions π¦ππ1 (0) = 0 gives
0 = βπΆ3 + πΆ4
πΆ3 = πΆ4
Therefore
π¦ππ1 β½ βπΆ3πβπ + πΆ3
β½ πΆ3 οΏ½1 β πβποΏ½
Now we have the leading order π¦ππ
π¦ππ (π) = π¦ππ0 + ππ¦ππ1= 1 + πΆ1 οΏ½1 β πβποΏ½ + ππΆ3 οΏ½1 β πβποΏ½ + π οΏ½π2οΏ½ (7)
155
3.3. HW3 CHAPTER 3. HWS
Now we are ready to do the matching between (7) and (3A)
limπββ
1 + πΆ1 οΏ½1 β πβποΏ½ + ππΆ3 οΏ½1 β πβποΏ½ βΌ
limπ₯β0οΏ½
2π
ππ₯22
β1 + π₯2
ββββββββ1 + π
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ
ββββββββ
Or
1+πΆ1+ππΆ3 βΌοΏ½2π
limπ₯β0
ππ₯22
β1 + π₯2+οΏ½2ππ limπ₯β0
ππ₯22
β1 + π₯2
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ
But limπ₯β0ππ₯22
β1+π₯2β 1, limπ₯β0
3π₯
4οΏ½1+π₯2οΏ½2 β 0 limπ₯β0
7π₯8οΏ½1+π₯2οΏ½
β 0 therefore the above becomes
1 + πΆ1 + ππΆ3 βΌοΏ½2π+οΏ½2ππ οΏ½54β932ποΏ½
Hence
1 + πΆ1 = οΏ½2π
πΆ1 = οΏ½2πβ 1
πΆ3 = οΏ½2π οΏ½54β932ποΏ½
This means that
π¦ππ (π) βΌ 1 + πΆ1 οΏ½1 β πβποΏ½ + ππΆ3 οΏ½1 β πβποΏ½
βΌ 1 +ββββββοΏ½
2πβ 1ββββββ οΏ½1 β π
βποΏ½ + ποΏ½2π οΏ½54β932ποΏ½ οΏ½1 β πβποΏ½
Therefore
π¦uniform(π₯) βΌ π¦ππ (π) + π¦ππ’π‘ (π₯) β π¦πππ‘πβ
βΌ 1 +ββββββοΏ½
2πβ 1ββββββ οΏ½1 β π
βποΏ½ + ποΏ½2π οΏ½54β932ποΏ½ οΏ½1 β πβποΏ½
+οΏ½2π
ππ₯22
β1 + π₯2
ββββββββ1 + π
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ
ββββββββ
βββββββοΏ½
2π+οΏ½2ππ οΏ½54β932ποΏ½
ββββββ
156
3.3. HW3 CHAPTER 3. HWS
Or (replacing π by π₯π and simplifying)
π¦uniform(π₯) βΌ 1 +ββββββοΏ½
2πβ 1ββββββ οΏ½1 β π
βποΏ½ β πβπποΏ½2π οΏ½54β932ποΏ½
+οΏ½2π
ππ₯22
β1 + π₯2
ββββββββ1 + π
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ
ββββββββ
βοΏ½2π
Or
π¦uniform(π₯) βΌ βοΏ½2ππβπ + πβπ β πβππ
οΏ½2π οΏ½54β932ποΏ½
+οΏ½2π
ππ₯22
β1 + π₯2
ββββββββ1 + π
ββββββββ54β932π β π₯ +
3π₯
4 οΏ½1 + π₯2οΏ½2 β
7π₯8 οΏ½1 + π₯2οΏ½
+98
arctan (π₯)
ββββββββ
ββββββββ
To check validity of the above solution, the approximate solution is plotted against thenumerical solution for diοΏ½erent values of π = {0.1, 0.05, 0.01}. This shows very good agreementwith the numerical solution. At π = 0.01 the solutions are almost the same.
ClearAll[x, y, Ο΅];
Ο΅ = 0.01;
r =2
Exp[1];
ode = Ο΅ y''[x] + x2 + 1 y'[x] - x3 y[x] β©΅ 0;
sol = First@NDSolve[{ode, y[0] β©΅ 1, y[1] β©΅ 1}, y, {x, 0, 1}];
p1 = Plot[Evaluate[y[x] /. sol], {x, 0, 1}];
mysol[x_, Ο΅_] := -r Exp-x
Ο΅ + Exp
-x
Ο΅ - Ο΅ r
5
4-
9
32Ο + r
Exp x2
2
1 + x2
1 + Ο΅5
4-
9
32Ο - x +
3 x
4 1 + x2
2-
7 x
8 1 + x2
+9
8ArcTan[x] ;
p2 = Plot[{Evaluate[y[x] /. sol], mysol[x, Ο΅]}, {x, 0, 1}, Frame β True, PlotStyle β {Red, Blue},
FrameLabel β {{"y(x)", None}, {"x", Row[{"Comparing numerical and apprxoimation, using Ο΅ = ", N@Ο΅}]}}, GridLines β Automatic,
GridLinesStyle β LightGray, PlotLegends β {"Approximation", "Numerical"}, BaseStyle β 14, ImageSize β 400]
157
3.3. HW3 CHAPTER 3. HWS
Out[145]=
0.0 0.2 0.4 0.6 0.8 1.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
x
y(x)
Comparing numerical and apprxoimation, using Ο΅ = 0.1
Approximation
Numerical
Out[153]=
0.0 0.2 0.4 0.6 0.8 1.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
x
y(x)
Comparing numerical and apprxoimation, using Ο΅ = 0.05
Approximation
Numerical
158
3.3. HW3 CHAPTER 3. HWS
Out[161]=
0.0 0.2 0.4 0.6 0.8 1.0
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
x
y(x)
Comparing numerical and apprxoimation, using Ο΅ = 0.01
Approximation
Numerical
3.3.6 problem 9.19
Problem Find lowest order uniform approximation to boundary value problem
ππ¦β²β² + (sin π₯) π¦β² + π¦ sin (2π₯) = 0π¦ (0) = ππ¦ (π) = 0
Solution
We expect a boundary layer at left end at π₯ = 0. Therefore, we need to find π¦ππ (π) , π¦ππ’π‘ (π₯) ,where π is an inner variable defined by π = π₯
ππ .
159
3.3. HW3 CHAPTER 3. HWS
x0 Ο
ΞΆ Ξ·
leftboundarylayer
outer region
matching at these locations
rightboundarylayer
Finding π¦ππ (π)
At π₯ = 0, we introduce inner variable π = π₯ππ and then express the original ODE using this
new variable. We also need to determine π in the above expression. Since ππ¦ππ₯ =
ππ¦ππ
ππππ₯ then
ππ¦ππ₯ =
ππ¦πππ
βπ. Hence πππ₯ β‘ π
βπ πππ
π2
ππ₯2=πππ₯
πππ₯
= οΏ½πβπππποΏ½ οΏ½
πβπππποΏ½
= πβ2ππ2
ππ2
Therefore π2π¦ππ₯2 = π
β2π π2π¦ππ2 and the ODE ππ¦β²β² + (sin π₯) π¦β² + π¦ sin (2π₯) = 0 now becomes
ππβ2ππ2π¦ππ2
+ (sin (πππ)) πβπ ππ¦ππ
+ sin (2πππ) π¦ = 0
π1β2ππ2π¦ππ2
+ (sin (πππ)) πβπ ππ¦ππ
+ sin (2πππ) π¦ = 0
160
3.3. HW3 CHAPTER 3. HWS
Expanding the sin terms in the above, in Taylor series around zero, sin (π₯) = π₯ β π₯3
3! +β― gives
π1β2ππ2π¦ππ2
+βββββπππ β
(πππ)3
3!+β―
βββββ πβπ
ππ¦ππ
+βββββ2πππ β
(2πππ)3
3!+β―
βββββ π¦ = 0
π1β2ππ2π¦ππ2
+ οΏ½π βπ3π2π
3!+β―οΏ½
ππ¦ππ
+βββββ2πππ β
(2πππ)3
3!+β―
βββββ π¦ = 0
Then the largest terms are οΏ½π1β2π, 1οΏ½, therefore 1 β 2π = 0 or
π = 12
The ODE now becomes
π¦β²β² + οΏ½π βπ3π3!
+β―οΏ½ π¦β² +
ββββββββ2πβπ β
οΏ½2πβποΏ½3
3!+β―
ββββββββ π¦ = 0 (1)
Assuming that
π¦ππππ‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Then (1) becomes
οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + οΏ½π βπ3π3!
+β―οΏ½ οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ +
ββββββββ2πβπ β
οΏ½2πβποΏ½3
3!+β―
ββββββββ οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0
Collecting terms in π (1) gives the balance
π¦β²β²0 (π) βΌ βππ¦β²0 (π)π¦0 (0) = π
Assuming π§ = π¦β²0, then
π§β² βΌ βππ§ππ§π§
βΌ βπ
ln |π§| βΌ βπ2
2+ πΆ1
π§ βΌ πΆ1πβπ22
Therefore π¦β²0 βΌ πΆ1πβπ22 . Hence
π¦0 (π) βΌ πΆ1οΏ½π
0πβπ 22 ππ + πΆ2
With boundary conditions π¦ (0) = π. Hence
π = πΆ2
161
3.3. HW3 CHAPTER 3. HWS
And the solution becomes
π¦ππ0 (π) βΌ πΆ1οΏ½π
0πβπ 22 ππ + π (2)
Now we need to find π¦ππ’π‘ (π₯). Assuming that
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Then ππ¦β²β² + (sin π₯) π¦β² + π¦ sin (2π₯) = 0 becomes
π οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + sin (π₯) οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + sin (2π₯) οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0Collecting terms in π (1) gives the balance
sin (π₯) π¦β²0 (π₯) βΌ β sin (2π₯) π¦0 (π₯)ππ¦0π¦0
βΌ βsin (2π₯)sin (π₯) ππ₯
ln οΏ½π¦0οΏ½ βΌ βοΏ½sin (2π₯)sin (π₯) ππ₯
βΌ βοΏ½2 sin π₯ cos π₯
sin (π₯) ππ₯
βΌ βοΏ½2 cos π₯ππ₯
βΌ β2 sin π₯ + πΆ5
Hence
π¦ππ’π‘0 (π₯) βΌ π΄πβ2 sin π₯
π¦0 (π) = 0
Therefore π΄ = 0 and π¦ππ’π‘0 (π₯) = 0.Now that we found all solutions, we can do the matching.The matching on the left side gives
limπββ
π¦ππ (π) = limπ₯β0
π¦ππ’π‘ (π₯)
limπββ
πΆ1οΏ½π
0πβπ 22 ππ + π = lim
π₯β0πΆ5πβ2 sin π₯
limπββ
πΆ1οΏ½π
0πβπ 22 ππ + π = 0 (3)
But
οΏ½π
0πβπ 22 ππ =
οΏ½π2
erf οΏ½π
β2οΏ½
162
3.3. HW3 CHAPTER 3. HWS
And limπββ erf οΏ½ π
β2οΏ½ = 1, hence (3) becomes
πΆ1οΏ½π2+ π = 0
πΆ1 = βποΏ½2π
= ββ2π (4)
Therefore from (2)
π¦ππ0 (π) βΌ ββ2ποΏ½π
0πβπ 22 ππ + π (5)
Near π₯ = π, using π = πβπ₯ππ . Expansion π¦
ππ οΏ½ποΏ½ βΌ π¦0 οΏ½ποΏ½ + ππ¦1 οΏ½ποΏ½ + π οΏ½π2οΏ½ gives π =12 . Hence
π (1) terms gives
π¦β²β²0 οΏ½ποΏ½ βΌ ππ¦β²0 οΏ½ποΏ½
π¦ππ0 (0) = 0
π¦ππ0 οΏ½ποΏ½ βΌ π·οΏ½π
0πβπ 22 ππ
And matching on the right side gives
limπββ
π¦ππ οΏ½ποΏ½ = limπ₯βπ
π¦ππ’π‘ (π₯)
limπββ
π·οΏ½π
0πβπ 22 ππ = 0
π· = 0 (6)
Therefore the solution is
π¦ (π₯) βΌ π¦ππ (π) + π¦ππ οΏ½ποΏ½ + π¦ππ’π‘ (π₯) β π¦πππ‘πβ
βΌ ββ2ποΏ½π
0πβπ 22 ππ + π + 0
βΌ ββ2ποΏ½π2
erf οΏ½π
β2οΏ½ + π
βΌ ββ2ποΏ½π2
erf οΏ½π₯
β2ποΏ½ + π
βΌ π β π erf οΏ½π₯
β2ποΏ½ (7)
The following plot compares exact solution with (7) for π = 0.1, 0.05. We see from theseresults, that as π decreased, the approximation solution improved.
163
3.3. HW3 CHAPTER 3. HWS
In[37]:= ClearAll[x, Ο΅, y]
mySol[x_, Ο΅_] := Pi - Pi Erfx
2 Ο΅
;
Ο΅ = 0.1;
ode = Ο΅ y''[x] + Sin[x] y'[x] + y[x] Sin[2 x] β©΅ 0;
sol = NDSolve[{ode, y[0] β©΅ Ο, y'[Ο] β©΅ 0}, y, {x, 0, Ο}];
Plot[{mySol[x, Ο΅], Evaluate[y[x] /. sol]}, {x, 0, Pi}, Frame β True,
FrameLabel β {{"y(x)", None}, {"x", Row[{"Numerical vs. approximation for Ο΅=", Ο΅}]}}, GridLines β Automatic,
GridLinesStyle β LightGray, BaseStyle β 16, ImageSize β 500, PlotLegends β {"Approximation", "Numerical"},
PlotStyle β {Red, Blue}, PlotRange β All]
Out[42]=
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
1
2
3
4
x
y(x)
Numerical vs. approximation for Ο΅=0.1
Approximation
Numerical
Out[54]=
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
x
y(x)
Numerical vs. approximation for Ο΅=0.05
Approximation
Numerical
164
3.3. HW3 CHAPTER 3. HWS
3.3.7 key solution of selected problems
3.3.7.1 section 9 problem 9
Problem 9.9 asks us to use boundary layer theory to find the leading order solution to the initialvalue problem Ξ΅yβ²β²(x) + ayβ²(x) + by(x) = 0 with y(0) = yβ²(0) = 1 and a > 0. Then we are to compareto the exact solution. The problem is ambiguous as to whether a and b are functions or constants.For clarityβs sake, we assume a and b are constants, but all the following work can be generalized fornonconstant a and b as well.
Since a > 0, the boundary layer occurs at x = 0, where the initial conditions are specified. We setx = Ρξ,1 and then in the inner region,
1
Ξ΅yβ²β²in(ΞΎ) +
a
Ξ΅yβ²in(ΞΎ) + byin(ΞΎ) = 0. (1)
Thus, to leading order, yβ²β²in(ΞΎ) βΌ βayβ²in(ΞΎ), which has solution yin(ΞΎ) = C0 + C1eβaΞΎ. Solving for C0 and
C1, we see that y(0) = 1 =β C0 + C1 = 1, and
yβ²(x)
β£β£β£β£x=0
= 1 =β yβ²(ΞΎ)
β£β£β£β£ΞΎ=0
= Ξ΅ =β βaC1 = Ξ΅ =β C1 = βΞ΅/a = O(Ξ΅).
This means that C1eβaΞΎ = O(Ξ΅) shouldnβt appear at this order in the expansion, and C1 = 0. We
should throw this information out because we have already thrown out information at O(Ξ΅) in solvingthe equation, and we have no guarantee that the O(Ξ΅) value for C1 is actually correct to O(Ξ΅).
So there is no boundary layer at leading order! The inner solution yin(x) = C0 = 1 does not changerapidly, and it will cancel when we match, just leaving the outer solution. This is okay and happensoccasionally when you get lucky.
In the outer region, we have yout βΌ β bayout, so yout(x) = Ceβbx/a + O(Ξ΅). Matching to the inner
solution, C = 1, so yuniform(x) = eβbx/a +O(Ξ΅). We note that yβ²uniform(0) = β ba6= 1 in general.
Although the problem does not ask for it, we can also go to the next order in our asymptoticexpansion. And even though no boundary layer appeared at leading order, one will appear at O(Ξ΅).
Going back to the inner region, let yin(ΞΎ) = Y0(ΞΎ) + Ξ΅Y1(ΞΎ) + O(Ξ΅2). We already computed thatY0(ΞΎ) = 1. Now looking at (1) at O(1), Y β²β²1 (ΞΎ) +aY β²1(ΞΎ) + bY0(ΞΎ) = 0 =β Y β²β²1 (ΞΎ) +aY β²1(ΞΎ) = βb. Solving,Y1(ΞΎ) = C2 + C3e
βaΞΎ β baΞΎ. We have initial conditions Y1(0) = 0, but since yβ²(0) = 1 was not satisfied,
we note that Y β²1(0) = 1. Thus, C2 + C3 = 0 and βaC3 β ba
= 1, so
C3 = β1
aβ b
a2and C2 = βC3 =
1
a+
b
a2.
Therefore,
yin(ΞΎ) = 1 + Ξ΅
((1
a+
b
a2
)(1β eβaΞΎ
)β b
aΞΎ
)+O(Ξ΅2),
so
yin(x) = 1 + Ξ΅
((1
a+
b
a2
)(1β eβax/Ξ΅
)β b
a
x
Ξ΅
)+O(Ξ΅2).
Turning to the outer solution, let yout(x) = y0(x) + Ξ΅y1(x) + O(Ξ΅2). We already found that y0(x) =eβbx/a. At order Ξ΅, the outer equation reads:
yβ²β²0(x) + ayβ²1(x) + by1(x) = 0 =β yβ²1(x) +b
ay1(x) = β1
aeβbx/a.
1We know that we can use Ξ΅ instead of Ξ΅p for some unknown constant p because a is positive near zero, and p 6= 1 onlyoccurs when a(x)β 0 at the boundary layer.
1
165
3.3. HW3 CHAPTER 3. HWS
Therefore,
y1(x) = C4eβbx/a β x
aeβbx/a,
andyout(x) = eβbx/a + Ξ΅
((C4 β
x
a
)eβbx/a
)+O(Ξ΅2).
Now we match. As xβ 0+, the outer solution goes to
yout(x) βΌ 1 + C4Ξ΅+O(Ξ΅2),
while as ΞΎ ββ, the inner solution approaches
yin(x) βΌ 1 +
(1
a+
b
a2
)Ξ΅+O(Ξ΅2),
where I have cheated slightly.2 Matching,
C4 =1
a+
b
a2,
and thus,
yuniform(x) = eβbx/a + Ξ΅
(((1
a+
b
a2
)β x
a
)eβbx/a β
(1
a+
b
a2
)eβax/Ξ΅
)+O(Ξ΅2).
Let us now graphically compare the asymptotic solutions accurate to O(1) and O(Ξ΅) with the exactsolution:
yexact(x) =1
2βa2 β 4bΞ΅
(βaeβ(a/Ξ΅+
βa2β4bΞ΅/Ξ΅)x/2 β 2Ξ΅eβ(a/Ξ΅+
βa2β4bΞ΅/Ξ΅)x/2 +
βa2 β 4bΞ΅eβ(a/Ξ΅+
βa2β4bΞ΅/Ξ΅)x/2
+aeβ(a/Ξ΅ββa2β4bΞ΅/Ξ΅)x/2 + 2Ξ΅eβ(a/Ξ΅β
βa2β4bΞ΅/Ξ΅)x/2 +
βa2 β 4bΞ΅eβ(a/Ξ΅β
βa2β4bΞ΅/Ξ΅)x/2
).
For simplicity, take a = b = 1.
2I ignored the term which was linear in ΞΎ, which blows up as ΞΎ ββ. This term came from ignoring the outer expansion,which changes on the same order, and is called a secular term. This occurs because we ignored the outer expansion, as isconventional when working with the boundary layer, but the outer solution, when Taylor expanded about zero, matchesthis term exactly at order x, and so we do not have any problems. The outer solution changes at a slow rate compared toΞΎ, and so the appearance of two time scales in the boundary layer causes problems with the match (which I swept underthe rug by cheating). This problem is therefore far better suited for multiscale methods, which form the subject matterof Chapter 11.
2
166
3.3. HW3 CHAPTER 3. HWS
For the first graph, Ξ΅ = 0.1:
The blue curve represents the exact solution, the orange curve the uniform solution to leading order, andthe green curve the uniform solution accurate to O(Ξ΅). We see that the differences between the curvesremains small always, and that the higher order approximation is much closer to the exact solution. Theleading order solution never differs by more than about 0.1 β Ξ΅, while the next order solution differsfrom the exact solution by a much smaller amount (approximately O(Ξ΅2)).
Now let Ξ΅ = 0.05:
The color scheme is the same as before. We notice that the same qualitative observations from beforehold for this graph as well. The difference between the orange and blue curves is even half as much, ingood agreement with our O(Ξ΅) error estimation. This also validates our conclusion that the boundarylayer only appears at O(Ξ΅). It is somewhat more difficult to verify pictorially that the error for the greencurve is O(Ξ΅2), but it is also clear that the error in this plot is less than in the first plot, and by a greaterfactor than two.
3
167
3.3. HW3 CHAPTER 3. HWS
3.3.7.2 section 9 problem 15
168
3.3. HW3 CHAPTER 3. HWS
169
3.3. HW3 CHAPTER 3. HWS
170
3.3. HW3 CHAPTER 3. HWS
171
3.3. HW3 CHAPTER 3. HWS
3.3.7.3 section 9 problem 19
Here is an outline for the solution to BO 9.19.
The boundary value problem is
Η«yβ²β² + sin(x)yβ² + 2 sin(x) cos(x)y = 0, y(0) = Ο, y(Ο) = 0
using sin(2x) = 2 sin(x) cos(x). Our heuristic argument based on the 1D advection-diffusion equation suggests a boundary layer on the left at x = 0. An outer expansionyout(x) βΌ yo(x) + Η«y1(x) +O(Η«2) gives
O(1) : yβ²o βΌ β2 cos(x)yo
with solution
yo(x) βΌ A exp[β2 sin(x)]
which cannot satisfy y(Ο) = 0 unless A = 0 (see below).
For the inner solution near x = 0 let ΞΎ = x/Η«p. The expansion yin(ΞΎ) βΌ yo(ΞΎ)+Η«y1(ΞΎ)+O(Η«2)leads to p = 1/2 and
O(1) : yβ²β²o (ΞΎ) βΌ βΞΎyβ²o(ΞΎ), y(ΞΎ = 0) = Ο
with solution
yo(ΞΎ) βΌ Ο + C
β« ΞΎ
0
exp(βt2/2)dt
yo(ΞΎ) βΌ Ο +β2C
β« ΞΎ/β2
0
exp(βs2)ds = Ο +β2C
βΟ
2erf(ΞΎ/
β2).
On the other side near x = Ο, let Ξ· = (Οβ x)/Η«p. The expansion yin(Ξ·) βΌ yo(Ξ·)+ Η«y1(Ξ·)+O(Η«2) leads to p = 1/2 and
O(1) : yβ²β²o (Ξ·) βΌ Ξ·yβ²o(Ξ·), y(Ξ· = 0) = 0
using sin(Ο β Η«1/2Ξ·) = sin(Η«1/2Ξ·) βΌ Η«1/2Ξ· for Η«1/2Ξ· β 0. Then
yo(Ξ·) βΌ D
β« Ξ·
0
exp(t2/2)dt
Now matching
limΞΎββ
yin(ΞΎ) = limxβ0
yout(x)
limΞ·ββ
yin(Ξ·) = limxβΟ
yout(x)
gives A = D = 0 and C = ββ2Ο. Thus we obtain
y(x) βΌ Ο β Ο erf(x/β2Η«).
1
172
3.3. HW3 CHAPTER 3. HWS
3.3.7.4 section 9 problem 4b
NEEP 548: Engineering Analysis II 2/27/2011
Problem 9.4-b: Bender & Orszag
Instructor: Leslie Smith
1 Problem Statement
Find the boundary layer solution for
Η«yβ²β² + (x2 + 1)yβ² β x3y = 0 (1)
2 Solution
We can expect a boundary layer at x = 0.
2.1 Outer Solution
(x2 + 1)yβ²o β x3yo = 0
yβ²o β
(
x3
x2 + 1
)
yo = 0
This is in a form that we can find an integrating factor for,
Β΅ = exp
(β«
x3
x2 + 1dx
)
= exp
[
β1
2(ln[x2 + 1]β x2)
]
= (x2 + 1)β1/2eβx2/2
=β [yo(x2 + 1)β1/2eβx2/2]β² = 0
yo = C1(x2 + 1)β1/2ex
2/2
B.C.: y(1) = 1 =β 1 = C1(2)β1/2e1/2 =β C1 =
β2eβ1/2. Therefore, the outer solution is:
yo =β2eβ1/2(x2 + 1)β1/2ex
2/2 (2)
1
173
3.3. HW3 CHAPTER 3. HWS
2.2 Inner Solution
Define ΞΎ = x/Η« such that when x βΌ Η«, ΞΎ βΌ 1
βy
βx=
dy
dΞΎ
dΞΎ
dx
β2y
βx2=
1
Η«2d2y
dΞΎ2
Plug this into the D. E. to get,
1
Η«
d2y
dΞΎ2+ Η«ΞΎ2
dy
dΞΎ+
1
Η«
dy
dΞΎβ ΞΎ2Η«2y = 0 (3)
The let the inner solution take the form,
yin = yo(ΞΎ) + Η«y1(ΞΎ) + Η«2y2(ΞΎ) + ...
Balance terms, order by order,
O
(
1
Η«
)
: yβ²β²o + yβ²o = 0 y(0) = 1
O(1) : yβ²β²1 + yβ²1 = 0 y(0) = 0
To the lowest order,
yβ²β²in + yβ²in = 0 let z = yβ²in
=β ln z = βΞΎ + C2
ln yβ²in = βΞΎ + C2
yβ²in = C3eβΞΎ
yin = C3eΞΎ + C4
We can solve for one of the constants by imposing the boundary conditions, yin(0) = 1 =β 1 =βC3 + C4 =β C4 = 1 + C3
yin = 1 + C(
1β eβΞΎ)
(4)
The other constant we get by matching the βouterβ and βinnerβ solutions:
2
174
3.3. HW3 CHAPTER 3. HWS
limxβ0
yo = limΞΎββ
yin
limxβ0
β2eβ1/2(x2 + 1)β1/2ex
2/2 = limΞΎββ
[1 +C(1β eβΞΎ)]
β2eβ1/2 = 1 +C
=β C =β2eβ1/2 β 1
Note: ymatch =β2eβ1/2
ytot = yin + yout β ymatch (5)
ytot =β2eβ1/2(x2 + 1)β1/2ex
2/2 + eβx/Η«(1ββ2eβ1/2) (6)
3
175
3.3. HW3 CHAPTER 3. HWS
3.3.7.5 section 9 problem 6
Here is an outline for the solution to BO 9.6.
Write the equation as
yβ² = (1 + Η«xβ2)y2 β 2y + 1, y(1) = 1
and we want to solve this on 0 β€ x β€ 1. For the outer solution, the expansion yout(x) βΌyo(x) + Η«y1(x) +O(Η«2) gives
O(1) : yβ²o βΌ (yo β 1)2, yo(1) = 1
O(Η«) : yβ²1βΌ 2yoy1 + y2ox
β2β 2y1, y1(1) = 0
with solutions
yo(x) βΌ 1, y1(x) βΌ 1β xβ1
valid away from zero. So yout(x) βΌ 1 + Η«(1β xβ1) +O(Η«2).
For the inner solution let ΞΎ = x/Η«p. The expansion yout(ΞΎ) βΌ yo(ΞΎ) + Η«y1(ΞΎ) +O(Η«2) leadsto ΞΎ β Η« (so we take ΞΎ = Η« for convenience). The biggest terms are:
O(Η«β1) : yβ²o βΌ y2oΞΎβ2
with solution
yo(ΞΎ) βΌ ΞΎ(1β AΞΎ)β1.
Now matching
limΞΎββ
yo(ΞΎ) +O(Η«) = limxβ0
yo(x) +O(Η«)
limΞΎββ
ΞΎ(1β AΞΎ)β1 +O(Η«) = limxβ0
1 +O(Η«)
limΞΎββ
βAβ1 +O(ΞΎβ1) +O(Η«) = limxβ0
1 +O(Η«)
gives A = β1 with matching region Η« βͺ x βͺ 1.
The next-order problem is
O(1) : yβ²1βΌ y2o + 2yoy1ΞΎ
β2β 2yo + 1.
Use yo(ΞΎ) from above and the integrating factor method to find
y1(ΞΎ) βΌ βΞΎ(1 + ΞΎ)β2 + CΞΎ2(1 + ΞΎ)β2.
Now matching
1
176
3.3. HW3 CHAPTER 3. HWS
limΞΎββ
yo(ΞΎ) + Η«y1(ΞΎ) +O(Η«2) = limxβ0
yo(x) + Η«y1(x) +O(Η«2)
limΞΎββ
ΞΎ(1 + ΞΎ)β1 + Η«
(
CΞΎ2(1 + ΞΎ)β2β ΞΎ(1 + ΞΎ)β2
)
+O(Η«2) = limxβ0
1 + Η«
(
1β xβ1
)
+O(Η«2)
limΞΎββ
1
1 + ΞΎβ1+ Η«
(
CΞΎ2(1 + ΞΎ)β2β ΞΎ(1 + ΞΎ)β2
)
+O(Η«2) = limxβ0
1 + Η«βΗ«
x+O(Η«2)
limΞΎββ
1β ΞΎβ1 +O(ΞΎβ2) + Η«C +O(Η«ΞΎβ1) +O(Η«2) = limxβ0
1 + Η«βΗ«
x+O(Η«2)
and so we choose C = 1.
Question for you: what is the matching region? (I think Η«1/2 βͺ x βͺ 1). Is this correct?
Now we form the uniform approximation y βΌ yin + yout β ymatch +O(Η«2):
y βΌ 1 + Η«(1β xβ1) + ΞΎ(1 + ΞΎ)β1 + Η«
(
ΞΎ2(1 + ΞΎ)β2β ΞΎ(1 + ΞΎ)β2
)
β
(
1 + Η«βΗ«
x
)
+O(Η«2)
with ΞΎ = x/Η« and simplify. Notice that there is no singularity at x = 0 the solution isuniformly valid on 0 β€ x β€ 1.
More questions for you: Does the solution above satisfy the initial condition y(1) = 1?If not, can you add an O(Η«2) correction so that y(1) = 1? Will you still have a solutionuniformly valid to O(Η«2)?
I encourage you to make plots so that you can visualize the solution!
2
177
3.4. HW4 CHAPTER 3. HWS
3.4 HW4
3.4.1 problem 10.5 (page 540)
problem Use WKB to obtain solution to
ππ¦β²β² + π (π₯) π¦β² + π (π₯) π¦ = 0 (1)
with π (π₯) > 0, π¦ (0) = π΄, π¦ (1) = π΅ correct to order π.
solution
Assuming
π¦ (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ πΏ β 0
Therefore
π¦β² (π₯) βΌ1πΏ
βοΏ½π=0
πΏππβ²π (π₯) exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½
π¦β²β² (π₯) βΌ1πΏ
βοΏ½π=0
πΏππβ²β²π (π₯) exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ + οΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½2
exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½
Substituting the above into (1) and simplifying gives (writing = instead of βΌ for simplicityfor now)
πβ‘β’β’β’β’β£1πΏ
βοΏ½π=0
πΏππβ²β²π (π₯) + οΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½2β€β₯β₯β₯β₯β¦ +
ππΏ
βοΏ½π=0
πΏππβ²π (π₯) + π = 0
ππΏ
βοΏ½π=0
πΏππβ²β²π (π₯) +ππΏ2 οΏ½
βοΏ½π=0
πΏππβ²π (π₯)βοΏ½π=0
πΏππβ²π (π₯)οΏ½ +ππΏ
βοΏ½π=0
πΏππβ²π (π₯) + π = 0
Expanding gives
ππΏοΏ½πβ²β²0 + πΏπβ²β²1 + πΏ2πβ²β²2 +β―οΏ½
+ππΏ2οΏ½οΏ½πβ²0 + πΏπβ²1 + πΏ2πβ²2 +β―οΏ½ οΏ½πβ²0 + πΏπβ²1 + πΏ2πβ²2 +β―οΏ½οΏ½
+ππΏοΏ½πβ²0 + πΏπβ²1 + πΏ2πβ²2 +β―οΏ½ + π = 0
Simplifying
οΏ½ππΏπβ²β²0 + ππβ²β²1 + ππΏπβ²β²2 +β―οΏ½
+ οΏ½ππΏ2οΏ½πβ²0οΏ½
2+2ππΏοΏ½πβ²0πβ²1οΏ½ + π οΏ½2πβ²0πβ²2 + οΏ½πβ²1οΏ½
2οΏ½ +β―οΏ½
+ οΏ½ππΏπβ²0 + ππβ²1 + ππΏπβ²2 +β―οΏ½ + π = 0 (1A)
The largest terms in the left are ππΏ2οΏ½πβ²0οΏ½
2and π
πΏπβ²0. By dominant balance these must be equal in
magnitude. Hence ππΏ2 = ποΏ½
1πΏοΏ½ or π
πΏ = π (1). Therefore πΏ is proportional to π and for simplicity
178
3.4. HW4 CHAPTER 3. HWS
π is taken as equal to πΏ, hence (1A) becomes
οΏ½πβ²β²0 + ππβ²β²1 + π2πβ²β²2 +β―οΏ½
+ οΏ½πβ1 οΏ½πβ²0οΏ½2+ 2πβ²0πβ²1 + π οΏ½2πβ²0πβ²2 + οΏ½πβ²1οΏ½
2οΏ½ +β―οΏ½
+ οΏ½ππβ1πβ²0 + ππβ²1 + πππβ²2 +β―οΏ½ + π = 0
Terms of ποΏ½πβ1οΏ½ give
οΏ½πβ²0οΏ½2+ ππβ²0 = 0 (2)
And terms of π (1) give
πβ²β²0 + 2πβ²0πβ²1 + ππβ²1 + π = 0 (3)
And terms of π (π) give
2πβ²0πβ²2 + ππβ²2 + οΏ½πβ²1οΏ½2+ πβ²β²1 = 0
πβ²2 = βοΏ½πβ²1οΏ½
2+ πβ²β²1
οΏ½π + 2πβ²0οΏ½(4)
Starting with (2)
πβ²0 οΏ½πβ²0 + ποΏ½ = 0
There are two cases to consider.
case 1 πβ²0 = 0. This means that π0 (π₯) = π0. A constant. Using this result in (3) gives an ODEto solve for π1 (π₯)
ππβ²1 + π = 0
πβ²1 = βπ (π₯)π (π₯)
π1 βΌ βοΏ½π₯
0
π (π‘)π (π‘)
ππ‘ + π1
Using this result in (4) gives an ODE to solve for π2 (π₯)
πβ²2 = βοΏ½β π(π₯)
π(π₯)οΏ½2+ οΏ½β π(π₯)
π(π₯)οΏ½β²
π (π₯)
= β
π2(π₯)π2(π₯) β οΏ½
πβ²(π₯)π(π₯) β
π(π₯)πβ²(π₯)π2(π₯)
οΏ½
π (π₯)
= βπ2(π₯)π2(π₯) β
π(π₯)πβ²(π₯)π2(π₯) + πβ²(π₯)π(π₯)
π2(π₯)
π (π₯)
=π (π₯) πβ² (π₯) β π2 (π₯) β πβ² (π₯) π (π₯)
π3 (π₯)
179
3.4. HW4 CHAPTER 3. HWS
Therefore
π2 = οΏ½π₯
0
π (π‘) πβ² (π‘) β π2 (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘ + π2
For case one, the solution becomes
π¦1 (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ πΏ β 0
βΌ exp οΏ½1ποΏ½π0 (π₯) + ππ1 (π₯) + π2π2 (π₯)οΏ½οΏ½ π β 0+
βΌ exp οΏ½1ππ0 (π₯) + π1 (π₯) + ππ2 (π₯)οΏ½
βΌ exp οΏ½1ππ0 βοΏ½
π₯
0
π (π‘)π (π‘)
ππ‘ + π1 + ποΏ½π₯
0
π (π‘) πβ² (π‘) β π2 (π‘) β π (π‘)π3 (π‘)
ππ‘ + π2οΏ½
βΌ πΆ1 exp οΏ½βοΏ½π₯
0
π (π‘)π (π‘)
ππ‘ + ποΏ½π₯
0
π (π‘) πβ² (π‘) β π2 (π‘) β π (π‘)π3 (π‘)
ππ‘οΏ½ (5)
Where πΆ1 is a constant that combines all π1π π0+π1+π2 constants into one. Equation (5) gives
the first WKB solution of order π (π) for case one. Case 2 now is considered.
case 2 In this case πβ²0 = βπ, therefore
π0 = βοΏ½π₯
0π (π‘) ππ‘ + π0
Equation (3) now gives
πβ²β²0 + 2πβ²0πβ²1 + ππβ²1 + π = 0βπβ² β 2ππβ²1 + ππβ²1 + π = 0
βππβ²1 = πβ² β π
πβ²1 =π β πβ²
π
πβ²1 =ππβπβ²
πIntegrating the above results in
π1 = οΏ½π₯
0
π (π‘)π (π‘)
ππ‘ β ln (π) + π1
180
3.4. HW4 CHAPTER 3. HWS
π2 (π₯) is now found from (4)
πβ²2 = βοΏ½πβ²1οΏ½
2+ πβ²β²1
οΏ½π + 2πβ²0οΏ½
= βοΏ½ πβπ
β²
ποΏ½2+ οΏ½ πβπ
β²
ποΏ½β²
(π + 2 (βπ))
= βπ2+(πβ²)2β2ππβ²
π2 + πβ²βπβ²β²
π β πβ²πβ(πβ²)2
π2
βπ
=π2 + (πβ²)2 β 2ππβ² + ππβ² β ππβ²β² β πβ²π β (πβ²)2
π3
=π2 β 2ππβ² + ππβ² β ππβ²β² β πβ²π
π3Hence
π2 = οΏ½π₯
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘ + π2
Therefore for this case the solution becomes
π¦2 (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ πΏ β 0
βΌ exp οΏ½1ποΏ½π0 (π₯) + ππ1 (π₯) + π2π2 (π₯)οΏ½οΏ½ π β 0+
βΌ exp οΏ½1ππ0 (π₯) + π1 (π₯) + ππ2 (π₯)οΏ½
Or
π¦2 (π₯) βΌ exp οΏ½β1π οΏ½
π₯
0π (π‘) ππ‘ + π0οΏ½ exp οΏ½οΏ½
π₯
0
π (π‘)π (π‘)
ππ‘ β ln (π) + π1οΏ½
exp οΏ½ποΏ½π₯
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘ + π2οΏ½
Which simplifies to
π¦2 (π₯) βΌπΆ2π
exp οΏ½β1π οΏ½
π₯
0π (π‘) ππ‘ +οΏ½
π₯
0
π (π‘)π (π‘)
ππ‘ + ποΏ½π₯
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
(6)
Where πΆ2 is new constant that combines π0, π1, π2 constants. The general solution is linearcombinations of π¦1, π¦2
π¦ (π₯) βΌ π΄π¦1 (π₯) + π΅π¦2 (π₯)
181
3.4. HW4 CHAPTER 3. HWS
Or
π¦ (π₯) βΌ πΆ1 exp οΏ½βοΏ½π₯
0
π (π‘)π (π‘)
ππ‘ + ποΏ½π₯
0
π (π‘) πβ² (π‘) β π2 (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
+πΆ2π (π₯)
exp οΏ½β1π οΏ½
π₯
0π (π‘) ππ‘ +οΏ½
π₯
0
π (π‘)π (π‘)
ππ‘ + ποΏ½π₯
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
Now boundary conditions are applied to find πΆ1, πΆ2. Using π¦ (0) = π΄ in the above gives
π΄ = πΆ1 +πΆ2π (0)
(7)
And using π¦ (1) = π΅ gives
π΅ = πΆ1 exp οΏ½βοΏ½1
0
π (π‘)π (π‘)
ππ‘ + ποΏ½1
0
π (π‘) πβ² (π‘) β π2 (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
+πΆ2π (1)
exp οΏ½β1π οΏ½
1
0π (π‘) ππ‘ +οΏ½
1
0
π (π‘)π (π‘)
ππ‘ + ποΏ½1
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
Neglecting exponentially small terms involving πβ1π the above becomes
π΅ = πΆ1 exp οΏ½βοΏ½1
0
π (π‘)π (π‘)
ππ‘οΏ½ exp οΏ½ποΏ½1
0
π (π‘) πβ² (π‘) β π2 (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
+πΆ2π (1)
exp οΏ½οΏ½1
0
π (π‘)π (π‘)
ππ‘οΏ½ exp οΏ½ποΏ½1
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½ (8)
To simplify the rest of the solution which finds πΆ1, πΆ2, let
π§1 = οΏ½1
0
π (π‘)π (π‘)
ππ‘
π§2 = οΏ½1
0
π (π‘) πβ² (π‘) β π2 (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘
π§3 = οΏ½1
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘
Hence (8) becomes
π΅ = πΆ1πβπ§1πππ§2 +πΆ2π (1)
ππ§1πππ§3 (8A)
182
3.4. HW4 CHAPTER 3. HWS
From (7) πΆ2 = π (0) (π΄ β πΆ1). Substituting this in (8A) gives
π΅ = πΆ1πβπ§1πππ§2 +π (0) (π΄ β πΆ1)
π (1)ππ§1πππ§3
= πΆ1πβπ§1πππ§2 +π (0)π (1)
π΄ππ§1πππ§3 βπ (0)π (1)
πΆ1ππ§1πππ§3
π΅ = πΆ1 οΏ½πβπ§1πππ§2 βπ (0)π (1)
ππ§1πππ§3οΏ½ + π΄π (0)π (1)
ππ§1πππ§3
πΆ1 =π΅ β π΄ π(0)
π(1)ππ§1πππ§3
πβπ§1πππ§2 β π(0)π(1)π
π§1πππ§3
=π (1) π΅ β π΄π (0) ππ§1πππ§3
π (1) πβπ§1πππ§2 β π (0) ππ§1πππ§3(9)
Using (7), now πΆ2 is found
π΄ = πΆ1 +πΆ2π (0)
π΄ =π (1) π΅ β π΄π (0) ππ§1πππ§3
π (1) πβπ§1πππ§2 β π (0) ππ§1πππ§3+
πΆ2π (0)
πΆ2 = π (0) οΏ½π΄ βπ (1) π΅ β π΄π (0) ππ§1πππ§3
π (1) πβπ§1πππ§2 β π (0) ππ§1πππ§3 οΏ½(10)
The constants πΆ1, πΆ2, are now found, hence the solution is now complete.
Summary of solution
π¦ (π₯) βΌ πΆ1 exp οΏ½βοΏ½π₯
0
π (π‘)π (π‘)
ππ‘ + ποΏ½π₯
0
π (π‘) πβ² (π‘) β π2 (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
+πΆ2π (π₯)
exp οΏ½β1π οΏ½
π₯
0π (π‘) ππ‘ +οΏ½
π₯
0
π (π‘)π (π‘)
ππ‘ + ποΏ½π₯
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘οΏ½
Where
πΆ1 =π (1) π΅ β π΄π (0) ππ§1πππ§3
π (1) πβπ§1πππ§2 β π (0) ππ§1πππ§3
πΆ2 = π (0) οΏ½π΄ βπ (1) π΅ β π΄π (0) ππ§1πππ§3
π (1) πβπ§1πππ§2 β π (0) ππ§1πππ§3 οΏ½
And
π§1 = οΏ½1
0
π (π‘)π (π‘)
ππ‘
π§2 = οΏ½1
0
π (π‘) πβ² (π‘) β π2 (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘
π§3 = οΏ½1
0
π2 (π‘) β 2π (π‘) πβ² (π‘) + π (π‘) πβ² (π‘) β π (π‘) πβ²β² (π‘) β πβ² (π‘) π (π‘)π3 (π‘)
ππ‘
183
3.4. HW4 CHAPTER 3. HWS
3.4.2 problem 10.6
problem Use second order WKB to derive formula which is more accurate than (10.1.31)
for the ππ‘β eigenvalue of the Sturm-Liouville problem in 10.1.27. Let π (π₯) = (π₯ + π)4 andcompare your formula with value of πΈπ in table 10.1
solution
Problem 10.1.27 is
π¦β²β² + πΈπ (π₯) π¦ = 0
With π (π₯) = (π₯ + π)4 and boundary conditions π¦ (0) = 0, π¦ (π) = 0. Letting
πΈ =1π
Then the ODE becomes
ππ¦β²β² (π₯) + (π₯ + π)4 π¦ (π₯) = 0 (1)
Physical optics approximation is obtained when π β β or π β 0+. Since the ODE is linear,and the highest derivative is now multiplied by a very small parameter π, WKB can be usedto solve it. Assuming the solution is
π¦ (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ πΏ β 0
Then
π¦β² (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ οΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½
π¦β²β² (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ οΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½2
+ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ οΏ½1πΏ
βοΏ½π=0
πΏππβ²β²π (π₯)οΏ½
Substituting these into (1) and canceling the exponential terms gives
πββββββοΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½2
+1πΏ
βοΏ½π=0
πΏππβ²β²π (π₯)ββββββ βΌ β (π₯ + π)4
ππΏ2οΏ½πβ²0 + πΏπβ²1 + πΏ2πβ²2 +β―οΏ½ οΏ½πβ²0 + πΏπβ²1 + πΏ2πβ²2 +β―οΏ½ +
ππΏοΏ½πβ²β²0 + πΏπβ²β²1 + πΏ2πβ²β²2 +β―οΏ½ βΌ β (π₯ + π)4
ππΏ2οΏ½οΏ½πβ²0οΏ½
2+ πΏ οΏ½2πβ²1πβ²0οΏ½ + πΏ2 οΏ½2πβ²0πβ²2 + οΏ½πβ²0οΏ½
2οΏ½ +β―οΏ½ +
ππΏοΏ½πβ²β²0 + πΏπβ²β²1 + πΏ2πβ²β²2 +β―οΏ½ βΌ β (π₯ + π)4
οΏ½ππΏ2οΏ½πβ²0οΏ½
2+ππΏοΏ½2πβ²1πβ²0οΏ½ + π οΏ½2πβ²0πβ²2 + οΏ½πβ²0οΏ½
2οΏ½ +β―οΏ½ + οΏ½
ππΏπβ²β²0 + π2πβ²β²1 +β―οΏ½ βΌ β (π₯ + π)4 (2)
The largest term in the left side is ππΏ2οΏ½πβ²0οΏ½
2. By dominant balance, this term has the same
order of magnitude as right side β (π₯ + π)4. Hence πΏ2 is proportional to π and for simplicity,πΏ can be taken equal to βπ or
πΏ = βπ
184
3.4. HW4 CHAPTER 3. HWS
Equation (2) becomes
οΏ½οΏ½πβ²0οΏ½2+ βπ οΏ½2πβ²1πβ²0οΏ½ + π οΏ½2πβ²0πβ²2 + οΏ½πβ²0οΏ½
2οΏ½ +β―οΏ½ + οΏ½βππβ²β²0 + ππβ²β²1 +β―οΏ½ βΌ β (π₯ + π)4
Balance of π (1) gives
οΏ½πβ²0οΏ½2βΌ β (π₯ + π)4 (3)
And Balance of ποΏ½βποΏ½ gives
2πβ²1πβ²0 βΌ βπβ²β²0 (4)
Balance of π (π) gives
2πβ²0πβ²2 + οΏ½πβ²1οΏ½2βΌ βπβ²β²1 (5)
Equation (3) is solved first in order to find π0 (π₯). Therefore
πβ²0 βΌ Β±π (π₯ + π)2
Hence
π0 (π₯) βΌ Β±ποΏ½π₯
0(π‘ + π)2 ππ‘ + πΆΒ±
βΌ Β±π οΏ½π‘3
3+ ππ‘2 + π2π‘οΏ½
π₯
0+ πΆΒ±
βΌ Β±π οΏ½π₯3
3+ ππ₯2 + π2π₯οΏ½ + πΆΒ± (6)
π1 (π₯) is now found from (4), and since πβ²β²0 = Β±2π (π₯ + π) therefore
πβ²1 βΌ β12πβ²β²0πβ²0
βΌ β12Β±2π (π₯ + π)Β±π (π₯ + π)2
βΌ β1
π₯ + πHence
π1 (π₯) βΌ βοΏ½π₯
0
1π‘ + π
ππ‘
βΌ β ln οΏ½π + π₯π
οΏ½ (7)
π2 (π₯) is now solved from from (5)
2πβ²0πβ²2 + οΏ½πβ²1οΏ½2βΌ βπβ²β²1
πβ²2 βΌβπβ²β²1 β οΏ½πβ²1οΏ½
2
2πβ²0
185
3.4. HW4 CHAPTER 3. HWS
Since πβ²1 βΌ β 1π₯+π , then π
β²β²1 βΌ 1
(π₯+π)2and the above becomes
πβ²2 βΌβ 1(π₯+π)2
β οΏ½β 1π₯+π
οΏ½2
2 οΏ½Β±π (π₯ + π)2οΏ½
βΌβ 1(π₯+π)2
β 1(π₯+π)2
Β±2π (π₯ + π)2
βΌ Β±π1
(π₯ + π)4
Hence
π2 βΌ Β±π οΏ½οΏ½π₯
0
1(π‘ + π)4
ππ‘οΏ½ + πΒ±
βΌ Β±π3 οΏ½
1π3 β
1(π + π₯)3
οΏ½ + πΒ±
Therefore the solution becomes
π¦ (π₯) βΌ exp οΏ½1
βποΏ½π0 (π₯) + βππ1 (π₯) + ππ2 (π₯)οΏ½οΏ½
βΌ exp οΏ½1
βππ0 (π₯) + π1 (π₯) + βππ2 (π₯)οΏ½
But πΈ = 1π , hence βπ =
1
βπΈ, and the above becomes
π¦ (π₯) βΌ exp οΏ½βπΈπ0 (π₯) + π1 (π₯) +1
βπΈπ2 (π₯)οΏ½
Therefore
π¦ (π₯) βΌ exp οΏ½Β±πβπΈ οΏ½π₯3
3+ ππ₯2 + π2π₯οΏ½ + πΆΒ± β ln οΏ½π + π₯
ποΏ½ Β± π
1
βπΈ13 οΏ½
1π3 β
1(π + π₯)3
οΏ½ + πΒ±οΏ½
Which can be written as
π¦ (π₯) βΌ οΏ½π + π₯π
οΏ½β1πΆ exp οΏ½π οΏ½βπΈ οΏ½
π₯3
3+ ππ₯2 + π2π₯οΏ½ +
1
βπΈ13 οΏ½
1π3 β
1(π + π₯)3
οΏ½οΏ½οΏ½
β πΆ οΏ½π + π₯π
οΏ½β1
exp οΏ½βπ οΏ½βπΈ οΏ½π₯3
3+ ππ₯2 + π2π₯οΏ½ +
1
βπΈ13 οΏ½
1π3 β
1(π + π₯)3
οΏ½οΏ½οΏ½
Where all constants combined into Β±πΆ. In terms of sin / cos the above becomes
π¦ (π₯) βΌππ΄π + π₯
cos οΏ½βπΈ οΏ½π₯3
3+ ππ₯2 + π2π₯οΏ½ +
1
βπΈ13 οΏ½
1π3 β
1(π + π₯)3
οΏ½οΏ½
+ππ΅π + π₯
sin οΏ½βπΈ οΏ½π₯3
3+ ππ₯2 + π2π₯οΏ½ +
1
βπΈ13 οΏ½
1π3 β
1(π + π₯)3
οΏ½οΏ½ (8)
186
3.4. HW4 CHAPTER 3. HWS
Boundary conditions π¦ (0) = 0 gives
0 βΌ π΄ cos οΏ½0 +1
βπΈοΏ½1π3 β
1π3 οΏ½οΏ½ + π΅ sin οΏ½0 +
1
βπΈ13 οΏ½
1π3 β
1π3 οΏ½οΏ½
βΌ π΄
Hence solution in (8) reduces to
π¦ (π₯) βΌππ΅π + π₯
sin οΏ½βπΈ οΏ½π₯3
3+ ππ₯2 + π2π₯οΏ½ +
1
βπΈ13 οΏ½
1π3 β
1(π + π₯)3
οΏ½οΏ½
Applying B.C. π¦ (π) = 0 gives
0 βΌππ΅π + π
sin οΏ½βπΈ οΏ½π3
3+ ππ2 + π2ποΏ½ +
1
βπΈ13 οΏ½
1π3 β
1(π + π)3
οΏ½οΏ½
βΌπ΅2
sin οΏ½βπΈ οΏ½73π3οΏ½ +
1
βπΈοΏ½7
24π3 οΏ½οΏ½
For non trivial solution, therefore
οΏ½πΈπ οΏ½73π3οΏ½ +
1
βπΈποΏ½7
24π3 οΏ½ = ππ π = 1, 2, 3,β―
Solving for βπΈπ. Let βπΈπ = π₯, then the above becomes
π₯2 οΏ½73π3οΏ½ + οΏ½
724π6 οΏ½ = π₯ππ
π₯2 β37π2π₯π +
18π6 = 0
Solving using quadratic formula and taking the positive root, since πΈπ > 0 gives
π₯ =1
28π3 οΏ½β2β18π2π2 β 49 + 6πποΏ½ π = 1, 2, 3,β―
οΏ½πΈπ =1
28π3 οΏ½β2β18π2π2 β 49 + 6πποΏ½
πΈπ = οΏ½β2β18π2π2 β 49 + 6πποΏ½2
Table 10.1 is now reproduced to compare the above more accurate πΈπ. The following tableshows the actual πΈπ values obtained this more accurate method. Values computed fromabove formula are in column 3.
187
3.4. HW4 CHAPTER 3. HWS
In[207]:= sol = x /. Last@Solvex2 -3
7 Pi2x n +
1
8 Pi^6β©΅ 0, x
Out[207]=3 n Ο +
-49+18 n2 Ο2
2
14 Ο3
In[244]:= lam[n_] := (Evaluate@sol)^2
nPoints = {1, 2, 3, 4, 5, 10, 20, 40};
book = {0.00188559, 0.00754235, 0.0169703, 0.0301694, 0.0471397, 0.188559, 0.754235, 3.01694};
exact = {0.00174401, 0.00734865, 0.0167524, 0.0299383, 0.0469006, 0.188395, 0.753977, 3.01668};
hw = Table[N@(lam[n]), {n, nPoints}];
data = Table[{nPoints[[i]], book[[i]], hw[[i]], exact[[i]]}, {i, 1, Length[nPoints]}];
data = Join[{{"n", "En using S0+S1 (book)", "En using S0+S1+S2 (HW)", "Exact"}}, data];
Style[Grid[data, Frame β All], 18]
Out[251]=
n En using S0+S1 (book) En using S0+S1+S2 (HW) Exact
1 0.0018856 0.0016151 0.001744
2 0.0075424 0.00728 0.0073487
3 0.01697 0.016709 0.016752
4 0.030169 0.029909 0.029938
5 0.04714 0.046879 0.046901
10 0.18856 0.1883 0.1884
20 0.75424 0.75398 0.75398
40 3.0169 3.0167 3.0167
The following table shows the relative error in place of the actual values of πΈπ to bettercompare how more accurate the result obtained in this solution is compared to the bookresult
In[252]:= data = Table[{nPoints[[i]],
100* Abs@(exact[[i]] - book[[i]])/ exact[[i]],
100* Abs@(exact[[i]] - hw[[i]])/ exact[[i]]}, {i, 1, Length[nPoints]}];
data = Join[{{"n", "En using S0+S1 (book) Rel error", "En using S0+S1+S2 (HW) Rel error"}}, data];
Style[Grid[data, Frame β All], 18]
Out[254]=
n En using S0+S1 (book) Rel error En using S0+S1+S2 (HW) Rel error
1 8.1181 7.3927
2 2.6359 0.9343
3 1.3007 0.2576
4 0.77192 0.098497
5 0.5098 0.045387
10 0.087051 0.051101
20 0.034219 0.00021643
40 0.0086187 0.000055619
The above shows clearly that adding one more term in the WKB series resulted in moreaccurate eigenvalue estimate.
188
3.5. HW5 CHAPTER 3. HWS
3.5 HW5
NE548 Problem: Similarity solution for the 1D homogeneous heat equation
Due Thursday April 13, 2017
1. (a) Non-dimensionalize the 1D homogeneous heat equation:
βu(x, t)
βt= Ξ½
β2u(x, t)
βx2(1)
with ββ < x < β, and u(x, t) bounded as x β Β±β.
(b) Show that the non-dimensional equations motivate a similarity variable ΞΎ = x/t1/2.
(c) Find a similarity solution u(x, t) = H(ΞΎ) by solving the appropriate ODE for H(ΞΎ).
(d) Show that the similarity solution is related to the solution we found in class on 4/6/17for initial condition
u(x, 0) = 0, x < 0 u(x, 0) = C, x > 0.
1
solution
3.5.1 Part a
Let οΏ½οΏ½ be the non-dimensional space coordinate and π‘ the non-dimensional time coordinate.Therefore we need
οΏ½οΏ½ =π₯π0
π‘ =π‘π‘0
οΏ½οΏ½ =π’π’0
Where π0 is the physical characteristic length scale (even if this infinitely long domain, π0is given) whose dimensions is [πΏ] and π‘0 of dimensions [π] is the characteristic time scaleand οΏ½οΏ½ (οΏ½οΏ½, π‘) is the new dependent variable, and π’0 characteristic value of π’ to scale against(typically this is the initial conditions) but this will cancel out. We now rewrite the PDEππ’ππ‘ = π
π2π’ππ₯2 in terms of the new dimensionless variables.
ππ’ππ‘
=ππ’ποΏ½οΏ½
ποΏ½οΏ½π π‘π π‘ππ‘
= π’0ποΏ½οΏ½π π‘
1π‘0
(1)
189
3.5. HW5 CHAPTER 3. HWS
Andππ’ππ₯
=ππ’ποΏ½οΏ½
ποΏ½οΏ½ποΏ½οΏ½ποΏ½οΏ½ππ₯
= π’0ποΏ½οΏ½ποΏ½οΏ½
1π0
Andπ2π’ππ₯2
= π’0π2οΏ½οΏ½ποΏ½οΏ½2
1π20
(2)
Substituting (1) and (2) into ππ’ππ‘ = π
π2π’ππ₯2 gives
π’0ποΏ½οΏ½π π‘
1π‘0= ππ’0
π2οΏ½οΏ½ποΏ½οΏ½2
1π20
ποΏ½οΏ½π π‘
= οΏ½ππ‘0π20οΏ½π2οΏ½οΏ½ποΏ½οΏ½2
The above is now non-dimensional. Since π has units οΏ½πΏ2
π οΏ½ andπ‘0π20also has units οΏ½ ππΏ2 οΏ½, therefore
the product π π‘0π20is non-dimensional quantity.
If we choose π‘0 to have same magnitude (not units) as π20, i.e. π‘0 = π20, thenπ‘0π20= 1 (with units
οΏ½ ππΏ2 οΏ½) and now we obtain the same PDE as the original, but it is non-dimensional. Where
now οΏ½οΏ½ β‘ οΏ½οΏ½ ( π‘, οΏ½οΏ½).
3.5.2 Part (b)
I Will use the Buckingham π theorem for finding expression for the solution in the form
π’ (π₯, π‘) = π (π) where π is the similarity variable. Starting with ππ’ππ‘ = ππ2π’
ππ₯2 , in this PDE, thediοΏ½usion substance is heat with units of Joule π½. Hence the concentration of heat, whichis what π’ represents, will have units of [π’] = π½
πΏ3 . (heat per unit volume). From physics, weexpect the solution π’ (π₯, π‘) to depend on π₯, π‘, π and initial conditions π’0 as these are the onlyrelevant quantities involved that can aοΏ½ect the diοΏ½usion. Therefore, by Buckingham theoremwe say
π’ β‘ π (π₯, π‘, π, π’0) (1)
190
3.5. HW5 CHAPTER 3. HWS
We have one dependent quantity π’ and 4 independent quantities.The units of each of theabove quantities is
[π’] =π½πΏ3
[π₯] = πΏ[π‘] = π
[π] =πΏ2
π
[π’0] =π½πΏ3
Hence using Buckingham theorem, we write
[π’] = οΏ½π₯ππ‘ππππ’π0οΏ½ (2)
We now determine π, π, π, π, by dimensional analysis. The above is
π½πΏ3
= πΏπππ οΏ½πΏ2
π οΏ½π
οΏ½π½πΏ3 οΏ½
π
(π½) οΏ½πΏβ3οΏ½ = οΏ½πΏπ+2πβ3ποΏ½ οΏ½ππβποΏ½ οΏ½π½ποΏ½
Comparing powers of same units on both sides, we see that
π = 1π β π = 0
π + 2π β 3π = β3
From second equation above, π = π, hence third equation becomes
π + 2π β 3π = β3π + 2π = 0
Since π = 1. Hence
π = βπ2
π = βπ2
Therefore, now that we found all the powers, (we have one free power π which we can setto any value), then from equation (1)
[π’] = οΏ½π₯ππ‘ππππ’π0οΏ½π’π’0
= οΏ½οΏ½ = π₯ππ‘πππ
Therefore οΏ½οΏ½ is function of all the variables in the RHS. Let this function be π (This is thesame as π» in problem statement). Hence the above becomes
οΏ½οΏ½ = π οΏ½π₯ππ‘βπ2πβ
π2 οΏ½
= π οΏ½π₯π
ππ2 π‘
π2οΏ½
191
3.5. HW5 CHAPTER 3. HWS
Since π is free variable, we can choose
π = 1 (2)
And obtain
οΏ½οΏ½ β‘ π οΏ½π₯
βππ‘οΏ½ (3)
In the above π₯
βππ‘is now non-dimensional quantity, which we call, the similarity variable
π =π₯
βππ‘(4)
Notice that another choice of π in (2), for example π = 2 would lead to π = π₯2
ππ‘ instead ofπ = π₯
βππ‘but we will use the latter for the rest of the problem.
3.5.3 Part (c)
Using π’ β‘ π (π) where π = π₯
βππ‘then
ππ’ππ‘
=ππππππππ‘
=ππππ
πππ‘ οΏ½
π₯
βππ‘οΏ½
= β12ππππ
βββββββ
π₯
βππ‘32
βββββββ
Andππ’ππ₯
=ππππππππ₯
=ππππ
πππ₯ οΏ½
π₯
βππ‘οΏ½
=ππππ
1
βππ‘
192
3.5. HW5 CHAPTER 3. HWS
Andπ2π’ππ₯2
=πππ₯ οΏ½
ππππ
1
βππ‘οΏ½
=1
βππ‘πππ₯ οΏ½
πππποΏ½
=1
βππ‘οΏ½π2πππ2
ππππ₯οΏ½
=1
βππ‘οΏ½π2πππ2
1
βππ‘οΏ½
=1ππ‘π2πππ2
Hence the PDE ππ’ππ‘ = π
π2π’ππ₯2 becomes
β12ππππ
βββββββ
π₯
βππ‘32
βββββββ = π
1ππ‘π2πππ2
1π‘π2πππ2
+12
π₯
βππ‘32
ππππ
= 0
π2πππ2
+12π₯
βππ‘ππππ
= 0
But π₯
βππ‘= π, hence we obtain the required ODE as
π2π (π)ππ2
+12πππ (π)ππ
= 0
πβ²β² +π2πβ² = 0
We now solve the above ODE for π (π). Let πβ² = π§, then the ODE becomes
π§β² +π2π§ = 0
Integrating factor is π = πβ«π2 ππ = π
π24 , hence
ππποΏ½π§ποΏ½ = 0
π§π = π1
π§ = π1πβπ24
Therefore, since πβ² = π§, then
πβ² = π1πβπ24
193
3.5. HW5 CHAPTER 3. HWS
Integrating gives
π (π) = π2 + π1οΏ½π
0πβπ 24 ππ
3.5.4 Part (d)
For initial conditions of step function
π’ (π₯, 0) =
β§βͺβͺβ¨βͺβͺβ©0 π₯ < 0πΆ π₯ > 0
The solution found in class was
π’ (π₯, π‘) =πΆ2+πΆ2
erf οΏ½π₯
β4ππ‘οΏ½ (1)
Where erf οΏ½ π₯
β4ππ‘οΏ½ = 2
βπβ«
π₯β4ππ‘
0πβπ§2ππ§. The solution found in part (c) earlier is
π (π) = π1οΏ½π
0πβπ 24 ππ + π2
Let π = β4π§, thenππ ππ§ = β4, when π = 0, π§ = 0 and when π = π, π§ = π
β4, therefore the integral
becomes
π (π) = π1β4οΏ½πβ4
0πβπ§2ππ§ + π2
But 2
βπβ«
πβ4
0πβπ§2ππ§ = erf οΏ½ π
β4οΏ½, hence β«
πβ4
0πβπ§2ππ§ = βπ
2 erf οΏ½ π
β4οΏ½ and the above becomes
π (π) = π1βπ erf οΏ½π
β4οΏ½ + π2
= π3 erf οΏ½π
β4οΏ½ + π2
Since π = π₯
βππ‘, then above becomes, when converting back to π’ (π₯, π‘)
π’ (π₯, π‘) = π3 erf οΏ½π₯
β4ππ‘οΏ½ + π2 (2)
Comparing (1) and (2), we see they are the same. Constants of integration are arbitraryand can be found from initial conditions.
194
3.6. HW6 CHAPTER 3. HWS
3.6 HW6
3.6.1 Problem 1
NE548 Problems
Due Tuesday April 25, 2017
1. Use the Method of Images to solve
βu
βt= Ξ½
β2u
βx2+Q(x, t), x > 0
(a)u(0, t) = 0, u(x, 0) = 0.
(b)u(0, t) = 1, u(x, 0) = 0.
(c)ux(0, t) = A(t), u(x, 0) = f(x).
2. (a) Solve by the Method of Characteristics:
β2u
βt2= c2
β2u
βx2, x β₯ 0, t β₯ 0
u(x, 0) = f(x),βu(x, 0)
βt= g(x),
βu(0, t)
βx= h(t)
(b) For the special case h(t) = 0, explain how you could use a symmetry argument to helpconstruct the solution.
(c) Sketch the solution if g(x) = 0, h(t) = 0 and f(x) = 1 for 4 β€ x β€ 5 and f(x) = 0otherwise.
1
Note, I will use π in place of π since easier to type.
3.6.1.1 Part (a)
ππ’ππ‘
= ππ2π’ππ₯2
+ π (π₯, π‘)
π₯ > 0π’ (0, π‘) = 0π’ (π₯, 0) = 0
Multiplying both sides by πΊ (π₯, π‘; π₯0, π‘0) and integrating over the domain gives (where in thefollowing πΊ is used instead of πΊ (π₯, π‘; π₯0, π‘0) for simplicity).
οΏ½β
π₯=0οΏ½
β
π‘=0πΊπ’π‘ ππ‘ππ₯ = οΏ½
β
π₯=0οΏ½
β
π‘=0ππ’π₯π₯πΊ ππ‘ππ₯ +οΏ½
β
π₯=0οΏ½
β
π‘=0ππΊ ππ‘ππ₯ (1)
For the integral on the LHS, we apply integration by parts once to move the time derivativefrom π’ to πΊ
οΏ½β
π₯=0οΏ½
β
π‘=0πΊπ’π‘ ππ‘ππ₯ = οΏ½
β
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
β
π₯=0οΏ½
β
π‘=0πΊπ‘π’ ππ‘ππ₯
195
3.6. HW6 CHAPTER 3. HWS
And the first integral in the RHS of (1) gives, after doing integration by parts two times
οΏ½β
π₯=0οΏ½
β
π‘=0ππ’π₯π₯πΊ ππ‘ππ₯ = οΏ½
β
π‘=0[π’π₯πΊ]
βπ₯=0 ππ‘ βοΏ½
β
π₯=0οΏ½
β
π‘=0ππ’π₯πΊπ₯ ππ‘ππ₯
= οΏ½β
π‘=0[π’π₯πΊ]
βπ₯=0 ππ‘ β οΏ½οΏ½
β
π‘=0[π’πΊπ₯]
βπ₯=0 ππ‘ βοΏ½
β
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯οΏ½
= οΏ½β
π‘=0οΏ½[π’π₯πΊ]
βπ₯=0 β [π’πΊπ₯]
βπ₯=0οΏ½ ππ‘ +οΏ½
β
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯
= οΏ½β
π‘=0[π’π₯πΊ β π’πΊπ₯]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯
= βοΏ½β
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯
Hence (1) becomes
οΏ½β
π₯=0[π’πΊ]βπ‘=0 ππ₯βοΏ½
β
π₯=0οΏ½
β
π‘=0πΊπ‘π’ ππ‘ππ₯ = οΏ½
β
π‘=0[π’π₯πΊ β π’πΊπ₯]
βπ₯=0 ππ‘+οΏ½
β
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯+οΏ½
β
π₯=0οΏ½
β
π‘=0πΊπ ππ‘ππ₯
Or
οΏ½β
π₯=0οΏ½
β
π‘=0βπΊπ‘π’ β ππ’πΊπ₯π₯ ππ‘ππ₯ = βοΏ½
β
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
β
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
β
π‘=0πΊπ ππ‘ππ₯
(2)
We now want to choose πΊ (π₯, π‘; π₯0, π‘0) such that
βπΊπ‘π’ β ππ’πΊπ₯π₯ = πΏ (π₯ β π₯0) πΏ (π‘ β π‘0)βπΊπ‘π’ = ππ’πΊπ₯π₯ + πΏ (π₯ β π₯0) πΏ (π‘ β π‘0) (3)
This way, the LHS of (2) becomes π’ (π₯0, π‘0). Hence (2) now becomes
π’ (π₯0, π‘0) = βοΏ½β
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
π‘0
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯ (4)
We now need to find the Green function which satisfies (3). But (3) is equivalent to solutionof problem of
βπΊπ‘π’ = ππ’πΊπ₯π₯
πΊ (π₯, 0) = πΏ (π₯ β π₯0) πΏ (π‘ β π‘0)ββ < π₯ < β
πΊ (π₯, π‘; π₯0, π‘0) = 0 π‘ > π‘0πΊ (Β±β, π‘; π₯0, π‘0) = 0πΊ (π₯, π‘0; π₯0, π‘0) = πΏ (π₯ β π₯0)
But the above problem has a known fundamental solution which we found, but for theforward heat PDE instead of the reverse heat PDE. The fundamental solution to the forwardheat PDE is
πΊ (π₯, π‘) =1
β4ππ (π‘ β π‘0)exp
ββββββ (π₯ β π₯0)
2
4π (π‘ β π‘0)
βββββ 0 β€ π‘0 β€ π‘
196
3.6. HW6 CHAPTER 3. HWS
Hence for the reverse heat PDE the above becomes
πΊ (π₯, π‘) =1
β4ππ (π‘0 β π‘)exp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ 0 β€ π‘ β€ π‘0 (5)
The above is the infinite space Green function and what we will use in (4). Now we goback to (4) and simplify the boundary conditions term. Starting with the term β«β
π₯=0[π’πΊ]βπ‘=0 ππ₯.
Since πΊ (π₯,β; π₯0, π‘0) = 0 then upper limit is zero. At π‘ = 0 we are given that π’ (π₯, 0) = 0, hencethis whole term is zero. So now (4) simplifies to
π’ (π₯0, π‘0) = βοΏ½β
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
β
π‘=0πΊπ ππ‘ππ₯ (6)
We are told that π’ (0, π‘) = 0. Hence
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = (π’ (β, π‘) πΊπ₯ (β, π‘) β π’π₯ (β, π‘) πΊ (β, π‘)) β (π’ (0, π‘) πΊπ₯ (0, π‘) β π’π₯ (0, π‘) πΊ (0, π‘))
= (π’ (β, π‘) πΊπ₯ (β, π‘) β π’π₯ (β, π‘) πΊ (β, π‘)) + π’π₯ (0, π‘) πΊ (0, π‘)
We also know that πΊ (Β±β, π‘; π₯0, π‘0) = 0, Hence πΊ (β) = 0 and also πΊπ₯ (β) = 0, hence the abovesimplifies
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = π’π₯ (0, π‘) πΊ (0, π‘)
To make πΊ (0, π‘) = 0 we place an image impulse at βπ₯0 with negative value to the impulse atπ₯0. This will make πΊ at π₯ = 0 zero.
xx = 0x0
βx0
+
β
Therefore the Green function to use is, from (5) becomes
πΊ (π₯, π‘) =1
β4ππ (π‘0 β π‘)
βββββexp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ β exp
ββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ 0 β€ π‘ β€ π‘0
Therefore the solution, from (4) becomes
π’ (π₯0, π‘0) = οΏ½β
π₯=0οΏ½
π‘0
π‘=0
1
β4ππ (π‘0 β π‘)
βββββexp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ β exp
ββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ π (π₯, π‘) ππ‘ππ₯ (7)
Switching the order of π₯0, π‘0 with π₯, π‘ gives
π’ (π₯, π‘) = οΏ½β
π₯0=0οΏ½
π‘
π‘0=0
1
β4ππ (π‘ β π‘0)
βββββexp
ββββββ (π₯0 β π₯)
2
4π (π‘ β π‘0)
βββββ β exp
ββββββ (π₯0 + π₯)
2
4π (π‘ β π‘0)
βββββ
βββββ π (π₯0, π‘0) ππ‘0ππ₯0 (8)
Notice, for the terms (π₯0 β π₯)2 , (π₯0 + π₯)
2, since they are squared, the order does not matter,so we might as well write the above as
π’ (π₯, π‘) = οΏ½β
π₯0=0οΏ½
π‘
π‘0=0
1
β4ππ (π‘ β π‘0)
βββββexp
ββββββ (π₯ β π₯0)
2
4π (π‘ β π‘0)
βββββ β exp
ββββββ (π₯ + π₯0)
2
4π (π‘ β π‘0)
βββββ
βββββ π (π₯0, π‘0) ππ‘0ππ₯0 (8)
197
3.6. HW6 CHAPTER 3. HWS
3.6.1.2 Part (b)
ππ’ππ‘
= ππ2π’ππ₯2
+ π (π₯, π‘)
π₯ > 0π’ (0, π‘) = 1π’ (π₯, 0) = 0
Everything follows the same as in part (a) up to the point where boundary condition termsneed to be evaluated.
π’ (π₯0, π‘0) = βοΏ½β
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
π‘0
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯ (4)
Where
πΊ (π₯, π‘) =1
β4ππ (π‘0 β π‘)exp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ 0 β€ π‘ β€ π‘0 (5)
Starting with the term β«β
π₯=0[π’πΊ]βπ‘=0 ππ₯. Since πΊ (π₯,β; π₯0, π‘0) = 0 then upper limit is zero. At
π‘ = 0 we are given that π’ (π₯, 0) = 0, hence this whole term is zero. So now (4) simplifies to
π’ (π₯0, π‘0) = βοΏ½β
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
β
π‘=0πΊπ ππ‘ππ₯ (6)
Now
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = (π’ (β, π‘) πΊπ₯ (β, π‘) β π’π₯ (β, π‘) πΊ (β, π‘)) β (π’ (0, π‘) πΊπ₯ (0, π‘) β π’π₯ (0, π‘) πΊ (0, π‘))
We are told that π’ (0, π‘) = 1, we also know that πΊ (Β±β, π‘; π₯0, π‘0) = 0, Hence πΊ (β, π‘) = 0 andalso πΊπ₯ (β, π‘) = 0, hence the above simplifies
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = βπΊπ₯ (0, π‘) + π’π₯ (0, π‘) πΊ (0, π‘)
To make πΊ (0) = 0 we place an image impulse at βπ₯0 with negative value to the impulse atπ₯0. This is the same as part(a)
xx = 0x0
βx0
+
β
πΊ (π₯, π‘) =1
β4ππ (π‘0 β π‘)
βββββexp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ β exp
ββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ 0 β€ π‘ β€ π‘0
Now the boundary terms reduces to just
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = βπΊπ₯ (0, π‘)
We need now to evaluate πΊπ₯ (0, π‘), the only remaining term. Since we know what πΊ (π₯, π‘) from198
3.6. HW6 CHAPTER 3. HWS
the above, then
πππ₯πΊ (π₯, π‘) =
1
β4ππ (π‘0 β π‘)
ββββββ (π₯ β π₯0)2π (π‘0 β π‘)
expββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ +
2 (π₯ + π₯0)4π (π‘0 β π‘)
expββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ
And at π₯ = 0 the above simplifies to
πΊπ₯ (0, π‘) =1
β4ππ (π‘0 β π‘)οΏ½
π₯02π (π‘0 β π‘)
exp οΏ½βπ₯20
4π (π‘0 β π‘)οΏ½ +
π₯02π (π‘0 β π‘)
exp οΏ½βπ₯20
4π (π‘0 β π‘)οΏ½οΏ½
=1
β4ππ (π‘0 β π‘)οΏ½
π₯0π (π‘0 β π‘)
exp οΏ½βπ₯20
4π (π‘0 β π‘)οΏ½οΏ½
Hence the complete solution becomes from (4)
π’ (π₯0, π‘0) = βοΏ½β
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
π‘0
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯
= βοΏ½π‘0
π‘=0βπΊπ₯ (0, π‘) ππ‘ +οΏ½
β
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯
Substituting πΊ and πΊπ₯ in the above gives
π’ (π₯0, π‘0) = οΏ½π‘0
π‘=0
1
β4ππ (π‘0 β π‘)οΏ½
π₯0π (π‘0 β π‘)
exp οΏ½βπ₯20
4π (π‘0 β π‘)οΏ½οΏ½ ππ‘
+οΏ½β
π₯=0οΏ½
π‘0
π‘=0
1
β4ππ (π‘0 β π‘)
βββββexp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ β exp
ββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ π (π₯, π‘) ππ‘ππ₯
Switching the order of π₯0, π‘0 with π₯, π‘
π’ (π₯, π‘) = οΏ½π‘
π‘0=0
1
β4ππ (π‘ β π‘0)οΏ½
π₯π (π‘ β π‘0)
exp οΏ½βπ₯2
4π (π‘ β π‘0)οΏ½οΏ½ ππ‘0
+οΏ½β
π₯0=0οΏ½
π‘
π‘0=0
1
β4ππ (π‘ β π‘0)
βββββexp
ββββββ (π₯0 β π₯)
2
4π (π‘ β π‘0)
βββββ β exp
ββββββ (π₯0 + π₯)
2
4π (π‘ β π‘0)
βββββ
βββββ π (π₯0, π‘0) ππ‘0ππ₯0 (7)
But
οΏ½π‘
π‘0=0
1
β4ππ (π‘ β π‘0)οΏ½
π₯π (π‘ β π‘0)
exp οΏ½βπ₯2
4π (π‘ β π‘0)οΏ½οΏ½ ππ‘0 = erfc οΏ½
π₯2βπ‘
οΏ½
Hence (7) becomes
π’ (π₯, π‘) = erfc οΏ½π₯2βπ‘
οΏ½ +οΏ½β
π₯0=0οΏ½
π‘
π‘0=0
1
β4ππ (π‘ β π‘0)
βββββexp
ββββββ (π₯0 β π₯)
2
4π (π‘ β π‘0)
βββββ β exp
ββββββ (π₯0 + π₯)
2
4π (π‘ β π‘0)
βββββ
βββββ π (π₯0, π‘0) ππ‘0ππ₯0
The only diοΏ½erence between this solution and part(a) solution is the extra term erfc οΏ½ π₯2βπ‘οΏ½
due to having non-zero boundary conditions in this case.
199
3.6. HW6 CHAPTER 3. HWS
3.6.1.3 part(c)
ππ’ππ‘
= ππ2π’ππ₯2
+ π (π₯, π‘)
π₯ > 0ππ’ (0, π‘)ππ₯
= π΄ (π‘)
π’ (π₯, 0) = π (π₯)
Everything follows the same as in part (a) up to the point where boundary condition termsneed to be evaluated.
π’ (π₯0, π‘0) = βοΏ½β
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
π‘0
π‘=0[π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 ππ‘ +οΏ½
β
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯ (4)
Starting with the term β«β
π₯=0[π’πΊ]βπ‘=0 ππ₯. Since πΊ (π₯,β; π₯0, π‘0) = 0 then upper limit is zero. At
π‘ = 0 we are given that π’ (π₯, 0) = π (π₯), hence
οΏ½β
π₯=0[π’πΊ]βπ‘=0 ππ₯ = οΏ½
β
π₯=0βπ’ (π₯, 0) πΊ (π₯, 0) ππ₯
= οΏ½β
π₯=0βπ (π₯)πΊ (π₯, 0) ππ₯
Looking at the second term in RHS of (4)
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = (π’ (β, π‘) πΊπ₯ (β, π‘) β π’π₯ (β, π‘) πΊ (β, π‘)) β (π’ (0, π‘) πΊπ₯ (0, π‘) β π’π₯ (0, π‘) πΊ (0, π‘))
We are told that π’π₯ (0, π‘) = π΄ (π‘), we also know that πΊ (Β±β, π‘; π₯0, π‘0) = 0, Hence πΊ (β, π‘) = 0 andalso πΊπ₯ (β, π‘) = 0. The above simplifies
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = β (π’ (0, π‘) πΊπ₯ (0, π‘) β π΄ (π‘) πΊ (0, π‘)) (5)
We see now from the above, that to get rid of the π’ (0, π‘) πΊπ₯ (0, π‘) term (since we do not knowwhat π’ (0, π‘) is), then we now need
πΊπ₯ (0, π‘) = 0
This means we need an image at βπ₯0 which is of same sign as at +π₯0 as shown in this diagram
xx = 0x0βx0
++
Which means the Green function we need to use is the sum of the Green function solutionsfor the infinite domain problem
πΊ (π₯, π‘) =1
β4ππ (π‘0 β π‘)
βββββexp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ 0 β€ π‘ β€ π‘0 (6)
200
3.6. HW6 CHAPTER 3. HWS
The above makes πΊπ₯ (0, π‘) = 0 and now equation (5) reduces to
[π’πΊπ₯ β π’π₯πΊ]βπ₯=0 = π΄ (π‘) πΊ (0, π‘)
= π΄ (π‘)βββββ
1
β4ππ (π‘0 β π‘)
βββββexp
ββββββ (βπ₯0)
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ
βββββ
=π΄ (π‘)
β4ππ (π‘0 β π‘)οΏ½exp οΏ½
βπ₯204π (π‘0 β π‘)
οΏ½ + exp οΏ½βπ₯20
4π (π‘0 β π‘)οΏ½οΏ½
=π΄ (π‘)
βππ (π‘0 β π‘)exp οΏ½
βπ₯204π (π‘0 β π‘)
οΏ½
We now know all the terms needed to evaluate the solution. From (4)
π’ (π₯0, π‘0) = βοΏ½β
π₯=0βπ (π₯)πΊ (π₯, 0) ππ₯ βοΏ½
π‘0
π‘=0π΄ (π‘) πΊ (0, π‘) ππ‘ +οΏ½
β
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯ (7)
Using the Green function we found in (6), then (7) becomes
π’ (π₯0, π‘0) = οΏ½β
π₯=0π (π₯)
1
β4πππ‘0
βββββexp
ββββββ (π₯ β π₯0)
2
4ππ‘0
βββββ + exp
ββββββ (π₯ + π₯0)
2
4ππ‘0
βββββ
βββββ ππ₯
βοΏ½π‘0
π‘=0
π΄ (π‘)
βππ (π‘0 β π‘)exp οΏ½
βπ₯204π (π‘0 β π‘)
οΏ½ ππ‘
+οΏ½β
π₯=0οΏ½
π‘0
π‘=0
1
β4ππ (π‘0 β π‘)
βββββexp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ
βββββ π ππ‘ππ₯
Changing the roles of π₯, π‘ and π₯0, π‘0 the above becomes
π’ (π₯, π‘) =1
β4πππ‘οΏ½
β
π₯0=0π (π₯0)
βββββexp
ββββββ (π₯0 β π₯)
2
4ππ‘
βββββ + exp
ββββββ (π₯0 + π₯)
2
4ππ‘
βββββ
βββββ ππ₯0
βοΏ½π‘
π‘0=0
π΄ (π‘0)
βππ (π‘ β π‘0)exp οΏ½
βπ₯2
4π (π‘ β π‘0)οΏ½ ππ‘0
+οΏ½β
π₯0=0οΏ½
π‘
π‘0=0
1
β4ππ (π‘ β π‘0)
βββββexp
ββββββ (π₯0 β π₯)
2
4π (π‘ β π‘0)
βββββ + exp
ββββββ (π₯0 + π₯)
2
4π (π‘ β π‘0)
βββββ
βββββ π (π₯0, π‘0) ππ‘0ππ₯0
Summary
π’ (π₯, π‘) =1
β4πππ‘Ξ1 β Ξ2 + Ξ3
Where
Ξ1 = οΏ½β
π₯0=0π (π₯0)
βββββexp
ββββββ (π₯0 β π₯)
2
4ππ‘
βββββ + exp
ββββββ (π₯0 + π₯)
2
4ππ‘
βββββ
βββββ ππ₯0
Ξ2 = οΏ½π‘
π‘0=0
π΄ (π‘0)
βππ (π‘ β π‘0)exp οΏ½
βπ₯2
4π (π‘ β π‘0)οΏ½ ππ‘0
Ξ3 = οΏ½β
π₯0=0οΏ½
π‘
π‘0=0
1
β4ππ (π‘ β π‘0)
βββββexp
ββββββ (π₯0 β π₯)
2
4π (π‘ β π‘0)
βββββ + exp
ββββββ (π₯0 + π₯)
2
4π (π‘ β π‘0)
βββββ
βββββ π (π₯0, π‘0) ππ‘0ππ₯0
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3.6. HW6 CHAPTER 3. HWS
Where Ξ1 comes from the initial conditions and Ξ2 comes from the boundary conditionsand Ξ3 comes from for forcing function. It is also important to note that Ξ1 is valid for onlyπ‘ > 0 and not for π‘ = 0.
3.6.2 Problem 2
NE548 Problems
Due Tuesday April 25, 2017
1. Use the Method of Images to solve
βu
βt= Ξ½
β2u
βx2+Q(x, t), x > 0
(a)u(0, t) = 0, u(x, 0) = 0.
(b)u(0, t) = 1, u(x, 0) = 0.
(c)ux(0, t) = A(t), u(x, 0) = f(x).
2. (a) Solve by the Method of Characteristics:
β2u
βt2= c2
β2u
βx2, x β₯ 0, t β₯ 0
u(x, 0) = f(x),βu(x, 0)
βt= g(x),
βu(0, t)
βx= h(t)
(b) For the special case h(t) = 0, explain how you could use a symmetry argument to helpconstruct the solution.
(c) Sketch the solution if g(x) = 0, h(t) = 0 and f(x) = 1 for 4 β€ x β€ 5 and f(x) = 0otherwise.
1
3.6.2.1 Part (a)
The general solution we will use as starting point is
π’ (π₯, π‘) = πΉ (π₯ β ππ‘) + πΊ (π₯ + ππ‘)
Where πΉ (π₯ β ππ‘) is the right moving wave and πΊ (π₯ + ππ‘) is the left moving wave. Applyingπ’ (π₯, 0) = π (π₯) gives
π (π₯) = πΉ (π₯) + πΊ (π₯) (1)
Andππ’ (π₯, π‘)ππ‘
=ππΉ
π (π₯ β ππ‘)π (π₯ β ππ‘)
ππ‘+
ππΊπ (π₯ + ππ‘)
π (π₯ + ππ‘)ππ‘
= βππΉβ² + ππΊβ²
Hence from second initial conditions we obtain
π (π₯) = βππΉβ² + ππΊβ² (2)
Equation (1) and (2) are for valid only for positive argument, which means for π₯ β₯ ππ‘.πΊ (π₯ + ππ‘) has positive argument always since π₯ β₯ 0 and π‘ β₯ 0, but πΉ (π₯ β ππ‘) can have negativeargument when π₯ < ππ‘. For π₯ < ππ‘, we will use the boundary conditions to define πΉ (π₯ β ππ‘).Therefore for π₯ β₯ ππ‘ we solve (1,2) for πΊ, πΉ and find
πΉ (π₯ β ππ‘) =12π (π₯ β ππ‘) β
12π οΏ½
π₯βππ‘
0π (π ) ππ (2A)
πΊ (π₯ + ππ‘) =12π (π₯ + ππ‘) +
12π οΏ½
π₯+ππ‘
0π (π ) ππ (2B)
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3.6. HW6 CHAPTER 3. HWS
This results in
π’ (π₯, π‘) =π (π₯ + ππ‘) + π (π₯ β ππ‘)
2+12π οΏ½
π₯+ππ‘
π₯βππ‘π (π ) ππ π₯ β₯ ππ‘ (3)
The solution (3) is only valid for arguments that are positive. This is not a problem forπΊ (π₯ + ππ‘) since its argument is always positive. But for πΉ (π₯ β ππ‘) its argument can becomenegative when 0 < π₯ < ππ‘. So we need to obtain a new solution for πΉ (π₯ β ππ‘) for the case whenπ₯ < ππ‘. First we find ππ’(π₯,π‘)
ππ₯
ππ’ (π₯, π‘)ππ₯
=ππΉ
π (π₯ β ππ‘)π (π₯ β ππ‘)ππ₯
+ππΊ
π (π₯ + ππ‘)π (π₯ + ππ‘)ππ₯
=ππΉ (π₯ β ππ‘)π (π₯ β ππ‘)
+ππΊ (π₯ + ππ‘)π (π₯ + ππ‘)
Hence at π₯ = 0
β (π‘) =ππΉ (βππ‘)π (βππ‘)
+ππΊ (ππ‘)π (ππ‘)
ππΉ (βππ‘)π (βππ‘)
= β (π‘) βππΊ (ππ‘)π (ππ‘)
(4)
Let π§ = βππ‘, therefore (4) becomes, where π‘ = βπ§π also
ππΉ (π§)ππ§
= β οΏ½βπ§ποΏ½ β
ππΊ (βπ§)π (βπ§)
π§ < 0
ππΉ (π§)ππ§
= β οΏ½βπ§ποΏ½ +
ππΊ (βπ§)ππ§
π§ < 0
To find πΉ (π§), we integrate the above which gives
οΏ½π§
0
ππΉ (π )ππ
ππ = οΏ½π§
0β οΏ½β
π ποΏ½ ππ +οΏ½
π§
0
ππΊ (βπ )ππ
ππ
πΉ (π§) β πΉ (0) = οΏ½π§
0β οΏ½β
π ποΏ½ ππ + πΊ (βπ§) β πΊ (0)
Ignoring the constants of integration πΉ (0) , πΊ (0) gives
πΉ (π§) = οΏ½π§
0β οΏ½β
π ποΏ½ ππ + πΊ (βπ§) (4A)
Replacing π§ = π₯ β ππ‘ in the above gives
πΉ (π₯ β ππ‘) = οΏ½π₯βππ‘
0β οΏ½β
π ποΏ½ ππ + πΊ (ππ‘ β π₯)
Let π = β π π then when π = 0, π = 0 and when π = π₯ β ππ‘ then π = βπ₯βππ‘
π = ππ‘βπ₯π . And ππ
ππ = β1π .
Using these the integral in the above becomes
πΉ (π₯ β ππ‘) = οΏ½ππ‘βπ₯π
0β (π) (βπππ) + πΊ (ππ‘ β π₯)
= βποΏ½π‘β π₯
π
0β (π) ππ + πΊ (ππ‘ β π₯) (5)
203
3.6. HW6 CHAPTER 3. HWS
The above is πΉ (β ) when its argument are negative. But in the above πΊ (ππ‘ β π₯) is the same aswe found above in (2b), which just replace it argument in 2B which was π₯+ ππ‘ with ππ‘ β π₯ andobtain
πΊ (ππ‘ β π₯) =12π (ππ‘ β π₯) +
12π οΏ½
ππ‘βπ₯
0π (π ) ππ π₯ < ππ‘
Therefore (5) becomes
πΉ (π₯ β ππ‘) = βποΏ½π‘β π₯
π
0β (π) ππ +
12π (ππ‘ β π₯) +
12π οΏ½
ππ‘βπ₯
0π (π ) ππ π₯ < ππ‘ (7)
Hence for π₯ < ππ‘
π’ (π₯, π‘) = πΉ (π₯ β ππ‘) + πΊ (π₯ + ππ‘) (8)
But in the above πΊ (π₯ + ππ‘) do not change, and we use the same solution for πΊ for π₯ β₯ ππ‘which is in (2B), given again below
πΊ (π₯ + ππ‘) =12π (π₯ + ππ‘) +
12π οΏ½
π₯+ππ‘
0π (π ) ππ (9)
Hence, plugging (7,9) into (8) gives
π’ (π₯, π‘) = βποΏ½ππ‘βπ₯π
0β (π ) ππ +
12π (ππ‘ β π₯) +
12π οΏ½
ππ‘βπ₯
0π (π ) ππ +
12π (π₯ + ππ‘) +
12π οΏ½
π₯+ππ‘
0π (π ) ππ π₯ < ππ‘
= βποΏ½ππ‘βπ₯π
0β (π ) ππ +
π (ππ‘ β π₯) + π (π₯ + ππ‘)2
+12π οΏ½οΏ½
ππ‘βπ₯
0π (π ) ππ +οΏ½
π₯+ππ‘
0π (π ) ππ οΏ½
The above for π₯ < ππ‘. Therefore the full solution is
π’ (π₯, π‘) =
β§βͺβͺβͺβ¨βͺβͺβͺβ©
π(π₯+ππ‘)+π(π₯βππ‘)2 + 1
2πβ«π₯+ππ‘
π₯βππ‘π (π ) ππ π₯ β₯ ππ‘
βπβ«π‘β π₯
π0
β (π ) ππ + π(ππ‘βπ₯)+π(π₯+ππ‘)2 + 1
2ποΏ½β«ππ‘βπ₯
0π (π ) ππ + β«
π₯+ππ‘
0π (π ) ππ οΏ½ π₯ < ππ‘
(10)
3.6.2.2 Part(b)
From (10) above, for β (π‘) = 0 the solution becomes
π’ (π₯, π‘) =
β§βͺβͺβͺβ¨βͺβͺβͺβ©
π(π₯+ππ‘)+π(π₯βππ‘)2 + 1
2πβ«π₯+ππ‘
π₯βππ‘π (π ) ππ π₯ β₯ ππ‘
π(ππ‘βπ₯)+π(π₯+ππ‘)2 + 1
2ποΏ½β«ππ‘βπ₯
0π (π ) ππ + β«
π₯+ππ‘
0π (π ) ππ οΏ½ π₯ < ππ‘
(1)
The idea of symmetry is to obtain the same solution (1) above but by starting from dβAlmbertsolution (which is valid only for positive arguments)
π’ (π₯, π‘) =π (π₯ + ππ‘) + π (π₯ β ππ‘)
2+12π οΏ½
π₯+ππ‘
π₯βππ‘π (π ) ππ π₯ > ππ‘ (1A)
But by using πππ₯π‘, πππ₯π‘ in the above instead of π, π, where the dβAlmbert solution becomesvalid for π₯ < ππ‘ when using πππ₯π‘, πππ₯π‘
π’ (π₯, π‘) =πππ₯π‘ (π₯ + ππ‘) + πππ₯π‘ (π₯ β ππ‘)
2+12π οΏ½
π₯+ππ‘
π₯βππ‘πππ₯π‘ (π ) ππ π₯ < ππ‘ (2)
204
3.6. HW6 CHAPTER 3. HWS
Then using (1A) and (2) we show it is the same as (1). We really need to show that (2) leadsto the second part of (1), since (1A) is the same as first part of (1).
The main issue is how to determine πππ₯π‘, πππ₯π‘ and determine if they should be even or oddextension of π, π . From boundary conditions, in part (a) equation (4A) we found that
πΉ (π§) = οΏ½π§
0β οΏ½β
π ποΏ½ ππ + πΊ (βπ§)
But now β (π‘) = 0, hence
πΉ (π§) = πΊ (βπ§) (3)
Now, looking at the first part of the solution in (1). we see that for positive argument thesolution has π (π₯ β ππ‘) for π₯ > ππ‘ and it has π (ππ‘ β π₯) when π₯ < ππ‘. So this leads us to pick πππ₯π‘being even as follows. Let
π§ = π₯ β ππ‘
Then we see that
π (π₯ β ππ‘) = π (β (π₯ β ππ‘))π (π§) = π (βπ§)
Therefore we need πππ₯π‘ to be an even function.
πππ₯π‘ (π§) =
β§βͺβͺβ¨βͺβͺβ©π (π§) π§ > 0π (βπ§) π§ < 0
But since πΉ (π§) = πΊ (βπ§) from (3), then πππ₯π‘ is also even function, which means
πππ₯π‘ (π§) =
β§βͺβͺβ¨βͺβͺβ©π (π§) π§ > 0π (βπ§) π§ < 0
Now that we found πππ₯π‘, πππ₯π‘ extensions, we go back to (2). For negative argument
π’ (π₯, π‘) =πππ₯π‘ (π₯ + ππ‘) + πππ₯π‘ (π₯ β ππ‘)
2+12π οΏ½
π₯+ππ‘
π₯βππ‘πππ₯π‘ (π ) ππ π₯ < ππ‘
Since πππ₯π‘, πππ₯π‘ are even, then using πππ₯π‘ (π§) = π (βπ§) since now π§ < 0 and using πππ₯π‘ (π§) = π (βπ§)since now π§ < 0 the above becomes
π’ (π₯, π‘) =π (β (π₯ + ππ‘)) + π (β (π₯ β ππ‘))
2+12π οΏ½
π₯+ππ‘
π₯βππ‘πππ₯π‘ (π ) ππ π₯ < ππ‘
But π (β (π₯ + ππ‘)) = π (π₯ + ππ‘) since even and π (β (π₯ β ππ‘)) = π (ππ‘ β π₯), hence the above becomes
π’ (π₯, π‘) ==π (π₯ + ππ‘) + π (ππ‘ β π₯)
2+12π οΏ½οΏ½
0
π₯βππ‘π (βπ ) ππ +οΏ½
π₯+ππ‘
0π (π ) ππ οΏ½ π₯ < ππ‘
Let π = βπ , then ππππ = β1. When π = π₯ β ππ‘, π = ππ‘ β π₯ and when π = 0, π = 0. Then the first
205
3.6. HW6 CHAPTER 3. HWS
integral on the RHS above becomes
π’ (π₯, π‘) =π (π₯ + ππ‘) + π (ππ‘ β π₯)
2+12π οΏ½οΏ½
0
ππ‘βπ₯π (π) (βππ) +οΏ½
π₯+ππ‘
0π (π ) ππ οΏ½
=π (π₯ + ππ‘) + π (ππ‘ β π₯)
2+12π οΏ½οΏ½
ππ‘βπ₯
0π (π) ππ +οΏ½
π₯+ππ‘
0π (π ) ππ οΏ½
Relabel π back to π , then
π’ (π₯, π‘) =π (π₯ + ππ‘) + π (ππ‘ β π₯)
2+12π οΏ½οΏ½
ππ‘βπ₯
0π (π ) ππ +οΏ½
π₯+ππ‘
0π (π ) ππ οΏ½ π₯ < ππ‘ (5)
Looking at (5) we see that this is the same solution in (1) for the case of π₯ < ππ‘. Hence (1A)and (5) put together give
π’ (π₯, π‘) =
β§βͺβͺβͺβ¨βͺβͺβͺβ©
π(π₯+ππ‘)+π(π₯βππ‘)2 + 1
2πβ«π₯+ππ‘
π₯βππ‘π (π ) ππ π₯ β₯ ππ‘
π(ππ‘βπ₯)+π(π₯+ππ‘)2 + 1
2ποΏ½β«ππ‘βπ₯
0π (π ) ππ + β«
π₯+ππ‘
0π (π ) ππ οΏ½ π₯ < ππ‘
(1)
Which is same solution obtain in part(a).
3.6.2.3 Part(c)
For π (π₯) = 0, β (π‘) = 0 and π (π₯) as given, the solution in equation (10) in part(a) becomes
π’ (π₯, π‘) =
β§βͺβͺβ¨βͺβͺβ©
π(π₯+ππ‘)+π(π₯βππ‘)2 π₯ β₯ ππ‘
π(ππ‘βπ₯)+π(π₯+ππ‘)2 π₯ < ππ‘
(10)
A small program we written to make few sketches important time instantces. The left movingwave πΊ (π₯ + ππ‘) hits the boundary at π₯ = 0 but do not reflect now as the case with Dirichletboundary conditions, but instead it remains upright and turns around as shown.
206
3.6. HW6 CHAPTER 3. HWS
time = 0.00
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 0.25
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 0.40
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 0.55
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 0.59
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 0.80
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 1.20
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 2.50
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 3.90
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 4.20
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 5.40
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
time = 7.00
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
207
3.6. HW6 CHAPTER 3. HWS
208
Chapter 4
Study notes
Local contents4.1 question asked 2/7/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104.2 note on Airy added 1/31/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.3 note p1 added 2/3/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2134.4 note added 1/31/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2204.5 note p3 figure 3.4 in text (page 92) reproduced in color . . . . . . . . . . . . . . . 2254.6 note p4 added 2/15/17 animations of the solutions to ODE from lecture 2/9/17
for small parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.7 note p5 added 2/15/17 animation figure 9.5 in text page 434 . . . . . . . . . . . . 2314.8 note p7 added 2/15/17, boundary layer problem solved in details . . . . . . . . . 2374.9 note p9. added 2/24/17, asking question about problem in book . . . . . . . . . . 2434.10 note p11. added 3/7/17, Showing that scaling for normalization is same for any π 2454.11 note p12. added 3/8/17, comparing exact solution to WKB solution using Math-
ematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.12 Convert ODE to Liouville form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2524.13 note p13. added 3/15/17, solving π2π¦β³(π₯) = (π + π₯ + π₯2)π¦(π₯) in Maple . . . . . . . . 254
209
4.1. question asked 2/7/2017 CHAPTER 4. STUDY NOTES
4.1 question asked 2/7/2017
Book at page 80 says "it is usually true that πβ²β² β (πβ²)2" as π₯ β π₯0. What are the exceptionsto this? Since it is easy to find counter examples.
For π¦β²β² = βπ₯π¦, as π₯ β 0+. Using π¦ (π₯) = ππ0(π₯) and substituting, gives the ODE
πβ²β²0 + οΏ½πβ²0οΏ½2= βπ₯ (1)
If we follow the book, and drop πβ²β²0 relative to (πβ²)2 then
οΏ½πβ²0οΏ½2= π₯
12
πβ²0 = ππ₯14
So, lets check the assumption. Since πβ²β²0 = π14π₯
β 34 . Therefore (ignoring all multiplicative
constants)
πβ²β²0?β οΏ½πβ²0οΏ½
2
1
π₯34
?β π₯
12
No. Did not check out. Since when π₯ β 0+ the LHS blow up while the RHS goes to zero.So in this example, this "rule of thumb" did not work out, and it is the other way around.Assuming I did not make a mistake, when the book says "it is usually true", it will good toknow under what conditions is this true or why did the book say this?
This will help in solving these problem.
210
4.2. note on Airy added 1/31/2017 CHAPTER 4. STUDY NOTES
4.2 note on Airy added 1/31/2017
Note on using CAS for working with book AiryAI/AiryBI
termsCAS such as Mathematica or Maple can actually handle very large numbers. It is possible to calculate
individual terms of the Airy series given by the book (but no sum it). Calculating individual terms, up to
10 million terms for x=1,100 and x=10000 is shown below as illustration. It takes few minutes to do each
term
define small function to calculate one term
bookAi[x_Integer, n_Integer] :=
3-2
3x3 n
9n Factorial[n] Gamman + 2 3- 3
-4
3x3 n+1
9n Factorial[n] Gamman + 4 3;
This is for x=1, N=10,000,000
In[3]:= bookAi[1, 10 000 000] // N
Out[3]= 5.7642115 Γ 10-140 856 542
This is for x=100, N=10,000,000
In[4]:= bookAi[100, 10 000 000] // N
Out[4]= 5.7583009 Γ 10-80 856 542
This is for x=10,000 and N=10,000,000
In[5]:= Timing[bookAi[10000, 10 000000] // N]
Out[5]= 582.3829332, 5.167240 Γ 10-20 856 542
The above took 582 seconds to complete! The largest number it can handle on my PC is
In[6]:= $MaxNumber
Out[6]= 1.605216761933662 Γ 101355718576 299 609
Let compare the above to Gamma[10,000,000]
In[8]:= Gamma[10 000000.0]
Out[8]= 1.2024234 Γ 1065657052
In[9]:= Gamma[10 000000.0] < $MaxNumber
Out[9]= True
We see that Gamma[10,000,000] is much less than the largest number it can handle. Here is Gamma
Printed by Wolfram Mathematica Student Edition
211
4.2. note on Airy added 1/31/2017 CHAPTER 4. STUDY NOTES
for 10 billion
In[10]:= Gamma[10 000000 000.0]
Out[10]= 2.3258 Γ 1095657055176
And 10 billion Factorial
In[11]:= Factorial[10 000000 000.0]
Out[11]= 2.3258 Γ 1095657055186
So CAS can handle these terms. But not the complete series summation as given in
the book
2 airy.nb
Printed by Wolfram Mathematica Student Edition
212
4.3. note p1 added 2/3/2017 CHAPTER 4. STUDY NOTES
4.3 note p1 added 2/3/2017
Current rules I am using in simplifications are
1. πβ²0 β πβ²1
2. πβ²0πβ²1 β οΏ½πβ²1οΏ½2
3. π0 β π1
4. (π0)π β π(π)0 (the first is power, the second is derivative order).
Verify the above are valid for π₯ β 0+ and well for π₯ β β. What can we say about (πβ²)compared to (πβ²)2?
4.3.1 problem (a), page 88
π¦β²β² =1π₯5π¦
Irregular singular point at π₯ β 0+. Let π¦ = ππ0(π₯) and the above becomes
π¦ (π₯) = ππ0(π₯)
π¦β² (π₯) = πβ²0ππ
π¦β²β² = πβ²β²0 ππ0 + οΏ½πβ²0οΏ½2ππ0
= οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ0
Substituting back into π2
ππ₯2π¦ = π₯β5π¦ gives
οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ = π₯β5ππ0(π₯)
πβ²β²0 + οΏ½πβ²0οΏ½2= π₯β5
Before solving for π0, we can do one more simplification. Using the approximation that
οΏ½πβ²0οΏ½2β πβ²β²0 for π₯ β π₯0, the above becomes
οΏ½πβ²0οΏ½2βΌ π₯β5
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Now we are ready to solve for π0
πβ²0 βΌ ππ₯β52
π0 βΌ ποΏ½π₯β52ππ₯
βΌ ππ₯β
32
β32
βΌ β23ππ₯β
32
To find leading behavior, let
π (π₯) = π0 (π₯) + π1 (π₯)
Then π¦ (π₯) = ππ0(π₯)+π1(π₯) and hence now
π¦β² (π₯) = (π0 (π₯) + π1 (π₯))β² ππ0+π1
π¦β²β² (π₯) = οΏ½(π0 + π1)β²οΏ½2ππ0+π1 + (π0 + π1)
β²β² ππ0+π1
Using the above, the ODE π2
ππ₯2π¦ = π₯β5π¦ now becomes
οΏ½(π0 + π1)β²οΏ½2ππ0+π1 + (π0 + π1)
β²β² ππ0+π1 βΌ π₯β5ππ0+π1
οΏ½(π0 + π1)β²οΏ½2+ (π0 + π1)
β²β² βΌ π₯β5
οΏ½πβ²0 + πβ²1οΏ½2+ πβ²β²0 + πβ²β²1 βΌ π₯β5
οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 βΌ π₯β5
But πβ²0 = ππ₯β 52 , found before, hence οΏ½πβ²0οΏ½
2= π₯β5 and the above simplifies to
οΏ½πβ²1οΏ½2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 = 0
Using approximation πβ²0πβ²1 β οΏ½πβ²1οΏ½2the above simplifies to
2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 = 0
Finally, using approximation πβ²β²0 β πβ²β²1 , the above becomes
2πβ²0πβ²1 + πβ²β²0 = 0
πβ²1 βΌ βπβ²β²02πβ²0
π1 βΌ β12
ln πβ²0 + π
πβ²1 βΌ β12
ln π₯β52 + π
πβ²1 βΌ54
ln π₯ + π
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Hence, the leading behavior is
π¦ (π₯) = ππ0(π₯)+π1(π₯)
= exp οΏ½β23ππ₯β
32 +
54
ln π₯ + ποΏ½
= ππ₯54 exp οΏ½βπ
23π₯β
32 οΏ½ (1)
To verify, using the formula 3.4.28, which is
π¦ (π₯) βΌ ππ1βπ2π exp οΏ½ποΏ½
π₯π (π‘)
1π ππ‘οΏ½
In this case, π = 2, since the ODE π¦β²β² = π₯β5π¦ is second order. Here we have π (π₯) = π₯β5,therefore, plug-in into the above gives
π¦ (π₯) βΌ π οΏ½π₯β5οΏ½1β24 exp οΏ½ποΏ½
π₯οΏ½π‘β5οΏ½
12 ππ‘οΏ½
βΌ π οΏ½π₯β5οΏ½β14 exp οΏ½ποΏ½
π₯π‘β
52ππ‘οΏ½
βΌ ππ₯54 exp
βββββββπ
βββββββπ₯β
32
β32
βββββββ
βββββββ
βΌ ππ₯54 exp οΏ½βπ
23π₯β
32 οΏ½ (2)
Comparing (1) and (2), we see they are the same.
4.3.2 problem (b), page 88
π¦β²β²β² = π₯π¦
Irregular singular point at π₯ β +β. Let π¦ = ππ0(π₯) and the above becomes
π¦ (π₯) = ππ0(π₯)
π¦β² (π₯) = πβ²0ππ0
π¦β²β² = πβ²β²0 ππ0 + οΏ½πβ²0οΏ½2ππ0
= οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½ ππ0
π¦β²β²β² = οΏ½πβ²β²0 + οΏ½πβ²0οΏ½2οΏ½β²ππ0 + οΏ½πβ²β²0 + οΏ½πβ²0οΏ½
2οΏ½ πβ²0ππ0
= οΏ½πβ²β²β²0 + 2πβ²0πβ²β²0 οΏ½ ππ0 + οΏ½πβ²0πβ²β²0 + οΏ½πβ²0οΏ½3οΏ½ ππ0
= οΏ½πβ²β²β²0 + 3πβ²0πβ²β²0 + οΏ½πβ²0οΏ½3οΏ½ ππ0
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4.3. note p1 added 2/3/2017 CHAPTER 4. STUDY NOTES
Substituting back into π¦β²β²β² = π₯π¦ gives
οΏ½πβ²β²β²0 + 3πβ²0πβ²β²0 + οΏ½πβ²0οΏ½3οΏ½ ππ0 = π₯ππ0(π₯)
πβ²β²β²0 + 3πβ²0πβ²β²0 + οΏ½πβ²0οΏ½3= π₯
Before solving for π0, we can do one more simplification. Using the approximation that
οΏ½πβ²0οΏ½3β πβ²β²β²0 for π₯ β π₯0, the above becomes
3πβ²0πβ²β²0 + οΏ½πβ²0οΏ½3βΌ π₯
In addition, since πβ²0 β πβ²β²0 then we can use the approximation οΏ½πβ²0οΏ½3β πβ²0πβ²β²0 and the above
becomes
οΏ½πβ²0οΏ½3βΌ π₯
πβ²0 βΌ ππ₯13
π0 βΌ ποΏ½π₯13ππ₯
π0 βΌ π34π₯43
To find leading behavior, let
π (π₯) = π0 (π₯) + π1 (π₯)
Then π¦ (π₯) = ππ0(π₯)+π1(π₯) and hence now
π¦β² (π₯) = (π0 (π₯) + π1 (π₯))β² ππ0+π1
π¦β²β² (π₯) = οΏ½(π0 + π1)β²οΏ½2ππ0+π1 + (π0 + π1)
β²β² ππ0+π1
= οΏ½πβ²0 + πβ²1οΏ½2ππ0+π1 + οΏ½πβ²β²0 + πβ²β²1 οΏ½ ππ0+π1
= οΏ½οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1οΏ½ ππ0+π1 + οΏ½πβ²β²0 + πβ²β²1 οΏ½ ππ0+π1
= οΏ½οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 οΏ½ ππ0+π1
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4.3. note p1 added 2/3/2017 CHAPTER 4. STUDY NOTES
We can take the third derivative
π¦β²β²β² (π₯) βΌ οΏ½οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 οΏ½
β²ππ0+π1
+ οΏ½οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 οΏ½ (π0 + π1)
β² ππ0+π1
βΌ οΏ½2πβ²0πβ²β²0 + 2πβ²1πβ²β²1 + 2πβ²β²0 πβ²1 + 2πβ²0πβ²β²1 + πβ²β²β²0 + πβ²β²β²1 οΏ½ ππ0+π1
+ οΏ½οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
2+ 2πβ²0πβ²1 + πβ²β²0 + πβ²β²1 οΏ½ οΏ½πβ²0 + πβ²1οΏ½ ππ0+π
βΌ οΏ½2πβ²0πβ²β²0 + 2πβ²1πβ²β²1 + 2πβ²β²0 πβ²1 + 2πβ²0πβ²β²1 + πβ²β²β²0 + πβ²β²β²1 οΏ½ ππ0+π1
+ οΏ½οΏ½πβ²0οΏ½3+ πβ²0 οΏ½πβ²1οΏ½
2+ 2 οΏ½πβ²0οΏ½
2πβ²1 + πβ²0πβ²β²0 + πβ²0πβ²β²1 οΏ½ ππ0+π1
+ οΏ½πβ²1 οΏ½πβ²0οΏ½2+ οΏ½πβ²1οΏ½
3+ 2πβ²0 οΏ½πβ²1οΏ½
2+ πβ²1πβ²β²0 + πβ²1πβ²β²1 οΏ½ ππ0+π1
βΌ οΏ½2πβ²0πβ²β²0 + 2πβ²1πβ²β²1 + 2πβ²β²0 πβ²1 + 2πβ²0πβ²β²1 + πβ²β²β²0 + πβ²β²β²1 οΏ½ ππ0+π1
+ οΏ½οΏ½πβ²0οΏ½3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + πβ²0πβ²β²0 + πβ²0πβ²β²1 + οΏ½πβ²1οΏ½
3+ πβ²1πβ²β²0 + πβ²1πβ²β²1 οΏ½ ππ0+π1
βΌ οΏ½3πβ²0πβ²β²0 + 3πβ²1πβ²β²1 + 3πβ²β²0 πβ²1 + 3πβ²0πβ²β²1 + πβ²β²β²0 + πβ²β²β²1 + οΏ½πβ²0οΏ½3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + οΏ½πβ²1οΏ½
3οΏ½ ππ0+π1
βΌ οΏ½οΏ½πβ²0οΏ½3+ οΏ½πβ²1οΏ½
3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 + 3πβ²1πβ²β²1 + 3πβ²β²0 πβ²1 + 3πβ²0πβ²β²1 + πβ²β²β²0 + πβ²β²β²1 οΏ½ ππ0+π1
Lets go ahead and plug-in this into the ODE
οΏ½πβ²0οΏ½3+ οΏ½πβ²1οΏ½
3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 + 3πβ²1πβ²β²1 + 3πβ²β²0 πβ²1 + 3πβ²0πβ²β²1 + πβ²β²β²0 + πβ²β²β²1 βΌ π₯
Now we do some simplification. οΏ½πβ²0οΏ½3β πβ²β²β²0 and οΏ½πβ²1οΏ½
3β πβ²β²β²1 , hence above becomes
οΏ½πβ²0οΏ½3+ οΏ½πβ²1οΏ½
3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 + 3πβ²1πβ²β²1 + 3πβ²β²0 πβ²1 + 3πβ²0πβ²β²1 βΌ π₯
Also, since πβ²β²0 β πβ²β²1 then 3πβ²0πβ²β²0 β 3πβ²0πβ²β²1
οΏ½πβ²0οΏ½3+ οΏ½πβ²1οΏ½
3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 + 3πβ²1πβ²β²1 + 3πβ²β²0 πβ²1 βΌ π₯
Also, since οΏ½πβ²0οΏ½2β πβ²β²0 then 3 οΏ½πβ²0οΏ½
2πβ²1 β 3πβ²β²0 πβ²1
οΏ½πβ²0οΏ½3+ οΏ½πβ²1οΏ½
3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 + 3πβ²1πβ²β²1 βΌ π₯
Also since πβ²1 β πβ²β²1 then 3 οΏ½πβ²0οΏ½2πβ²1 β 3πβ²1πβ²β²1
οΏ½πβ²0οΏ½3+ οΏ½πβ²1οΏ½
3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 βΌ π₯
But πβ²0 βΌ π₯13 henceοΏ½πβ²0οΏ½
3βΌ π₯ and the above simplies to
οΏ½πβ²1οΏ½3+ 3πβ²0 οΏ½πβ²1οΏ½
2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 = 0
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Using 3πβ²0 οΏ½πβ²1οΏ½2β οΏ½πβ²1οΏ½
3since πβ²0 β πβ²1 then
3πβ²0 οΏ½πβ²1οΏ½2+ 3 οΏ½πβ²0οΏ½
2πβ²1 + 3πβ²0πβ²β²0 = 0
Using 3 οΏ½πβ²0οΏ½2πβ²1 β πβ²0 οΏ½πβ²1οΏ½
2since οΏ½πβ²0οΏ½
2β πβ²0 then
3 οΏ½πβ²0οΏ½2πβ²1 + 3πβ²0πβ²β²0 = 0
No more simplification. We are ready to solve for π1.
πβ²1 βΌβπβ²0πβ²β²0οΏ½πβ²0οΏ½
2
βΌβπβ²β²0πβ²0
Hence
π1 βΌ βοΏ½πβ²β²0πβ²0ππ₯
βΌ β ln πβ²0 + π
Since πβ²0 βΌ π₯13 then the above becomes
π1 βΌ β ln π₯13 + π
π1 βΌ β13
ln π₯ + π
Hence, the leading behavior is
π¦ (π₯) = ππ0(π₯)+π1(π₯)
= exp οΏ½π34π₯43 β
13
ln π₯ + ποΏ½
= ππ₯β13 exp οΏ½π
34π₯43 οΏ½ (1)
To verify, using the formula 3.4.28, which is
π¦ (π₯) βΌ ππ1βπ2π exp οΏ½ποΏ½
π₯π (π‘)
1π ππ‘οΏ½
In this case, π = 3, since the ODE π¦β²β²β² = π₯π¦ is third order. Here we have π (π₯) = π₯, therefore,
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4.3. note p1 added 2/3/2017 CHAPTER 4. STUDY NOTES
plug-in into the above gives
π¦ (π₯) βΌ π (π₯)1β36 exp οΏ½ποΏ½
π₯(π‘)
13 ππ‘οΏ½
βΌ ππ₯β13 exp οΏ½π
43π₯43 οΏ½
Comparing (1) and (2), we see they are the same.
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4.4. note added 1/31/2017 CHAPTER 4. STUDY NOTES
4.4 note added 1/31/2017
This note solves in details the ODE
π₯3π¦β²β² (π₯) = π¦ (π₯)
Using asymptoes method using what is called the dominant balance submethod where it isassumed that π¦ (π₯) = ππ(π₯).
4.4.1 Solution
π₯ = 0 is an irregular singular point. The solution is assumeed to be π¦ (π₯) = ππ(π₯). Thereforeπ¦β² = πβ²ππ(π₯) and π¦β²β² = πβ²β²ππ(π₯) + (πβ²)2 ππ(π₯) and the given ODE becomes
π₯3 οΏ½πβ²β² + (πβ²)2οΏ½ = 1 (1)
Assuming that
πβ² (π₯) βΌ ππ₯πΌ
Hence πβ²β² βΌ ππΌπ₯πΌβ1. and (1) becomes
π₯3 οΏ½ππΌπ₯πΌβ1 + (ππ₯πΌ)2οΏ½ βΌ 1ππΌπ₯πΌ+2 + π2π₯2πΌ+3 βΌ 1
Term ππΌπ₯πΌ+2 β π2π₯2πΌ+3, hence we set πΌ = β32 to remove the subdominant term. Therefore
the above becomes, after substituting for the found πΌπ₯β0
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β32ππ₯
12 + π2 βΌ 1
π2 = 1
Therefore π = Β±1. The result so far is πβ² (π₯) βΌ ππ₯β32 . Now another term is added. Let
πβ² (π₯) βΌ ππ₯β32 + π΄ (π₯)
Now we will try to find π΄ (π₯). Hence πβ²β² (π₯) βΌ β32 ππ₯
β52 +π΄β² and π₯3 οΏ½πβ²β² + (πβ²)2οΏ½ = 1 now becomes
π₯3 οΏ½β32ππ₯
β52 + π΄β² + οΏ½ππ₯
β32 + π΄οΏ½
2
οΏ½ βΌ 1
π₯3 οΏ½β32ππ₯
β52 + π΄β² + π2π₯β3 + π΄2 + 2π΄ππ₯
β32 οΏ½ βΌ 1
οΏ½β32ππ₯
12 + π₯3π΄β² + π2 + π₯3π΄2 + 2π΄ππ₯
32 οΏ½ βΌ 1
Since π2 = 1 from the above, thenβ32ππ₯
12 + π₯3π΄β² + π₯3π΄2 + 2π΄ππ₯
32 βΌ 0
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4.4. note added 1/31/2017 CHAPTER 4. STUDY NOTES
Dominant balance says to keep dominant term (but now looking at those terms in π΄ only).From the above, since π΄β π΄2 and π΄ β π΄β² then from the above, we can cross out π΄2 andπ΄β² resulting in
β32ππ₯
12 + 2π΄ππ₯
32 βΌ 0
Hence we just need to find π΄ to balance the aboveβ32ππ₯
12 + 2π΄ππ₯
32 βΌ 0
2π΄ππ₯32 βΌ
32ππ₯
12
π΄ βΌ34π₯
We found π΄ (π₯) for the second term. Therefore, so far we have
πβ² (π₯) = ππ₯β32 +
34π₯
Or
π (π₯) = β2ππ₯β12 +
34
ln π₯ + πΆ0
But πΆ0 can be dropped (subdominant to ln π₯ when π₯ β 0) and so far then we can write thesolution as
π¦ (π₯) = ππ(π₯)π(π₯)
= ππ(π₯)βοΏ½π=0
πππ₯ππ
= exp οΏ½β2ππ₯β12 +
34
ln π₯οΏ½βοΏ½π=0
πππ₯ππ
= πβ2ππ₯β12 π₯
34
βοΏ½π=0
πππ₯ππ
= πβ 2πβπ₯
βοΏ½π=0
πππ₯ππ+ 3
4
= πΒ± 2βπ₯
βοΏ½π=0
πππ₯ππ+ 3
4
Since π = Β±1. We can now try adding one more term to π (π₯). Let
πβ² (π₯) = ππ₯β32 +
34π₯+ π΅ (π₯)
Hence
πβ²β² =β32ππ₯
β52 β
34π₯2
+ π΅β² (π₯)
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4.4. note added 1/31/2017 CHAPTER 4. STUDY NOTES
And π₯3 οΏ½πβ²β² + (πβ²)2οΏ½ βΌ 1 now becomes
π₯3ββββββοΏ½β32ππ₯
β52 β
34π₯2
+ π΅β² (π₯)οΏ½ + οΏ½ππ₯β32 +
34π₯+ π΅ (π₯)οΏ½
2ββββββ βΌ 1
π₯3 οΏ½π2
π₯3β316π₯β2 + 2ππ΅π₯β
32 +
32π΅π₯β1 + π΅2 + π΅β²οΏ½ βΌ 1
ββββββ=1βπ2 β
316π₯ + 2ππ΅π₯
32 +
32π΅π₯2 + π₯3π΅2 + π₯3π΅β²
ββββββ βΌ 1
β316π₯ + 2ππ΅π₯
32 +
32π΅π₯2 + π₯3π΅2 + π₯3π΅β² βΌ 0
From the above, since π΅ (π₯) β π΅2 (π₯) and π΅ (π₯) β π΅β² (π₯) and for small π₯, then we can crossout terms with π΅2 and π΅β² from above, and we are left with
β316π₯ + 2ππ΅π₯
32 +
32π΅π₯2 β½ 0
Between 2ππ΅π₯32 and 3
2π΅π₯2, for small π₯, then 2ππ΅π₯
32 β 3
2π΅π₯2, so we can cross out 3
2π΅π₯2 from
above
β316π₯ + 2ππ΅π₯
32 β½ 0
2ππ΅π₯32 β½
316π₯
π΅ β½332π
π₯β12
We found π΅ (π₯), Hence now we have
πβ² (π₯) = ππ₯β32 +
34π₯+
332π
π₯β12
Or
π (π₯) = β2ππ₯β12 +
34
ln π₯ + 316π
π₯12 + πΆ1
But πΆ1 can be dropped (subdominant to ln π₯ when π₯ β 0) and so far then we can write thesolution as
π¦ (π₯) = ππ(π₯)π(π₯)
= ππ(π₯)βοΏ½π=0
πππ₯ππ
= exp οΏ½β2ππ₯β12 +
34
ln π₯ + 316π
π₯12 οΏ½
βοΏ½π=0
πππ₯ππ
= πβ2ππ₯β12 + 3
16ππ₯12 π₯
34
βοΏ½π=0
πππ₯ππ
= πβ2ππ₯β12 + 3
16ππ₯12
βοΏ½π=0
πππ₯ππ+ 3
4
222
4.4. note added 1/31/2017 CHAPTER 4. STUDY NOTES
For π = 1
π¦1 (π₯) = πβ2π₯
β12 + 3
16π₯12
βοΏ½π=0
πππ₯ππ+ 3
4
For π = β1
π¦2 (π₯) = π2π₯
β12 β 3
16π₯12
βοΏ½π=0
πππ₯ππ+ 3
4
Hence
π¦ (π₯) βΌ π΄π¦1 (π₯) + π΅π¦2 (π₯)
Reference
1. Page 80-82 Bender and Orszag textbook.
2. Lecture notes, Lecture 5, Tuesday Janurary 31, 2017. EP 548, University of Wisconsin,Madison by Professor Smith.
3. Lecture notes from http://www.damtp.cam.ac.uk/
223
4.4. note added 1/31/2017 CHAPTER 4. STUDY NOTES
224
4.5. note p3 figure 3.4 in text (page 92) . . . CHAPTER 4. STUDY NOTES
4.5 note p3 οΏ½gure 3.4 in text (page 92) reproduced incolor
figure 3.4 in text (page 92) was not too clear (it is black and white in my text book that I am using). So I
reproduced it in color. This type of plot is needed to check what happens with the improvement in
approximation for problem 3.42 (a) as more terms are added to leading behavior.
It is done in Wolfram Mathematica. Used 300 terms to find y(x). We see the red line, which is leading
behavior/y(x) ratio going to 1 for large x as would be expected.
In[1]:= leading[x] := 1 2 Pi^-1 2 x^-1 4 Exp2 x^1 2;
y[x_, max_] := Sumx^n Factorial[n]^2, {n, 0, max};
LogLinearPlotEvaluateleading[x] y[x, 300], y[x, 10] y[x, 300], {x, 0.001, 70 000},
PlotRange β All, Frame β True, GridLines β Automatic, GridLinesStyle β LightGray,
PlotLegends β {"leading behavior/y(x)", "truncated Taylor/y(x)"}, FrameLabel β
{{None, None}, {"x", "Reproducing figure 3.4 in text book"}}, PlotStyle β {Red, Blue}
Out[3]=
0.01 1 100 104
0.0
0.5
1.0
1.5
x
Reproducing figure 3.4 in text book
leading behavior/y(x)
truncated Taylor/y(x)
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225
4.5. note p3 figure 3.4 in text (page 92) . . . CHAPTER 4. STUDY NOTES
226
4.6. note p4 added 2/15/17 animations of . . . CHAPTER 4. STUDY NOTES
4.6 note p4 added 2/15/17 animations of the solutions toODE from lecture 2/9/17 for small parameter
These are small animations of the solutions to the 2 ODEβs from lecture 2/9/17 for the small parameter.
Nasser M. Abbasi
2/20/2017, EP 548, Univ. Wisconsin Madison.
Manipulate
Labeled
Plot-1 + β x/z
-1 + β 1
z
, {x, 0, 2}, PlotRange β {{0, 1.1}, {0, 1}},
GridLines β Automatic, GridLinesStyle β LightGray, PlotStyle β Red,
Row[{"solution to ", TraditionalForm[Ο΅ y''[x] - y'[x] β©΅ 0]}], Top,
{{z, 0.1, "Ο΅"}, .1, 0.001, -0.001, Appearance β "Labeled"}
Ο΅ 0.1
solution to Ο΅ yβ²β²(x) - yβ²(x) 0
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
For eps*yββ+y=0 ode, we notice global variations showing up in frequency as well as in amplitude as
eps become very small
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227
4.6. note p4 added 2/15/17 animations of . . . CHAPTER 4. STUDY NOTES
Manipulate
Labeled
Plot
Sin x
z
Sin 1
z
, {x, 0, 2}, PlotRange β {{0, 2}, {-10, 10}},
GridLines β Automatic, GridLinesStyle β LightGray, PlotStyle β Red,
Style[Row[{"solution to ", TraditionalForm[z y''[x] + y[x] β©΅ 0]}], 16], Top,
{{z, 0.01, "Ο΅"}, 0.01, 0.0001, -0.0001}
Ο΅
solution to 0.01 yβ²β²(x) + y(x) 0
0.5 1.0 1.5 2.0
-10
-5
0
5
10
2 p4.nb
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228
4.6. note p4 added 2/15/17 animations of . . . CHAPTER 4. STUDY NOTES
First animation
second animation
229
4.6. note p4 added 2/15/17 animations of . . . CHAPTER 4. STUDY NOTES
230
4.7. note p5 added 2/15/17 animation . . . CHAPTER 4. STUDY NOTES
4.7 note p5 added 2/15/17 animation οΏ½gure 9.5 in textpage 434
Animation of figure 9.5Nasser M. Abbasi, Feb 2017, EP 548 course
Animation of figure 9.5 in text page 434, compare higher order boundary layer solutions
This is an animation of the percentage error between boundary layer solution and the
exact solution for Ο΅yβ²β²+(1+x)yβ²+y=0 for different orders. Orders are given in text up y0,y1,y2,y3. We
see the error is smaller when order increases as what one would expect.
The percentage error is largest around x=0.1 than other regions. Why? Location of matching between
yin and yout. Due
to approximation made when doing matching? But all boundary layer solutions underestimate the exact
solution since error is negative.
The red color is the most accurate. 3rd order.
First animation
In[45]:= Clear["Global`*"];
yout[x_] := 2 1 + x;
yin[x_, eps_] :=
Plot[2 - Exp[-x / eps], {x, 0, 1}, AxesOrigin β {0, 1}, PlotStyle β Black];
p1 = Plot[yout[x], {x, 0, 1}, PlotStyle β Blue];
combine[x_, eps_] := Plot[yout[x] - Exp[-x / eps], {x, 0, 1}, PlotStyle β {Thick, Red},
AxesOrigin β {0, 1}, GridLines β Automatic, GridLinesStyle β LightGray];
Animate
GridTraditionalForm
NumberForm[eps, {Infinity, 4}] HoldFormy''[x] + 1 + x y'[x] + y[x] β©΅ 0,
{Show[
Legended[combine[x, eps], Style["combinbed solution", Red]],
Legended[p1, Style["Outer solution", Blue]],
Legended[yin[x, eps], Style["Inner solution", Black]]
, PlotRange β {{0, 1}, {1, 2}}, ImageSize β 400]
}, {eps, 0.001, .1, .00001}
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231
4.7. note p5 added 2/15/17 animation . . . CHAPTER 4. STUDY NOTES
second animation, y0 vs. exactIn[51]:= Clear["Global`*"];
yout[x_] := 2 1 + x;
sol =
y[x] /. First@DSolveΟ΅ y''[x] + 1 + x y'[x] + y[x] β©΅ 0, y[0] β©΅ 1, y[1] β©΅ 1, y[x], x;
exact[x_, Ο΅_] := Evaluate@sol;
yin[x_, eps_] :=
Plot[2 - Exp[-x / eps], {x, 0, 1}, AxesOrigin β {0, 1}, PlotStyle β Black];
p1 = Plot[yout[x], {x, 0, 1}, PlotStyle β Blue];
combine[x_, eps_] := Plot[yout[x] - Exp[-x / eps], {x, 0, 1}, PlotStyle β {Thick, Red},
AxesOrigin β {0, 1}, GridLines β Automatic, GridLinesStyle β LightGray];
Animate
Grid
TraditionalForm
NumberForm[Ο΅, {Infinity, 4}] HoldFormy''[x] + 1 + x y'[x] + y[x] β©΅ 0,
{Row[{"exact solution ", TraditionalForm[sol]}]},
{Show[
Legended[combine[x, Ο΅], Style["Boundary layer solution, zero order", Red]],
Legended[Plot[Evaluate@exact[x, Ο΅], {x, 0, 1}, AxesOrigin β {0, 0}, PlotStyle β Blue],
Style["Exact analytical solution", Blue]]
, PlotRange β {{0, 1}, {1, 2}}, ImageSize β 400]
}, {Ο΅, 0.0001, .1, .00001}
2 p5.nb
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232
4.7. note p5 added 2/15/17 animation . . . CHAPTER 4. STUDY NOTES
add more terms
In[60]:= Clear["Global`*"];
sol =
y[x] /. First@DSolveΟ΅ y''[x] + 1 + x y'[x] + y[x] β©΅ 0, y[0] β©΅ 1, y[1] β©΅ 1, y[x], x;
exact[x_, Ο΅_] := Evaluate@sol;
boundary0[x_, eps_] :=2
1 + x- Exp[-x / eps] ;
boundary1[x_, eps_] :=
2
1 + x- Exp[-x / eps] + eps
2
1 + x3-
1
2 1 + x+
1
2(x / eps)2 -
3
2Exp[-x / eps] ;
boundary2[x_, eps_] :=2
1 + x- Exp[-x / eps] +
eps2
1 + x3-
1
2 1 + x+
1
2(x / eps)2 -
3
2Exp[-x / eps] +
eps26
1 + x5-
1
2 1 + x3-
1
4 1 + x-
1
8(x / eps)4 -
3
4(x / eps)2 +
21
4Exp[-x / eps] ;
boundary3[x_, eps_] :=2
1 + x- Exp[-x / eps] +
eps2
1 + x3-
1
2 1 + x+
1
2(x / eps)2 -
3
2Exp[-x / eps] +
eps26
1 + x5-
1
2 1 + x3-
1
4 1 + x-
1
8(x / eps)4 -
3
4(x / eps)2 +
21
4Exp[-x / eps] +
eps330
1 + x7-
3
2 1 + x5-
1
4 1 + x3-
5
16 1 + x+
1
48(x / eps)6 -
3
16(x / eps)4 +
21
8(x / eps)2 -
1949
72Exp[-x / eps] ;
p5.nb 3
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233
4.7. note p5 added 2/15/17 animation . . . CHAPTER 4. STUDY NOTES
In[67]:= Animate
ex = exact[x, Ο΅];
Grid
{Style[
"percentage relative error, exact vs. boundary layer for different order", 14]},
Show
LegendedPlot100 *boundary3[x, Ο΅] - ex
ex, {x, 0, 1},
PlotStyle β {Thick, Red}, AxesOrigin β {0, 1}, GridLines β Automatic,
GridLinesStyle β LightGray, PlotRange β {Automatic, {-4, .5}}, Frame β True,
Epilog β Text[Style[Row[{"Ο΅ = ", NumberForm[Ο΅, {Infinity, 4}]}], 16], {.8, -3}],
Style"Boundary layer 3rd order error", Red,
LegendedPlot100 *boundary2[x, Ο΅] - ex
ex, {x, 0, 1}, PlotStyle β {Thick, Black},
AxesOrigin β {0, 1}, GridLines β Automatic, GridLinesStyle β LightGray,
Style"Boundary layer 2nd order error", Black,
LegendedPlot100 *boundary1[x, Ο΅] - ex
ex, {x, 0, 1}, PlotStyle β {Thick, Blue},
AxesOrigin β {0, 1}, GridLines β Automatic, GridLinesStyle β LightGray,
Style"Boundary layer 1st order error", Blue,
LegendedPlot100 *boundary0[x, Ο΅] - ex
ex, {x, 0, 1}, PlotStyle β {Thick, Magenta},
AxesOrigin β {0, 1}, GridLines β Automatic, GridLinesStyle β LightGray,
Style["Boundary layer 0 order error", Magenta]
, PlotRange β {{0, 1}, {-4, .5}}, ImageSize β 400
, {Ο΅, 0.001, .1, .0001}
4 p5.nb
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234
4.7. note p5 added 2/15/17 animation . . . CHAPTER 4. STUDY NOTES
Out[67]=
Ο΅
percentage relative error, exact vs. boundary layer for different order
0.0 0.2 0.4 0.6 0.8 1.0-4
-3
-2
-1
0
Ο΅ = 0.0656
Boundary layer 3rd order error
Boundary layer 2nd order error
Boundary layer 1st order errorBoundary layer 0 order error
p5.nb 5
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4.7. note p5 added 2/15/17 animation . . . CHAPTER 4. STUDY NOTES
Animation of figure 9.5 in text page 434, compare higher order boundary layer solutions
This is an animation of the percentage error between boundary layer solution and the exactsolution for ππ¦β³+(1+π₯)π¦β²+π¦ = 0 for diοΏ½erent orders. Orders are given in text up π¦0, π¦1, π¦2, π¦3.We see the error is smaller when the order increases as what one would expect.
The percentage error is largest around π₯ = 0.1 than other regions. Why? Location of matchingis between π¦ππ and π¦ππ’π‘. Due to approximation made when doing matching? But all boundarylayer solutions underestimate the exact solution since error is negative.
The red color is the most accurate. 3rd order.
First animation
second animation
third animation
236
4.8. note p7 added 2/15/17, boundary layer . . . CHAPTER 4. STUDY NOTES
4.8 note p7 added 2/15/17, boundary layer problemsolved in details
This note solves
ππ¦β²β² (π₯) + (1 + π₯) π¦β² (π₯) + π¦ (π₯) = 0π¦ (0) = 1π¦ (1) = 1
Where π is small parameter, using boundary layer theory.
4.8.0.1 Solution
Since (1 + π₯) > 0 in the domain, we expect boundary layer to be on the left. Let π¦ππ’π‘ (π₯) bethe solution in the outer region. Starting with π¦ (π₯) = ββ
π=0 πππ¦π and substituting back into
the ODE gives
π οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + (1 + π₯) οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0
π (1) terms
Collecting all terms with zero powers of π(1 + π₯) π¦β²0 + π¦0 = 0
The above is solved using the right side conditions, since this is where the outer region islocated. Solving the above using π¦0 (1) = 1 gives
π¦ππ’π‘0 (π₯) =2
1 + π₯Now we need to find π¦ππ (π₯). To do this, we convert the ODE using transformation π = π₯
π .
Hence ππ¦ππ₯ = ππ¦
ππππππ₯ = ππ¦
ππ1π . Hence the operator π
ππ₯ β‘ 1π
πππ . This means the operator π2
ππ₯2 β‘
οΏ½1π
ππποΏ½ οΏ½1
πππποΏ½ = 1
π2π2
ππ2 . The ODE becomes
π1π2π2π¦ (π)ππ2
+ (1 + ππ)1πππ¦ (π)ππ
+ π¦ (π) = 0
1ππ¦β²β² + οΏ½
1π+ ποΏ½ π¦β² + π¦ = 0
Plugging π¦ (π) = ββπ=0 π
ππ¦ (π)π into the above gives
1ποΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + οΏ½
1π+ ποΏ½ οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0
1ποΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ +
1ποΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + π οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0
237
4.8. note p7 added 2/15/17, boundary layer . . . CHAPTER 4. STUDY NOTES
Collecting all terms with smallest power of π , which is πβ1 in this case, gives1ππ¦β²β²0 +
1ππ¦β²0 = 0
π¦β²β²0 + π¦β²0 = 0
Let π§ = π¦β²0, the above becomes
π§β² + π§ = 0
π οΏ½πππ§οΏ½ = 0πππ§ = ππ§ = ππβπ
Hence π¦β²0 (π) = ππβπ. Integrating
π¦ππ0 (π) = βππβπ + π1 (1A)
Since π is arbitrary constant, the negative sign can be removed, giving
π¦ππ0 (π) = ππβπ + π1 (1A)
This is the lowest order solution for the inner π¦ππ (π). We have two boundary conditions, butwe can only use the left side one, where π¦ππ (π) lives. Hence using π¦0 (π = 0) = 1, the abovebecomes
1 = π + π1π1 = 1 β π
The solution (1A) becomes
π¦0 (π) = ππβπ + (1 β π)
= 1 + π οΏ½πβπ β 1οΏ½
Let π = π΄0 to match the book notation.
π¦0 (π) = 1 + π΄0 οΏ½πβπ β 1οΏ½
To find π΄0, we match π¦ππ0 (π) with π¦ππ’π‘0 (π₯)
limπββ
1 + π΄0 οΏ½πβπ β 1οΏ½ = limπ₯β0+
21 + π₯
1 β π΄0 = 2π΄0 = β1
Hence
π¦ππ0 (π) = 2 β πβπ
π (π) terms
We now repeat the process to find π¦ππ1 (π) and π¦ππ’π‘1 (π₯) . Starting with π¦ππ’π‘ (π₯)
π οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + (1 + π₯) οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0
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4.8. note p7 added 2/15/17, boundary layer . . . CHAPTER 4. STUDY NOTES
Collecting all terms with π1 now
ππ¦β²β²0 + (1 + π₯) ππ¦β²1 + ππ¦1 = 0π¦β²β²0 + (1 + π₯) π¦β²1 + π¦1 = 0
But we know π¦0 =2
1+π₯ , from above. Hence π¦β²β²0 =4
(1+π₯)3and the above becomes
(1 + π₯) π¦β²1 + π¦1 = β4
(1 + π₯)3
π¦β²1 +π¦11 + π₯
= β4
(1 + π₯)4
Integrating factor π = πβ«1
1+π₯ππ₯ = πln(1+π₯) = 1 + π₯ and the above becomes
πππ₯οΏ½ππ¦1οΏ½ = βπ
4(1 + π₯)4
πππ₯οΏ½(1 + π₯) π¦1οΏ½ = β
4(1 + π₯)3
Integrating
(1 + π₯) π¦1 = βοΏ½4
(1 + π₯)3ππ₯ + π
=2
(1 + π₯)2+ π
Hence
π¦1 (π₯) =2
(1 + π₯)3+
π1 + π₯
Applying π¦ (1) = 0 (notice the boundary condition now becomes π¦ (1) = 0 and not π¦ (1) = 1,since we have already used π¦ (1) = 1 to find leading order). From now on, all boundaryconditions will be π¦ (1) = 0.
0 =2
(1 + 1)3+
π1 + 1
π = β12
Hence
π¦1 (π₯) =2
(1 + π₯)3β12
1(1 + π₯)
Now we need to find π¦ππ1 (π). To do this, starting from
1ποΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + οΏ½
1π+ ποΏ½ οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0
1ποΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ +
1ποΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + π οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0
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4.8. note p7 added 2/15/17, boundary layer . . . CHAPTER 4. STUDY NOTES
But now collecting all terms with π (1) order, (last time, we collected terms with ποΏ½πβ1οΏ½ ).
π¦β²β²1 + π¦β²1 + ππ¦β²0 + π¦0 = 0π¦β²β²1 + π¦β²1 = βππ¦β²0 β π¦0 (1)
But we found π¦ππ0 earlier which was
π¦ππ0 (π) = 1 + π΄0 οΏ½πβπ β 1οΏ½
Hence π¦β²0 = βπ΄0πβπ and the ODE (1) becomes
π¦β²β²1 + π¦β²1 = ππ΄0πβπ β οΏ½1 + π΄0 οΏ½πβπ β 1οΏ½οΏ½
We need to solve this with boundary conditions π¦1 (0) = 0. (again, notice change in B.C. aswas mentioned above). The solution is
π¦1 (π) = βπ + π΄0 οΏ½π β12π2πβποΏ½ + π΄1 οΏ½1 β πβποΏ½
= βπ + π΄0 οΏ½π β12π2πβποΏ½ β π΄1 οΏ½πβπ β 1οΏ½
Since π΄1 is arbitrary constant, and to match the book, we can call π΄2 = βπ΄1 and thenrename π΄2 back to π΄1 and obtain
π¦1 (π) = βπ + π΄0 οΏ½π β12π2πβποΏ½ + π΄1 οΏ½πβπ β 1οΏ½
This is to be able to follow the book. Therefore, this is what we have so far
π¦ππ’π‘ = π¦ππ’π‘0 + ππ¦ππ’π‘1
=2
1 + π₯+ π οΏ½
2(1 + π₯)3
β1
2 (1 + π₯)οΏ½
And
π¦ππ (π) = π¦ππ0 + ππ¦ππ1
= οΏ½1 + π΄0 οΏ½πβπ β 1οΏ½οΏ½ + π οΏ½βπ + π΄0 οΏ½π β12π2πβποΏ½ + π΄1 οΏ½πβπ β 1οΏ½οΏ½
= π΄0πβπ β ππ β ππ΄1 β π΄0 + ππ΄1πβπ + πππ΄0 β12π2ππ΄0πβπ + 1
To find π΄0, π΄1, we match π¦ππ with π¦ππ’π‘, therefore
limπββ
π¦ππ = limπ₯β0
π¦ππ’π‘
Or
limπββ
οΏ½π΄0πβπ β ππ β ππ΄1 β π΄0 + ππ΄1πβπ + πππ΄0 β12π2ππ΄0πβπ + 1οΏ½ =
limπ₯β0
21 + π₯
+ π οΏ½2
(1 + π₯)3β
12 (1 + π₯)οΏ½
Which simplifies to
βππ β ππ΄1 β π΄0 + πππ΄0 + 1 = limπ₯β0
21 + π₯
+ π οΏ½2
(1 + π₯)3β
12 (1 + π₯)οΏ½
240
4.8. note p7 added 2/15/17, boundary layer . . . CHAPTER 4. STUDY NOTES
It is easier now to convert the LHS to use π₯ instead of π so we can compare. Since π = π₯π ,
then the above becomes
βπ₯ β ππ΄1 β π΄0 + π₯π΄0 + 1 = limπ₯β0
21 + π₯
+ π οΏ½2
(1 + π₯)3β
12 (1 + π₯)οΏ½
Using Taylor series on the RHS
1 β π΄0 β π₯ + π΄0π₯ + π΄1π = limπ₯β0
2 οΏ½1 β π₯ + π₯2 +β―οΏ½
+ 2π οΏ½1 β π₯ + π₯2 +β―οΏ½ οΏ½1 β π₯ + π₯2 +β―οΏ½ οΏ½1 β π₯ + π₯2 +β―οΏ½ βπ2οΏ½1 β π₯ + π₯2 +β―οΏ½
Since we have terms on the the LHS of only π (1) ,π (π₯) , π (π), then we need to keep at leastterms with π (1) ,π (π₯) , π (π) on the RHS and drop terms with ποΏ½π₯2οΏ½ , π (ππ₯) , π οΏ½π2οΏ½ to be ableto do the matching. So in the above, RHS simplifies to
βπ₯ β ππ΄1 β π΄0 + π₯π΄0 + 1 = 2 (1 β π₯) + 2π βπ2
βπ₯ β ππ΄1 β π΄0 + π₯π΄0 + 1 = 2 β 2π₯ + 2π βπ2
βππ΄1 β π΄0 + π₯ (π΄0 β 1) + 1 = 2 β 2π₯ +32π
Comparing, we see that
π΄0 β 1 = β2π΄0 = β1
We notice this is the same π΄0 we found for the lowest order. This is how it should always comeout. If we get diοΏ½erent value, it means we made mistake. We could also match βπ΄0 + 1 = 2which gives π΄0 = β1 as well. Finally
βππ΄1 =32π
π΄1 = β32
So we have used matching to find all the constants for π¦ππ. Here is the final solution so far
π¦ππ’π‘ (π₯) =
π¦0οΏ½21+ π₯
+ π
π¦1οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
2(1 + π₯)3
β1
2 (1 + π₯)οΏ½
π¦ππ (π) =
π¦0οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½1 + π΄0 οΏ½πβπ β 1οΏ½ + π
π¦1οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½βπ + π΄0 οΏ½π β
12π2πβποΏ½ + π΄1 οΏ½πβπ β 1οΏ½οΏ½
= 1 β οΏ½πβπ β 1οΏ½ + π οΏ½βπ β οΏ½π β12π2πβποΏ½ β
32οΏ½πβπ β 1οΏ½οΏ½
=32π β πβπ β 2ππ β
32ππβπ +
12π2ππβπ + 2
241
4.8. note p7 added 2/15/17, boundary layer . . . CHAPTER 4. STUDY NOTES
In terms of π₯, since Since π = π₯π the above becomes
π¦ππ (π₯) =32π β πβ
π₯π β 2π₯ β
32ππβ
π₯π +
12π₯2
ππβ
π₯π + 2
= 2 β 2π₯ +32π + πβ
π₯π οΏ½12π₯2
πβ32π β 1οΏ½
Hence
π¦π’ππππππ = π¦ππ + π¦ππ’π‘ β π¦πππ‘πβWhere
π¦πππ‘πβ = limπββ
π¦ππ
= 2 β 2π₯ +32π
Hence
π¦π’ππππππ = 2 β 2π₯ +32π + πβ
π₯π οΏ½12π₯2
πβ32π β 1οΏ½ +
21 + π₯
+ π οΏ½2
(1 + π₯)3β
12 (1 + π₯)οΏ½
β οΏ½2 β 2π₯ +32ποΏ½
= πβπ₯π οΏ½12π₯2
πβ32π β 1οΏ½ +
21 + π₯
+ π οΏ½2
(1 + π₯)3β
12 (1 + π₯)οΏ½
= οΏ½2
1 + π₯β πβ
π₯π +
12π₯2
ππβ
π₯π οΏ½ + π οΏ½
2(1 + π₯)3
β1
2 (1 + π₯)β32πβ
π₯π οΏ½
Which is the same as
π¦π’ππππππ = οΏ½2
1 + π₯β πβπ +
12π₯2
ππβποΏ½ + π οΏ½
2(1 + π₯)3
β1
2 (1 + π₯)β32πβποΏ½
= οΏ½2
1 + π₯β πβποΏ½ + π οΏ½
2(1 + π₯)3
β1
2 (1 + π₯)β32πβπ +
12π2πβποΏ½
= οΏ½2
1 + π₯β πβποΏ½ + π οΏ½
2(1 + π₯)3
β1
2 (1 + π₯)+ οΏ½
12π2 β
32οΏ½πβποΏ½ (1)
Comparing (1) above, with book result in first line of 9.3.16, page 433, we see the sameresult.
4.8.0.2 References
1. Advanced Mathematica methods, Bender and Orszag. Chapter 9.
2. Lecture notes. Feb 16, 2017. By Professor Smith. University of Wisconsin. NE 548
242
4.9. note p9. added 2/24/17, asking . . . CHAPTER 4. STUDY NOTES
4.9 note p9. added 2/24/17, asking question aboutproblem in book
How did the book, on page 429, near the end, arrive at π΄1 = βπ ? I am not able to see it.This is what I tried. The book does things little diοΏ½erent than what we did in class. Thebook does not do
limπββ
π¦ππ βΌ limπ₯β0
π¦ππ’π‘
But instead, book replaces π₯ in the π¦ππ’π‘ (π₯) solution already obtained, with ππ, and rewritesπ¦ππ’π‘ (π₯), which is what equation (9.2.14) is. So following this, I am trying to verify the bookresult for π΄1 = βπ, but do not see how. Using the book notation, of using π in place of π,we have
π1 (π) = (π΄1 + π΄0) οΏ½1 β πβποΏ½ β ππ
Which is the equation in the book just below 9.2.14. The goal now is to find π΄1. Book alreadyfound π΄0 = π earlier. So we write
limπββ
π1(π)
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(π΄1 + π΄0) οΏ½1 β πβποΏ½ β ππ βΌ π¦ππ’π‘ (π₯)
limπββ
(π΄1 + π΄0) οΏ½1 β πβποΏ½ β ππ βΌ π οΏ½1 β ππ +π2π2
2!ββ―οΏ½
So far so good. But now the book says "comparing π1 (π₯) when π β β with the second termin 9.2.14 gives π΄1 = βπ". But how? If we take π β β on the LHS above, we get
limπββ
(π΄1 + π΄0) β ππ βΌ π οΏ½1 β ππ +ππ2
2!ββ―οΏ½
But π΄0 = π, so
limπββ
π΄1 + π β ππ βΌ π β πππ + πππ2
2!ββ―
limπββ
π΄1 β ππ βΌ βπππ + πππ2
2!ββ―
How does the above says that π΄1 = βπ ? If we move βππ to the right sides, it becomes
π΄1 βΌ ππ β πππ + πππ2
2!ββ―
π΄1 βΌ π (π β ππ) + πππ2
2!ββ―
I do not see how π΄1 = βπ. Does any one see how to get π΄1 = βπ?
Let redo this using the class method
243
4.9. note p9. added 2/24/17, asking . . . CHAPTER 4. STUDY NOTES
limπββ
π1(π)
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(π΄1 + π΄0) οΏ½1 β πβποΏ½ β ππ βΌ limπ₯β0
π¦ππ’π‘ (π₯)
limπββ
(π΄1 + π΄0) οΏ½1 β πβποΏ½ β ππ βΌ limπ₯β0
π οΏ½1 β π₯ +π₯2
2!ββ―οΏ½
limπββ
π΄1 + π β ππ βΌ π
limπββ
π΄1 βΌ ππ
How does the above says that π΄1 = βπ? and what happend to the limπββ of π which remainsthere?
244
4.10. note p11. added 3/7/17, Showing . . . CHAPTER 4. STUDY NOTES
4.10 note p11. added 3/7/17, Showing that scaling fornormalization is same for any π
This small computation verifies that normalization constant for the S-L from lecture 3/2/2017for making all eigenfunction orthonormal is the same for each π. Its numerical value is0.16627. Here is the table generated for π = 1, 2, β¦ , 6.
245
4.10. note p11. added 3/7/17, Showing . . . CHAPTER 4. STUDY NOTES
246
4.11. note p12. added 3/8/17, comparing . . . CHAPTER 4. STUDY NOTES
4.11 note p12. added 3/8/17, comparing exact solutionto WKB solution using Mathematica
Comparing Exact to WKB solution
for ODE in lecture 3/2/2017by Nasser M. Abbasi EP 548, Spring 2017.
This note shows how to obtain exact solution for the ODE given in lecture 3/2/2017, EP 548, and to
compare it to the WKB solution for different modes. This shows that the WKB becomes very close to
the exact solution for higher modes.
Obtain the exact solution, in terms of BesselJ functionsIn[16]:= ClearAll[n, c, y, m, lam];
lam[n_] :=9 n^2
49 Pi^4; (*eigenvalues from WKB solution*)
c =6
7 Pi^3; (*normalization value found for WKB*)
y[n_, x_] := c1
Pi + xSinn x^3 + 3 x^2 Pi + 3 Pi^2 x 7 Pi^2;
(*WKB solution found*)
Find exact solution
In[18]:= ode = y''[x] + lam x + Pi^4 y[x] β©΅ 0;
solExact = y[x] /. First@DSolve[{ode, y[0] β©΅ 0}, y[x], x] // TraditionalForm
Out[19]//TraditionalForm=
1
66
lam24
J 16
lam Ο3
3
c1 Ξ5
6 lam (x + Ο)4
8J 1
6
lam Ο3
3J-
16
lam x4 + 4 lam Ο x3 + 6 lam Ο2x
2 + 4 lam Ο3x + lam Ο4
34
3 lam4
-
J-
16
lam Ο3
3J 1
6
lam x4 + 4 lam Ο x3 + 6 lam Ο2x
2 + 4 lam Ο3x + lam Ο4
34
3 lam4
Make function which normalizes the exact solution eigenfunctions and plot each
mode eigenfunction with the WKB on the same plot
Printed by Wolfram Mathematica Student Edition
247
4.11. note p12. added 3/8/17, comparing . . . CHAPTER 4. STUDY NOTES
In[65]:= compare[modeNumber_] :=
Module{solExact1, int, cFromExact, eigenvalueFromHandSolution, flip},
eigenvalueFromHandSolution = lam[modeNumber];
solExact1 = solExact /. lam β eigenvalueFromHandSolution;
int = IntegratesolExact1^2 * x + Pi^4, {x, 0, Pi};
cFromExact = First@NSolve[int β©΅ 1, C[1]];
solExact1 = solExact1 /. cFromExact;
If[modeNumber > 5, flip = -1, flip = 1];
Plot {y[modeNumber, x], flip * solExact1}, {x, 0, Pi},
PlotStyle β {Red, Blue}, Frame -> True, FrameLabel β {{"y(x)", None},
{"x", Row[{"Comparing exact solution with WKB for mode ", modeNumber}]}},
GridLines β Automatic, GridLinesStyle β LightGray, BaseStyle β 12, ImageSize β 310,
FrameTicks β {Automatic, None}, 0, Pi 4, Pi 2, 3 4 Pi, Pi, None
;
Generate 4 plots, for mode 1, up to 6
These plots show that after mode 5 or 6, the two eigenfunctions are almost exact
2 p12.nb
Printed by Wolfram Mathematica Student Edition
248
4.11. note p12. added 3/8/17, comparing . . . CHAPTER 4. STUDY NOTES
In[66]:= plots = Table[compare[n], {n, 6}];
Grid[Partition[plots, 2]]
Out[67]=
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.01
0.02
0.03
x
y(x)
Comparing exact solution with WKB for mode 1
0 Ο
4
Ο
2
3Ο
4Ο
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
x
y(x)
Comparing exact solution with WKB for mode 2
0 Ο
4
Ο
2
3Ο
4Ο
-0.02
0.00
0.02
0.04
x
y(x)
Comparing exact solution with WKB for mode 3
0 Ο
4
Ο
2
3Ο
4Ο
-0.02
0.00
0.02
0.04
x
y(x)
Comparing exact solution with WKB for mode 4
0 Ο
4
Ο
2
3Ο
4Ο
-0.04
-0.02
0.00
0.02
0.04
x
y(x)
Comparing exact solution with WKB for mode 5
0 Ο
4
Ο
2
3Ο
4Ο
-0.04
-0.02
0.00
0.02
0.04
x
y(x)
Comparing exact solution with WKB for mode 6
p12.nb 3
Printed by Wolfram Mathematica Student Edition
249
4.11. note p12. added 3/8/17, comparing . . . CHAPTER 4. STUDY NOTES
Generate the above again, but now using relative error between the exact and
WKB for each mode, to make it more clear
In[68]:= compareError[modeNumber_] :=
Module{solExact1, eigenvalueFromHandSolution, int, cFromExact, flip},
eigenvalueFromHandSolution = lam[modeNumber];
solExact1 = solExact /. lam β eigenvalueFromHandSolution;
int = IntegratesolExact1^2 * x + Pi^4, {x, 0, Pi};
cFromExact = First@NSolve[int β©΅ 1, C[1]];
solExact1 = solExact1 /. cFromExact;
If[modeNumber > 5, flip = -1, flip = 1];
Plot 100 * Absflip * solExact1 - y[modeNumber, x], {x, 0, Pi}, PlotStyle β
{Red, Blue}, Frame -> True, FrameLabel β {{"relative error percentage", None},
{"x", Row[{"Absolute error. Exact solution vs WKB for mode ", modeNumber}]}},
GridLines β Automatic, GridLinesStyle β LightGray, BaseStyle β 12, ImageSize β 310,
FrameTicks β {Automatic, None}, 0, Pi 4, Pi 2, 3 4 Pi, Pi, None,
PlotRange β {Automatic, {0, 0.3}}
;
In[69]:= plots = Table[compareError[n], {n, 10}]; (*let do 10 modes*)
Grid[Partition[plots, 2]]
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 1
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 2
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 3
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 4
4 p12.nb
Printed by Wolfram Mathematica Student Edition
250
4.11. note p12. added 3/8/17, comparing . . . CHAPTER 4. STUDY NOTES
Out[69]=
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 5
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 6
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 7
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 8
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 9
0 Ο
4
Ο
2
3Ο
4Ο
0.00
0.05
0.10
0.15
0.20
0.25
0.30
x
rela
tive
erro
rpe
rcen
tage
Absolute error. Exact solution vs WKB for mode 10
p12.nb 5
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251
4.12. Convert ODE to Liouville form CHAPTER 4. STUDY NOTES
4.12 Convert ODE to Liouville form
Handy image to remember. Thanks to http://people.uncw.edu/hermanr/mat463/ODEBook/Book/SL.pdf
252
4.12. Convert ODE to Liouville form CHAPTER 4. STUDY NOTES
253
4.13. note p13. added 3/15/17, solving . . . CHAPTER 4. STUDY NOTES
4.13 note p13. added 3/15/17, solvingπ2π¦β³(π₯) = (π + π₯ + π₯2)π¦(π₯) in Maple
> >
> >
> >
> >
> >
Define the ODEassume(a>0 and 'real');ode:=epsilon^2*diff(y(x),x$2)=(x^2+a*x)*y(x);
Solve without giving any B.C.sol:=dsolve(ode,y(x));
Solve with one B.C. at infinity givensol:=dsolve({ode,y(infinity)=0},y(x));
Now solve giving B.C. at -infinitysol:=dsolve({ode,y(-infinity)=0},y(x));
Now solve by giving both B.C.at both endssol:=dsolve({ode,y(infinity)=0,y(-infinity)=0},y(x));
254
Chapter 5
Exams
Local contents5.1 Exam 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2565.2 Exam 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
255
5.1. Exam 1 CHAPTER 5. EXAMS
5.1 Exam 1
5.1.1 problem 3.26 (page 139)
Problem Perform local analysis solution to (π₯ β 1) π¦β²β² β π₯π¦β² + π¦ = 0 at π₯ = 1. Use the resultof this analysis to prove that a Taylor series expansion of any solution about π₯ = 0 has aninfinite radius of convergence. Find the exact solution by summing the series.
solution
Writing the ODE in standard form
π¦β²β² (π₯) + π (π₯) π¦β² (π₯) + π (π₯) π¦ (π₯) = 0 (1)
π¦β²β² βπ₯
(π₯ β 1)π¦β² +
1(π₯ β 1)
π¦ = 0 (2)
Where π (π₯) = βπ₯(π₯β1) , π (π₯) =
1(π₯β1) . The above shows that π₯ = 1 is singular point for both π (π₯)
and π (π₯). The next step is to classify the type of the singular point. Is it regular singularpoint or irregular singular point?
limπ₯β1
(π₯ β 1) π (π₯) = limπ₯β1
(π₯ β 1)βπ₯
(π₯ β 1)= β1
And
limπ₯β1
(π₯ β 1)2 π (π₯) = limπ₯β1
(π₯ β 1)21
(π₯ β 1)= 0
Because the limit exist, then π₯ = 1 is a regular singular point. Therefore solution is assumedto be a Frobenius power series given by
π¦ (π₯) =βοΏ½π=0
ππ (π₯ β 1)π+π
Substituting this in the original ODE (π₯ β 1) π¦β²β² β π₯π¦β² + π¦ = 0 gives
π¦β² (π₯) =βοΏ½π=0
(π + π) ππ (π₯ β 1)π+πβ1
π¦β²β² (π₯) =βοΏ½π=0
(π + π) (π + π β 1) ππ (π₯ β 1)π+πβ2
In order to move the (π₯ β 1) inside the summation, the original ODE (π₯ β 1) π¦β²β² β π₯π¦β² + π¦ = 0is first rewritten as
(π₯ β 1) π¦β²β² β (π₯ β 1) π¦β² β π¦β² + π¦ = 0 (3)
256
5.1. Exam 1 CHAPTER 5. EXAMS
Substituting the Frobenius series into the above gives
(π₯ β 1)βοΏ½π=0
(π + π) (π + π β 1) ππ (π₯ β 1)π+πβ2
β (π₯ β 1)βοΏ½π=0
(π + π) ππ (π₯ β 1)π+πβ1
ββοΏ½π=0
(π + π) ππ (π₯ β 1)π+πβ1
+βοΏ½π=0
ππ (π₯ β 1)π+π = 0
OrβοΏ½π=0
(π + π) (π + π β 1) ππ (π₯ β 1)π+πβ1
ββοΏ½π=0
(π + π) ππ (π₯ β 1)π+π
ββοΏ½π=0
(π + π) ππ (π₯ β 1)π+πβ1
+βοΏ½π=0
ππ (π₯ β 1)π+π = 0
Adjusting all powers of (π₯ β 1) to be the same by rewriting exponents and summation indicesgives
βοΏ½π=0
(π + π) (π + π β 1) ππ (π₯ β 1)π+πβ1
ββοΏ½π=1
(π + π β 1) ππβ1 (π₯ β 1)π+πβ1
ββοΏ½π=0
(π + π) ππ (π₯ β 1)π+πβ1
+βοΏ½π=1
ππβ1 (π₯ β 1)π+πβ1 = 0
Collecting terms with same powers in (π₯ β 1) simplifies the above toβοΏ½π=0
((π + π) (π + π β 1) β (π + π)) ππ (π₯ β 1)π+πβ1 β
βοΏ½π=1
(π + π β 2) ππβ1 (π₯ β 1)π+πβ1 = 0 (4)
Setting π = 0 gives the indicial equation((π + π) (π + π β 1) β (π + π)) π0 = 0
((π) (π β 1) β π) π0 = 0
Since π0 β 0 then the indicial equation is
(π) (π β 1) β π = 0π2 β 2π = 0π (π β 2) = 0
257
5.1. Exam 1 CHAPTER 5. EXAMS
The roots of the indicial equation are therefore
π1 = 2π2 = 0
Each one of these roots generates a solution to the ODE. The next step is to find the solutionπ¦1 (π₯) associated with π = 2. (The largest root is used first). Using π = 2 in equation (4) gives
βοΏ½π=0
((π + 2) (π + 1) β (π + 2)) ππ (π₯ β 1)π+1 β
βοΏ½π=1
πππβ1 (π₯ β 1)π+1 = 0
βοΏ½π=0
π (π + 2) ππ (π₯ β 1)π+1 β
βοΏ½π=1
πππβ1 (π₯ β 1)π+1 = 0 (5)
At π β₯ 1, the recursive relation is found and used to generate the coeοΏ½cients of the Frobeniuspower series
π (π + 2) ππ β πππβ1 = 0
ππ =π
π (π + 2)ππβ1
Few terms are now generated to see the pattern of the series and to determine the closedform. For π = 1
π1 =13π0
For π = 2
π2 =2
2 (2 + 2)π1 =
2813π0 =
112π0
For π = 3
π3 =3
3 (3 + 2)π2 =
315
112π0 =
160π0
For π = 4
π4 =4
4 (4 + 2)π3 =
16160π0 =
1360
π0
And so on. From the above, the first solution becomes
π¦1 (π₯) =βοΏ½π=0
ππ (π₯ β 1)π+2
= π0 (π₯ β 1)2 + π1 (π₯ β 1)
3 + π2 (π₯ β 1)4 + π3 (π₯ β 1)
4 + π4 (π₯ β 1)5 +β―
= (π₯ β 1)2 οΏ½π0 + π1 (π₯ β 1) + π2 (π₯ β 1)2 + π3 (π₯ β 1)
3 + π4 (π₯ β 1)4 +β―οΏ½
= (π₯ β 1)2 οΏ½π0 +13π0 (π₯ β 1) +
112π0 (π₯ β 1)
2 +160π0 (π₯ β 1)
3 +1360
π0 (π₯ β 1)4 +β―οΏ½
= π0 (π₯ β 1)2 οΏ½1 +
13(π₯ β 1) +
112(π₯ β 1)2 +
160(π₯ β 1)3 +
1360
(π₯ β 1)4 +β―οΏ½ (6)
To find closed form solution to π¦1 (π₯), Taylor series expansion of ππ₯ around π₯ = 1 is found
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5.1. Exam 1 CHAPTER 5. EXAMS
first
ππ₯ β π + π (π₯ β 1) +π2(π₯ β 1)2 +
π3!(π₯ β 1)3 +
π4!(π₯ β 1)4 +
π5!(π₯ β 1)5 +β―
β π + π (π₯ β 1) +π2(π₯ β 1)2 +
π6(π₯ β 1)3 +
π24(π₯ β 1)4 +
π120
(π₯ β 1)5 +β―
β π οΏ½1 + (π₯ β 1) +12(π₯ β 1)2 +
16(π₯ β 1)3 +
124(π₯ β 1)4 +
1120
(π₯ β 1)5 +β―οΏ½
Multiplying the above by 2 gives
2ππ₯ β π οΏ½2 + 2 (π₯ β 1) + (π₯ β 1)2 +13(π₯ β 1)3 +
112(π₯ β 1)4 +
160(π₯ β 1)5 +β―οΏ½
Factoring (π₯ β 1)2 from the RHS results in
2ππ₯ β π οΏ½2 + 2 (π₯ β 1) + (π₯ β 1)2 οΏ½1 +13(π₯ β 1) +
112(π₯ β 1)2 +
160(π₯ β 1)3 +β―οΏ½οΏ½ (6A)
Comparing the above result with the solution π¦1 (π₯) in (6), shows that the (6A) can be writtenin terms of π¦1 (π₯) as
2ππ₯ = π οΏ½2 + 2 (π₯ β 1) + (π₯ β 1)2 οΏ½π¦1 (π₯)
π0 (π₯ β 1)2 οΏ½οΏ½
Therefore
2ππ₯ = π οΏ½2 + 2 (π₯ β 1) +π¦1 (π₯)π0
οΏ½
2ππ₯β1 = 2 + 2 (π₯ β 1) +π¦1 (π₯)π0
2ππ₯β1 β 2 β 2 (π₯ β 1) =π¦1 (π₯)π0
Solving for π¦1 (π₯)
π¦1 (π₯) = π0 οΏ½2ππ₯β1 β 2 β 2 (π₯ β 1)οΏ½
= π0 οΏ½2ππ₯β1 β 2 β 2π₯ + 2οΏ½
= π0 οΏ½2ππ₯β1 β 2π₯οΏ½
=2π0πππ₯ β 2π0π₯
Let 2π0π = πΆ1 and β2π0 = πΆ2, then the above solution can be written as
π¦1 (π₯) = πΆ1ππ₯ + πΆ2π₯
Now that π¦1 (π₯) is found, which is the solution associated with π = 2, the next step is to findthe second solution π¦2 (π₯) associated with π = 0. Since π2 β π1 = 2 is an integer, the solutioncan be either case πΌπΌ(π) (π) or case πΌπΌ (π) (ππ) as given in the text book at page 72.
From equation (3.3.9) at page 72 of the text, using π = 2 since π = π2βπ1 and where π (π₯) = βπ₯and π (π₯) = 1 in this problem by comparing our ODE with the standard ODE in (3.3.2) at
259
5.1. Exam 1 CHAPTER 5. EXAMS
page 70 given by
π¦β²β² +π (π₯)(π₯ β π₯0)
π¦β² +π (π₯)
(π₯ β π₯0)2π¦ = 0
Expanding π (π₯) , π (π₯) in Taylor series
π (π₯) =βοΏ½π=0
ππ (π₯ β 1)π
π (π₯) =βοΏ½π=0
ππ (π₯ β 1)π
Since π (π₯) = βπ₯ in our ODE, then π0 = β1 and π1 = β1 and all other terms are zero. For π (π₯),which is just 1 in our ODE, then π0 = 1 and all other terms are zero. Hence
π0 = β1π1 = β1π0 = 1π = 2π = 0
The above values are now used to evaluate RHS of 3.3.9 in order to find which case it is.(book uses πΌ for π)
0ππ = βπβ1οΏ½π=0
οΏ½(π + π) ππβπ + ππβποΏ½ ππ (3.3.9)
Since π = 2 the above becomes
0π2 = β1οΏ½π=0
οΏ½(π + π) π2βπ + π2βποΏ½ ππ
Using π = 0, since this is the second root, gives
0π2 = β1οΏ½π=0
οΏ½ππ2βπ + π2βποΏ½ ππ
= β οΏ½οΏ½0π2β0 + π2β0οΏ½ π0 + οΏ½π2β1 + π2β1οΏ½ π1οΏ½
= β οΏ½οΏ½0π2 + π2οΏ½ π0 + οΏ½π1 + π1οΏ½ π1οΏ½
= β οΏ½0 + π2οΏ½ π0 β οΏ½π1 + π1οΏ½ π1Since π2 = 0, π1 = β1, π1 = 1, therefore
0π2 = β (0 + 0) π0 β (β1 + 1) π1= 0
The above shows that this is case πΌπΌ (π) (ππ), because the right side of 3.3.9 is zero. This meansthe second solution π¦2 (π₯) is also a Fronbenius series. If the above was not zero, the methodof reduction of order would be used to find second solution.
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5.1. Exam 1 CHAPTER 5. EXAMS
Assuming π¦2 (π₯) = βππ (π₯ β 1)π+π, and since π = 0, therefore
π¦2 (π₯) =βοΏ½π=0
ππ (π₯ β 1)π
Following the same method used to find the first solution, this series is now used in the ODEto determine ππ.
π¦β²2 (π₯) =βοΏ½π=0
πππ (π₯ β 1)πβ1 =
βοΏ½π=1
πππ (π₯ β 1)πβ1 =
βοΏ½π=0
(π + 1) ππ+1 (π₯ β 1)π
π¦β²β²2 (π₯) =βοΏ½π=0
π (π + 1) ππ+1 (π₯ β 1)πβ1 =
βοΏ½π=1
π (π + 1) ππ+1 (π₯ β 1)πβ1 =
βοΏ½π=0
(π + 1) (π + 2) ππ+2 (π₯ β 1)π
The ODE (π₯ β 1) π¦β²β² β (π₯ β 1) π¦β² β π¦β² + π¦ = 0 now becomes
(π₯ β 1)βοΏ½π=0
(π + 1) (π + 2) ππ+2 (π₯ β 1)π
β (π₯ β 1)βοΏ½π=0
(π + 1) ππ+1 (π₯ β 1)π
ββοΏ½π=0
(π + 1) ππ+1 (π₯ β 1)π
+βοΏ½π=0
ππ (π₯ β 1)π = 0
OrβοΏ½π=0
(π + 1) (π + 2) ππ+2 (π₯ β 1)π+1
ββοΏ½π=0
(π + 1) ππ+1 (π₯ β 1)π+1
ββοΏ½π=0
(π + 1) ππ+1 (π₯ β 1)π
+βοΏ½π=0
ππ (π₯ β 1)π = 0
HenceβοΏ½π=1
(π) (π + 1) ππ+1 (π₯ β 1)π β
βοΏ½π=1
πππ (π₯ β 1)π β
βοΏ½π=0
(π + 1) ππ+1 (π₯ β 1)π +
βοΏ½π=0
ππ (π₯ β 1)π = 0
π = 0 gives
β (π + 1) ππ+1 + ππ = 0βπ1 + π0 = 0
π1 = π0
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5.1. Exam 1 CHAPTER 5. EXAMS
π β₯ 1 generates the recursive relation to find all remaining ππ coeοΏ½cients
(π) (π + 1) ππ+1 β πππ β (π + 1) ππ+1 + ππ = 0(π) (π + 1) ππ+1 β (π + 1) ππ+1 = πππ β ππππ+1 ((π) (π + 1) β (π + 1)) = ππ (π β 1)
ππ+1 = ππ(π β 1)
(π) (π + 1) β (π + 1)Therefore the recursive relation is
ππ+1 =πππ + 1
Few terms are generated to see the pattern and to find the closed form solution for π¦2 (π₯).For π = 1
π2 = π112=12π0
For π = 2
π3 =π23=1312π0 =
16π0
For π = 3
π4 =π33 + 1
=1416π0 =
124π0
For π = 4
π5 =π44 + 1
=15124π0 =
1120
π0
And so on. Therefore, the second solution is
π¦2 (π₯) =βοΏ½π=0
ππ (π₯ β 1)π
= π0 + π1 (π₯ β 1) + π2 (π₯ β 1)2 +β―
= π0 + π0 (π₯ β 1) +12π0 (π₯ β 1)
2 +16π0 (π₯ β 1)
3 +124π0 (π₯ β 1)
4 +1120
π0 (π₯ β 1)5 +β―
= π0 οΏ½1 + (π₯ β 1) +12(π₯ β 1)2 +
16(π₯ β 1)3 +
124(π₯ β 1)4 +
1120
(π₯ β 1)5 +β―οΏ½ (7A)
The Taylor series for ππ₯ around π₯ = 1 is
ππ₯ β π + π (π₯ β 1) +π2(π₯ β 1)2 +
π6(π₯ β 1)3 +
π24(π₯ β 1)4 +
π120
(π₯ β 1)5 +β―
β π οΏ½1 + (π₯ β 1) +12(π₯ β 1)2 +
16(π₯ β 1)3 +
124(π₯ β 1)4 +
1120
(π₯ β 1)5 +β―οΏ½ (7B)
Comparing (7A) with (7B) shows that the second solution closed form is
π¦2 (π₯) = π0ππ₯
πLet π0
π be some constant, say πΆ3, the second solution above becomes
π¦2 (π₯) = πΆ3ππ₯
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5.1. Exam 1 CHAPTER 5. EXAMS
Both solutions π¦1 (π₯) , π¦2 (π₯) have now been found. The final solution is
π¦ (π₯) = π¦1 (π₯) + π¦2 (π₯)
=π¦1(π₯)
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½πΆ1ππ₯ + πΆ2π₯ +π¦2(π₯)οΏ½πΆ3ππ₯
= πΆ4ππ₯ + πΆ2π₯
Hence, the exact solution is
π¦ (π₯) = π΄ππ₯ + π΅π₯ (7)
Where π΄,π΅ are constants to be found from initial conditions if given. Above solution is nowverified by substituting it back to original ODE
π¦β² = π΄ππ₯ + π΅π¦β²β² = π΄ππ₯
Substituting these into (π₯ β 1) π¦β²β² β π₯π¦β² + π¦ = 0 gives(π₯ β 1)π΄ππ₯ β π₯ (π΄ππ₯ + π΅) + π΄ππ₯ + π΅π₯ = 0π₯π΄ππ₯ β π΄ππ₯ β π₯π΄ππ₯ β π₯π΅ + π΄ππ₯ + π΅π₯ = 0
βπ΄ππ₯ β π₯π΅ + π΄ππ₯ + π΅π₯ = 00 = 0
To answer the final part of the question, the above solution (7) is analytic around π₯ = 0 withinfinite radius of convergence since exp (β ) is analytic everywhere. Writing the solution as
π¦ (π₯) = οΏ½π΄βοΏ½π=0
π₯π
π! οΏ½+ π΅π₯
The function π₯ have infinite radius of convergence, since it is its own series. And the ex-ponential function has infinite radius of convergence as known, verified by using standardratio test
π΄ limπββ
οΏ½ππ+1ππ
οΏ½ = π΄ limπββ
οΏ½π₯π+1π!
(π + 1)!π₯π οΏ½= π΄ lim
πββοΏ½π₯π!
(π + 1)!οΏ½ = π΄ lim
πββοΏ½π₯
π + 1οΏ½ = 0
For any π₯. Since the ratio is less than 1, then the solution π¦ (π₯) expanded around π₯ = 0 hasan infinite radius of convergence.
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5.1. Exam 1 CHAPTER 5. EXAMS
5.1.2 problem 9.8 (page 480)
Problem Use boundary layer to find uniform approximation with error of order ποΏ½π2οΏ½ forthe problem ππ¦β²β² + π¦β² + π¦ = 0 with π¦ (0) = π, π¦ (1) = 1. Compare your solution to exact solution.Plot the solution for some values of π.
solution
ππ¦β²β² + π¦β² + π¦ = 0 (1)
Since π (π₯) = 1 > 0, then a boundary layer is expected at the left side, near π₯ = 0. Matchingwill fail if this was not the case. Starting with the outer solution near π₯ = 1. Let
π¦ππ’π‘ (π₯) =βοΏ½π=0
πππ¦π (π₯)
Substituting this into (1) gives
π οΏ½π¦β²β²0 + ππ¦β²β²1 + π2π¦β²β²2 +β―οΏ½ + οΏ½π¦β²0 + ππ¦β²1 + π2π¦β²2 +β―οΏ½ + οΏ½π¦0 + ππ¦1 + π2π¦2 +β―οΏ½ = 0 (2)
Collecting powers of ποΏ½π0οΏ½ results in the ODE
π¦β²0 βΌ βπ¦0ππ¦0π¦0
βΌ βππ₯
ln οΏ½π¦0οΏ½ βΌ βπ₯ + πΆ1
π¦ππ’π‘0 (π₯) βΌ πΆ1πβπ₯ + π (π) (3)
πΆ1 is found from boundary conditions π¦ (1) = 1. Equation (3) gives
1 = πΆ1πβ1
πΆ1 = π
Hence solution (3) becomes
π¦ππ’π‘0 (π₯) βΌ π1βπ₯
π¦ππ’π‘1 (π₯) is now found. Using (2) and collecting terms of ποΏ½π1οΏ½ gives the ODE
π¦β²1 + π¦1 βΌ βπ¦β²β²0 (4)
But
π¦β²0 (π₯) = βπ1βπ₯
π¦β²β²0 (π₯) = π1βπ₯
Using the above in the RHS of (4) gives
π¦β²1 + π¦1 βΌ βπ1βπ₯
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5.1. Exam 1 CHAPTER 5. EXAMS
The integrating factor is ππ₯, hence the above becomesπππ₯οΏ½π¦1ππ₯οΏ½ βΌ βππ₯π1βπ₯
πππ₯οΏ½π¦1ππ₯οΏ½ βΌ βπ
Integrating both sides gives
π¦1ππ₯ βΌ βππ₯ + πΆ2
π¦ππ’π‘1 (π₯) βΌ βπ₯π1βπ₯ + πΆ2πβπ₯ (5)
Applying boundary conditions π¦ (1) = 0 to the above gives
0 = β1 + πΆ2πβ1
πΆ2 = π
Hence the solution in (5) becomes
π¦ππ’π‘1 (π₯) βΌ βπ₯π1βπ₯ + π1βπ₯
βΌ (1 β π₯) π1βπ₯
Therefore the outer solution is
π¦ππ’π‘ (π₯) = π¦0 + ππ¦1= π1βπ₯ + π (1 β π₯) π1βπ₯ (6)
Now the boundary layer (inner) solution π¦ππ (π₯) near π₯ = 0 is found. Let π = π₯ππ be the inner
variable. The original ODE is expressed using this new variable, and π is found. Sinceππ¦ππ₯ =
ππ¦ππ
ππππ₯ then ππ¦
ππ₯ =ππ¦πππ
βπ. The diοΏ½erential operator is πππ₯ β‘ π
βπ πππ therefore
π2
ππ₯2=πππ₯
πππ₯
= οΏ½πβπππποΏ½ οΏ½
πβπππποΏ½
= πβ2ππ2
ππ2
Hence π2π¦ππ₯2 = π
β2π π2π¦ππ2 and ππ¦
β²β² + π¦β² + π¦ = 0 becomes
π οΏ½πβ2ππ2π¦ππ2 οΏ½
+ πβπππ¦ππ
+ π¦ = 0
π1β2ππ¦β²β² + πβππ¦β² + π¦ = 0 (7A)
The largest terms are οΏ½π1β2π, πβποΏ½, balance gives 1 β 2π = βπ or
π = 1
The ODE (7A) becomes
πβ1π¦β²β² + πβ1π¦β² + π¦ = 0 (7)
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5.1. Exam 1 CHAPTER 5. EXAMS
Assuming that solution is
π¦ππ (π₯) =βοΏ½π=0
πππ¦π = π¦0 + ππ¦1 + π2π¦2 +β―
Substituting the above into (7) gives
πβ1 οΏ½π¦β²β²0 + ππ¦β²β²1 +β―οΏ½ + πβ1 οΏ½π¦β²0 + ππ¦β²1 +β―οΏ½ + οΏ½π¦0 + ππ¦1 +β―οΏ½ = 0 (8)
Collecting terms with ποΏ½πβ1οΏ½ gives the first order ODE to solve
π¦β²β²0 βΌ βπ¦β²0Let π§ = π¦β²0, the above becomes
π§β² βΌ βπ§ππ§π§
βΌ βππ
ln |π§| βΌ βπ + πΆ4
π§ βΌ πΆ4πβπ
Hence
π¦β²0 βΌ πΆ4πβπ
Integrating
π¦ππ0 (π) βΌ πΆ4οΏ½πβπππ + πΆ5
βΌ βπΆ4πβπ + πΆ5 (9)
Applying boundary conditions π¦ (0) = π gives
π = βπΆ4 + πΆ5
πΆ5 = π + πΆ4
Equation (9) becomes
π¦ππ0 (π) βΌ βπΆ4πβπ + π + πΆ4
βΌ πΆ4 οΏ½1 β πβποΏ½ + π (10)
The next leading order π¦ππ1 (π) is found from (8) by collecting terms in ποΏ½π0οΏ½, which resultsin the ODE
π¦β²β²1 + π¦β²1 βΌ βπ¦0Since π¦ππ0 (π) βΌ πΆ4 οΏ½1 β πβποΏ½ + π, therefore π¦β²0 βΌ πΆ4πβπ and the above becomes
π¦β²β²1 + π¦β²1 βΌ βπΆ4πβπ
The homogenous solution is found first, then method of undetermined coeοΏ½cients is usedto find particular solution. The homogenous ODE is
π¦β²β²1,β βΌ π¦β²1,βThis was solved above for π¦ππ0 , and the solution is
π¦1,β βΌ βπΆ5πβπ + πΆ6
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5.1. Exam 1 CHAPTER 5. EXAMS
To find the particular solution, let π¦1,π βΌ π΄ππβπ, where π was added since πβπ shows up inthe homogenous solution. Hence
π¦β²1,π βΌ π΄πβπ β π΄ππβπ
π¦β²β²1,π βΌ βπ΄πβπ β οΏ½π΄πβπ β π΄ππβποΏ½
βΌ β2π΄πβπ + π΄ππβπ
Substituting these in the ODE π¦β²β²1,π + π¦β²1,π βΌ βπΆ4πβπ results in
β2π΄πβπ + π΄ππβπ + π΄πβπ β π΄ππβπ βΌ βπΆ4πβπ
βπ΄ = βπΆ4
π΄ = πΆ4
Therefore the particular solution is
π¦1,π βΌ πΆ4ππβπ
And therefore the complete solution is
π¦ππ1 (π) βΌ π¦1,β + π¦1,πβΌ βπΆ5πβπ + πΆ6 + πΆ4ππβπ
Applying boundary conditions π¦ (0) = 0 to the above gives
0 = βπΆ5 + πΆ6
πΆ6 = πΆ5
Hence the solution becomes
π¦ππ1 (π) βΌ βπΆ5πβπ + πΆ5 + πΆ4ππβπ
βΌ πΆ5 οΏ½1 β πβποΏ½ + πΆ4ππβπ (11)
The complete inner solution now becomes
π¦ππ (π) βΌ π¦ππ0 + ππ¦ππ1βΌ πΆ4 οΏ½1 β πβποΏ½ + π + π οΏ½πΆ5 οΏ½1 β πβποΏ½ + πΆ4ππβποΏ½ (12)
There are two constants that need to be determined in the above from matching with theouter solution.
limπββ
π¦ππ (π) βΌ limπ₯β0
π¦ππ’π‘ (π₯)
limπββ
πΆ4 οΏ½1 β πβποΏ½ + π + π οΏ½πΆ5 οΏ½1 β πβποΏ½ + πΆ4ππβποΏ½ βΌ limπ₯β0
π1βπ₯ + π (1 β π₯) π1βπ₯
πΆ4 + π + ππΆ5 βΌ π + ππ
The above shows that
πΆ5 = ππΆ4 + π = π
πΆ4 = 0
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5.1. Exam 1 CHAPTER 5. EXAMS
This gives the boundary layer solution π¦ππ (π) as
π¦ππ (π) βΌ π + ππ οΏ½1 β πβποΏ½
βΌ π οΏ½1 + π οΏ½1 β πβποΏ½οΏ½
In terms of π₯, since π = π₯π , the above can be written as
π¦ππ (π₯) βΌ π οΏ½1 + π οΏ½1 β πβπ₯π οΏ½οΏ½
The uniform solution is therefore
π¦uniform (π₯) βΌ π¦ππ (π₯) + π¦ππ’π‘ (π₯) β π¦πππ‘πβ
βΌ
π¦ππ
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½π οΏ½1 + π οΏ½1 β πβ
π₯π οΏ½οΏ½ +
π¦ππ’π‘
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½π1βπ₯ + π (1 β π₯) π1βπ₯ β (π + ππ)
βΌ π + ππ οΏ½1 β πβπ₯π οΏ½ + π1βπ₯ + (π β ππ₯) π1βπ₯ β (π + ππ)
βΌ π + ππ β ππ1βπ₯π + π1βπ₯ + ππ1βπ₯ β ππ₯π1βπ₯ β π β ππ
βΌ βππ1βπ₯π + π1βπ₯ + ππ1βπ₯ β ππ₯π1βπ₯
βΌ π1βπ₯ οΏ½βππβ1π + 1 + π β ππ₯οΏ½
Or
π¦uniform (π₯) βΌ π1βπ₯ οΏ½1 + π οΏ½1 β π₯ β πβ1π οΏ½οΏ½
With error ποΏ½π2οΏ½.
The above solution is now compared to the exact solution of ππ¦β²β² + π¦β² + π¦ = 0 with π¦ (0) =π, π¦ (1) = 1. Since this is a homogenous second order ODE with constant coeοΏ½cient, it iseasily solved using characteristic equation.
ππ2 + π + 1 = 0
The roots are
π =βπ2π
Β± βπ2 β 4ππ2π
=β12π
Β± β1 β 4π2π
Therefore the solution is
π¦ (π₯) = π΄ππ1π₯ + π΅ππ2π₯
= π΄ποΏ½β12π+
β1β4π2π οΏ½π₯
+ π΅ποΏ½β12π β
β1β4π2π οΏ½π₯
= π΄πβπ₯2π π
β1β4π2π π₯ + π΅π
βπ₯2π π
ββ1β4π2π π₯
= πβπ₯2π οΏ½π΄π
β1β4π2π π₯ + π΅π
ββ1β4π2π π₯οΏ½
268
5.1. Exam 1 CHAPTER 5. EXAMS
Applying first boundary conditions π¦ (0) = π to the above gives
π = π΄ + π΅π΅ = π β π΄
Hence the solution becomes
π¦ (π₯) = πβπ₯2π οΏ½π΄π
β1β4π2π π₯ + (π β π΄) π
ββ1β4π2π π₯οΏ½
= πβπ₯2π οΏ½π΄π
β1β4π2π π₯ + π1β
β1β4π2π π₯ β π΄π
ββ1β4π2π π₯οΏ½
= πβπ₯2π οΏ½π΄ οΏ½π
β1β4π2π π₯ β π
ββ1β4π2π π₯οΏ½ + π1β
β1β4π2π π₯οΏ½ (13)
Applying second boundary conditions π¦ (1) = 1 gives
1 = πβ12π οΏ½π΄ οΏ½π
β1β4π2π β π
ββ1β4π2π οΏ½ + π1β
β1β4π2π οΏ½
π12π = π΄οΏ½π
β1β4π2π β π
ββ1β4π2π οΏ½ + π1β
β1β4π2π
π΄ =π
12π β π1β
β1β4π2π
πβ1β4π2π β π
ββ1β4π2π
(14)
Substituting this into (13) results in
π¦exact (π₯) = πβπ₯2π οΏ½π΄ οΏ½π
β1β4π2π π₯ β π
ββ1β4π2π π₯οΏ½ + π1β
β1β4π2π π₯οΏ½
Where π΄ is given in (14). hence
π¦exact (π₯) = πβπ₯2π
βββββββββ
βββββββββπ
12π β π1β
β1β4π2π
πβ1β4π2π β π
ββ1β4π2π
βββββββββ οΏ½πβ1β4π2π π₯ β π
ββ1β4π2π π₯οΏ½ + π1β
β1β4π2π π₯
βββββββββ
In summary
exact solution asymptotic solution
πβπ₯2π
βββββββ
βββββββ
π12π βπ1β
β1β4π2π
πβ1β4π2π βπ
ββ1β4π2π
βββββββ οΏ½π
β1β4π2π π₯ β π
ββ1β4π2π π₯οΏ½ + π1β
β1β4π2π π₯
βββββββ π1βπ₯ + ππ1βπ₯ οΏ½1 β π₯ β πβ
1π οΏ½ + π οΏ½π2οΏ½
The following plot compares the exact solution with the asymptotic solution for π = 0.1
269
5.1. Exam 1 CHAPTER 5. EXAMS
Exact solution
Asymptotic
0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.5
2.0
2.5
3.0
x
y(x)
Exact solution vs. two terms asymptotic Ο΅ = 0.1
The following plot compares the exact solution with the asymptotic solution for π = 0.01.The diοΏ½erence was too small to notice in this case, the plot below is zoomed to be near π₯ = 0
Out[21]=
Exact solution
Asymptotic
0.00 0.05 0.10 0.15 0.20
2.3
2.4
2.5
2.6
2.7
x
y(x)
Exact solution vs. two terms asymptotic Ο΅ = 0.01
At π = 0.001, the diοΏ½erence between the exact and the asymptotic solution was not noticeable.Therefore, to better compare the solutions, the following plot shows the relative percentage
270
5.1. Exam 1 CHAPTER 5. EXAMS
error given by
100 οΏ½π¦exact β π¦uniform
π¦exact οΏ½%
For diοΏ½erent π.
Ο΅=0.1
Ο΅=0.05
Ο΅=0.01
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
x
%relativeerror
Percentage relative error between exact solution and asymptotic solution
Some observations: The above plot shows more clearly how the diοΏ½erence between theexact solution and the asymptotic solution became smaller as π became smaller. The plotalso shows that the boundary layer near π₯ = 0 is becoming more narrow are π becomessmaller as expected. It also shows that the relative error is smaller in the outer region thanin the boundary layer region. For example, for π = 0.05, the largest percentage error inthe outer region was less than 1%, while in the boundary layer, very near π₯ = 0, the errorgrows to about 5%. Another observation is that at the matching location, the relative errorgoes down to zero. One also notices that the matching location drifts towards π₯ = 0 as πbecomes smaller because the boundary layer is becoming more narrow. The following tablesummarizes these observations.
π % error near π₯ = 0 apparent width of boundary layer
0.1 10 0.20.05 5 0.120.01 1 0.02
271
5.1. Exam 1 CHAPTER 5. EXAMS
5.1.3 problem 3
Problem (a) Find physical optics approximation to the eigenvalue and eigenfunctions of theSturm-Liouville problem are π β β
βπ¦β²β² = π (sin (π₯) + 1)2 π¦π¦ (0) = 0π¦ (π) = 0
(b) What is the integral relation necessary to make the eigenfunctions orthonormal? Forsome reasonable choice of scaling coeοΏ½cient (give the value), plot the eigenfunctions forπ = 5, π = 20.
(c) Estimate how large π should be for the relative error of less than 0.1%
solution
5.1.3.1 Part a
Writing the ODE as
π¦β²β² + π (sin (π₯) + 1)2 π¦ = 0Let1
π = 1π2
Then the given ODE becomes
π2π¦β²β² (π₯) + (sin (π₯) + 1)2 π¦ (π₯) = 0 (1)
Physical optics approximation is obtained when π β β which implies π β 0+. Since theODE is linear and the highest derivative is now multiplied by a very small parameter π,WKB can therefore be used to solve it. WKB starts by assuming that the solution has theform
π¦ (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ πΏ β 0
Therefore, taking derivatives and substituting back in the ODE results in
π¦β² (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ οΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½
π¦β²β² (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ οΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½2
+ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ οΏ½1πΏ
βοΏ½π=0
πΏππβ²β²π (π₯)οΏ½
1π = 1πcould also be used. But the book uses π2.
272
5.1. Exam 1 CHAPTER 5. EXAMS
Substituting these into (1) and canceling the exponential terms gives
π2ββββββοΏ½1πΏ
βοΏ½π=0
πΏππβ²π (π₯)οΏ½2
+1πΏ
βοΏ½π=0
πΏππβ²β²π (π₯)ββββββ βΌ β (sin (π₯) + 1)2
π2
πΏ2οΏ½πβ²0 + πΏπβ²1 +β―οΏ½ οΏ½πβ²0 + πΏπβ²1 +β―οΏ½ +
π2
πΏοΏ½πβ²β²0 + πΏπβ²β²1 +β―οΏ½ βΌ β (sin (π₯) + 1)2
π2
πΏ2οΏ½οΏ½πβ²0οΏ½
2+ πΏ οΏ½2πβ²1πβ²0οΏ½ +β―οΏ½ +
π2
πΏοΏ½πβ²β²0 + πΏπβ²β²1 +β―οΏ½ βΌ β (sin (π₯) + 1)2
οΏ½π2
πΏ2οΏ½πβ²0οΏ½
2+2π2
πΏπβ²1πβ²0 +β―οΏ½ + οΏ½
π2
πΏπβ²β²0 + π2πβ²β²1 +β―οΏ½ βΌ β (sin (π₯) + 1)2 (2)
The largest term in the left side is π2
πΏ2οΏ½πβ²0οΏ½
2. By dominant balance, this term has the same
order of magnitude as the right side β (sin (π₯) + 1)2. This implies that πΏ2 is proportional toπ2. For simplicity (following the book) πΏ can be taken as equal to π
πΏ = π
Using the above in equation (2) results in
οΏ½οΏ½πβ²0οΏ½2+ 2ππβ²1πβ²0 +β―οΏ½ + οΏ½ππβ²β²0 + π2πβ²β²1 +β―οΏ½ βΌ β (sin (π₯) + 1)2
Balance of π (1) gives
οΏ½πβ²0οΏ½2βΌ β (sin (π₯) + 1)2 (3)
Balance of π (π) gives
2πβ²1πβ²0 βΌ βπβ²β²0 (4)
Equation (3) is solved first in order to find π0 (π₯).
πβ²0 βΌ Β±π (sin (π₯) + 1)Hence
π0 (π₯) βΌ Β±ποΏ½π₯
0(sin (π‘) + 1) ππ‘ + πΆΒ±
βΌ Β±π (π‘ β cos (π‘))π₯0 + πΆΒ±
βΌ Β±π (1 + π₯ β cos (π₯)) + πΆΒ± (5)
π1 (π₯) is now found from (4) and using πβ²β²0 = Β±π cos (π₯) gives
πβ²1 βΌ β12πβ²β²0πβ²0
βΌ β12
Β±π cos (π₯)Β±π (sin (π₯) + 1)
βΌ β12
cos (π₯)(sin (π₯) + 1)
Hence the solution is
π1 (π₯) βΌ β12
ln (1 + sin (π₯)) (6)
273
5.1. Exam 1 CHAPTER 5. EXAMS
Having found π0 (π₯) and π1 (π₯), the leading behavior is now obtained from
π¦ (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½
βΌ exp οΏ½1π(π0 (π₯) + ππ1 (π₯)) +β―οΏ½
βΌ exp οΏ½1ππ0 (π₯) + π1 (π₯) +β―οΏ½
The leading behavior is only the first two terms (called physical optics approximation inWKB), therefore
π¦ (π₯) βΌ exp οΏ½1ππ0 (π₯) + π1 (π₯)οΏ½
βΌ exp οΏ½Β±ππ(1 + π₯ β cos (π₯)) + πΆΒ± β
12
ln (1 + sin (π₯))οΏ½
βΌ1
β1 + sin π₯exp οΏ½Β±
ππ(1 + π₯ β cos (π₯)) + πΆΒ±οΏ½
Which can be written as
π¦ (π₯) βΌπΆ
β1 + sin π₯exp οΏ½
ππ(1 + π₯ β cos (π₯))οΏ½ β
πΆ
β1 + sin π₯exp οΏ½
βππ(1 + π₯ β cos (π₯))οΏ½
In terms of sin and cos the above becomes (using the standard Euler relation simplifications)
π¦ (π₯) βΌπ΄
β1 + sin π₯cos οΏ½
1π(1 + π₯ β cos (π₯))οΏ½ +
π΅
β1 + sin π₯sin οΏ½
1π(1 + π₯ β cos (π₯))οΏ½
Where π΄,π΅ are the new constants. But π = 1π2 , and the above becomes
π¦ (π₯) βΌπ΄
β1 + sin π₯cos οΏ½βπ (1 + π₯ β cos (π₯))οΏ½ + π΅
β1 + sin π₯sin οΏ½βπ (1 + π₯ β cos (π₯))οΏ½ (7)
Boundary conditions are now applied to determine π΄,π΅.
π¦ (0) = 0π¦ (π) = 0
First B.C. applied to (7) gives (where now βΌ is replaced by = for notation simplicity)
0 = π΄ cos οΏ½βπ (1 β cos (0))οΏ½ + π΅ sin οΏ½βπ (1 β cos (0))οΏ½
0 = π΄ cos (0) + π΅ sin (0)0 = π΄
Hence solution (7) becomes
π¦ (π₯) βΌπ΅
β1 + sin π₯sin οΏ½βπ (1 + π₯ β cos (π₯))οΏ½
274
5.1. Exam 1 CHAPTER 5. EXAMS
Applying the second B.C. π¦ (π) = 0 to the above results in
0 =π΅
β1 + sinπsin οΏ½βπ (1 + π β cos (π))οΏ½
0 = π΅ sin οΏ½βπ (1 + π + 1)οΏ½
= π΅ sin οΏ½(2 + π)βποΏ½
Hence, non-trivial solution implies that
(2 + π)οΏ½ππ = ππ π = 1, 2, 3,β―
οΏ½ππ =ππ2 + π
The eigenvalues are
ππ =π2π2
(2 + π)2π = 1, 2, 3,β―
Hence ππ β π2 for large π. The eigenfunctions are
π¦π (π₯) βΌπ΅π
β1 + sin π₯sin οΏ½οΏ½ππ (1 + π₯ β cos (π₯))οΏ½ π = 1, 2, 3,β―
The solution is therefore a linear combination of the eigenfunctions
π¦ (π₯) βΌβοΏ½π=1
π¦π (π₯)
βΌβοΏ½π=1
π΅πβ1 + sin π₯
sin οΏ½οΏ½ππ (1 + π₯ β cos (π₯))οΏ½ (7A)
This solution becomes more accurate for large π or large π.
5.1.3.2 Part b
For normalization, the requirement is that
οΏ½π
0π¦2π (π₯)
weight
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(sin (π₯) + 1)2ππ₯ = 1
Substituting the eigenfunction π¦π (π₯) solution obtained in first part in the above results in
οΏ½π
0οΏ½
π΅πβ1 + sin π₯
sin οΏ½οΏ½ππ (1 + π₯ β cos (π₯))οΏ½οΏ½2
(sin (π₯) + 1)2 ππ₯ βΌ 1
The above is now solved for constant π΅π. The constant π΅π will the same for each π fornormalization. Therefore any π can be used for the purpose of finding the scaling constant.
275
5.1. Exam 1 CHAPTER 5. EXAMS
Selecting π = 1 in the above gives
οΏ½π
0οΏ½
π΅
β1 + sin π₯sin οΏ½ π
2 + π(1 + π₯ β cos (π₯))οΏ½οΏ½
2
(sin (π₯) + 1)2 ππ₯ βΌ 1
π΅2οΏ½π
0
11 + sin π₯ sin2 οΏ½
π2 + π
(1 + π₯ β cos (π₯))οΏ½ (sin (π₯) + 1)2 ππ₯ βΌ 1
π΅2οΏ½π
0sin2 οΏ½
π2 + π
(1 + π₯ β cos (π₯))οΏ½ (sin (π₯) + 1) ππ₯ βΌ 1 (8)
Letting π’ = π2+π
(1 + π₯ β cos (π₯)), thenππ’ππ₯
=π
2 + π(1 + sin (π₯))
When π₯ = 0, then π’ = π2+π
(1 + 0 β cos (0)) = 0 and when π₯ = π then π’ = π2+π
(1 + π β cos (π)) =π
2+π(2 + π) = π, hence (8) becomes
π΅2οΏ½π
0sin2 (π’)
2 + ππ
ππ’ππ₯ππ₯ = 1
2 + ππ
π΅2οΏ½π
0sin2 (π’) ππ’ = 1
But sin2 (π’) = 12 β
12 cos 2π’, therefore the above becomes
2 + ππ
π΅2οΏ½π
0οΏ½12β12
cos 2π’οΏ½ ππ’ = 1
122 + ππ
π΅2 οΏ½π’ βsin 2π’2 οΏ½
π
0= 1
2 + π2π
π΅2 οΏ½οΏ½π βsin 2π2 οΏ½ β οΏ½0 β
sin 02 οΏ½οΏ½ = 1
2 + π2π
π΅2π = 1
π΅2 =2
2 + πTherefore
π΅ =οΏ½
2π + 2
= 0.62369
Using the above for each π΅π in the solution obtained for the eigenfunctions in (7A), andpulling this scaling constant out of the sum results in
π¦normalized βΌοΏ½
2π + 2
βοΏ½π=1
1
β1 + sin π₯sin οΏ½οΏ½ππ (1 + π₯ β cos (π₯))οΏ½ (9)
Where
οΏ½ππ =ππ2 + π
π = 1, 2, 3,β―
The following are plots for the normalized π¦π (π₯) for π values it asks to show.
276
5.1. Exam 1 CHAPTER 5. EXAMS
0 Ο
4Ο
23Ο4
Ο
-0.4
-0.2
0.0
0.2
0.4
x
y(x)
yn(x) for n =5
0 Ο
4Ο
23Ο4
Ο
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
x
y(x)
yn(x) for n =20
The following shows the π¦ (π₯) as more eigenfunctions are added up to 55.
277
5.1. Exam 1 CHAPTER 5. EXAMS
0 Ο
4Ο
23Ο4
Ο
0.0
0.5
1.0
1.5
2.0
x
y(x)
sum of first 5 eigenfunction
0 Ο
4Ο
23Ο4
Ο
0
1
2
3
4
5
6
xy(x)
sum of first 15 eigenfunction
0 Ο
4Ο
23Ο4
Ο
0
2
4
6
8
10
x
y(x)
sum of first 25 eigenfunction
0 Ο
4Ο
23Ο4
Ο
0
5
10
15
x
y(x)
sum of first 35 eigenfunction
0 Ο
4Ο
23Ο4
Ο
0
5
10
15
20
x
y(x)
sum of first 45 eigenfunction
0 Ο
4Ο
23Ο4
Ο
0
5
10
15
20
25
x
y(x)
sum of first 55 eigenfunction
278
5.1. Exam 1 CHAPTER 5. EXAMS
5.1.3.3 Part c
Since approximate solution is
π¦ (π₯) βΌ exp οΏ½1πΏ
βοΏ½π=0
πΏπππ (π₯)οΏ½ πΏ β 0
βΌ exp οΏ½1πΏπ0 (π₯) + π1 (π₯) + πΏπ2 (π₯) +β―οΏ½ (1)
And the physical optics approximation includes the first two terms in the series above, thenthe relative error between physical optics and exact solution is given by πΏπ2 (π₯). But πΏ = π.Hence (1) becomes
π¦ (π₯) βΌ exp οΏ½1ππ0 (π₯) + π1 (π₯) + ππ2 (π₯) +β―οΏ½
Hence the relative error must be such that
|ππ2 (π₯)|max β€ 0.001 (1A)
Now π2 (π₯) is found. From (2) in part(a)
π2
πΏ2οΏ½πβ²0 + πΏπβ²1 + πΏ2πβ²2 +β―οΏ½ οΏ½πβ²0 + πΏπβ²1 + πΏ2πβ²2 +β―οΏ½ +
π2
πΏοΏ½πβ²β²0 + πΏπβ²β²1 + πΏ2πβ²β²2 +β―οΏ½ βΌ β (sin (π₯) + 1)2
π2
πΏ2οΏ½οΏ½πβ²0οΏ½
2+ πΏ οΏ½2πβ²1πβ²0οΏ½ + πΏ2 οΏ½2πβ²0πβ²2 + οΏ½πβ²1οΏ½
2οΏ½ +β―οΏ½ +
π2
πΏοΏ½πβ²β²0 + πΏπβ²β²1 + πΏ2πβ²β²2 +β―οΏ½ βΌ β (sin (π₯) + 1)2
οΏ½οΏ½πβ²0οΏ½2+ π οΏ½2πβ²1πβ²0οΏ½ + π2 οΏ½2πβ²0πβ²2 + οΏ½πβ²1οΏ½
2οΏ½ +β―οΏ½ + οΏ½ππβ²β²0 + π2πβ²β²1 + π3πβ²β²2 +β―οΏ½ βΌ β (sin (π₯) + 1)2
A balance on ποΏ½π2οΏ½ gives the ODE to solve to find π2
2πβ²0πβ²2 βΌ β οΏ½πβ²1οΏ½2β πβ²β²1 (2)
But
πβ²0 βΌ Β±π (1 + sin (π₯))
οΏ½πβ²1οΏ½2βΌ οΏ½β
12
cos (π₯)(sin (π₯) + 1)οΏ½
2
βΌ14
cos2 (π₯)(1 + sin (π₯))2
πβ²β²1 βΌ β12πππ₯ οΏ½
cos (π₯)(1 + sin (π₯))οΏ½
βΌ12 οΏ½
11 + sin (π₯)οΏ½
279
5.1. Exam 1 CHAPTER 5. EXAMS
Hence (2) becomes
2πβ²0πβ²2 βΌ β οΏ½πβ²1οΏ½2β πβ²β²1
πβ²2 βΌ βοΏ½οΏ½πβ²1οΏ½
2+ πβ²β²1 οΏ½
2πβ²0
βΌ βοΏ½14
cos2(π₯)(1+sin(π₯))2
+ 12οΏ½ 11+sin(π₯)οΏ½οΏ½
Β±2π (1 + sin (π₯))
βΌ Β±π οΏ½14
cos2(π₯)(sin(π₯)+1)2
+ 12οΏ½ 11+sin(π₯)οΏ½οΏ½
2 (sin (π₯) + 1)
βΌ Β±π14 οΏ½
cos2(π₯)+2(1+sin(π₯))(sin(π₯)+1)2
οΏ½
2 (sin (π₯) + 1)
βΌ Β±π8
cos2 (π₯) + 2 (1 + sin (π₯))(1 + sin (π₯))3
Therefore
π2 (π₯) βΌ Β±π8 οΏ½
π₯
0
cos2 (π‘) + 2 (1 + sin (π‘))(1 + sin (π‘))3
ππ‘
βΌ Β±π8 οΏ½οΏ½
π₯
0
cos2 (π‘)(1 + sin (π‘))3
ππ‘ + 2οΏ½π₯
0
1(1 + sin (π‘))2
ππ‘οΏ½ (3)
To do β«π₯
0cos2(π‘)
(1+sin(π‘))3ππ‘, I used a lookup integration rule from tables which says β« cosπ (π‘) (π + sin π‘)π ππ‘ =
1(π)(π) cosπ+1 (π‘) (π + sin π‘)π, therefore using this rule the integral becomes, where now π =β3, π = 2, π = 1,
οΏ½π₯
0
cos2 π‘(1 + sin π‘)3
ππ‘ =1β3 οΏ½
cos3 π‘(1 + sin π‘)3
οΏ½π₯
0
=1β3 οΏ½
cos3 π₯(1 + sin π₯)3
β 1οΏ½
=13 οΏ½1 β
cos3 π₯(1 + sin π₯)3
οΏ½
And for β« 1(1+sin(π₯))2
ππ₯, half angle substitution can be used. I do not know what other substi-
tution to use. Using CAS for little help on this, I get
οΏ½π₯
0
1(1 + sin π‘)2
ππ‘ = οΏ½βcos π‘
3 (1 + sin π‘)2β13
cos π‘1 + sin π‘οΏ½
π₯
0
= οΏ½βcos π₯
3 (1 + sin π₯)2β13
cos π₯1 + sin π₯οΏ½ β οΏ½β
13β13οΏ½
=23β
cos π₯3 (1 + sin π₯)2
β13
cos π₯1 + sin π₯
280
5.1. Exam 1 CHAPTER 5. EXAMS
Hence from (3)
π2 (π₯) βΌ Β±π8 οΏ½
13 οΏ½1 β
cos3 (π₯)(1 + sin (π₯))3
οΏ½ + 2 οΏ½23β
cos π₯3 (1 + sin π₯)2
β13
cos π₯1 + sin π₯οΏ½οΏ½
βΌ Β±π8 οΏ½
13β13
cos3 (π₯)(1 + sin (π₯))3
+43β
2 cos π₯3 (1 + sin π₯)2
β23
cos π₯1 + sin π₯οΏ½
βΌ Β±π24 οΏ½
1 βcos3 (π₯)
(1 + sin (π₯))3+ 4 β
2 cos π₯(1 + sin π₯)2
β2 cos π₯1 + sin π₯οΏ½
Therefore, from (1A)
|ππ2 (π₯)|max β€ 0.001
οΏ½ππ24 οΏ½
1 βcos3 (π₯)
(1 + sin (π₯))3+ 4 β
2 cos π₯(1 + sin π₯)2
β2 cos π₯1 + sin π₯οΏ½οΏ½
maxβ€ 0.001
124 οΏ½
π οΏ½1 βcos3 (π₯)
(1 + sin (π₯))3+ 4 β
2 cos π₯(1 + sin π₯)2
β2 cos π₯1 + sin π₯οΏ½οΏ½
maxβ€ 0.001
οΏ½π οΏ½1 βcos3 (π₯)
(1 + sin (π₯))3+ 4 β
2 cos π₯(1 + sin π₯)2
β2 cos π₯1 + sin π₯οΏ½οΏ½
maxβ€ 0.024 (2)
The maximum value of οΏ½1 β cos3(π₯)(1+sin(π₯))3
+ 4 β 2 cos π₯(1+sin π₯)2
β 2 cos π₯1+sin π₯
οΏ½ between π₯ = 0 and π₯ = π is now
found and used to find π. A plot of the above shows the maximum is maximum at the end,at π₯ = π (Taking the derivative and setting it to zero to determine where the maximum iscan also be used).
In[324]:= myResult = 1 -Cos[x]3
(1 + Sin[x])3+ 4 -
2 Cos[x]
(1 + Sin[x])2-
2 Cos[x]
1 + Sin[x];
Plot[myResult, {x, 0, Pi}, PlotRange β All, Frame β True, GridLines β Automatic, GridLinesStyle β LightGray,
FrameLabel β {{"s2(x)", None}, {"x", "Finding where maximum S2(x) is, part(c)"}}, PlotStyle β Red, BaseStyle β 14,
FrameTicks β {Automatic, {{0, Pi / 4, Pi / 2, 3 / 4 Pi, Pi}, None}}, ImageSize β 400]
Out[325]=
0 Ο
4
Ο
2
3Ο
4Ο
0
2
4
6
8
10
x
s2(x)
Finding where maximum S2(x) is, part(c)
Therefore, at π₯ = π
οΏ½1 βcos3 π₯
(1 + sin π₯)3+ 4 β
2 cos π₯(1 + sin π₯)2
β2 cos π₯1 + sin π₯οΏ½
π₯=π
= οΏ½1 βcos3 (π)
(1 + sinπ)3+ 4 β
2 cosπ(1 + sinπ)2
β2 cosπ1 + sinποΏ½
= 10
281
5.1. Exam 1 CHAPTER 5. EXAMS
Hence (2) becomes
10π β€ 0.024π β€ 0.0024
But since π = 1π2 the above becomes
1
βπβ€ 0.0024
βπ β₯1
0.0024βπ β₯ 416.67
Hence
π β₯ 17351.1
To find which mode this corresponds to, since ππ =π2π2
(2+π)2, then need to solve for π
17351.1 =π2π2
(2 + π)2
π2π2 = (17351.1) (2 + π)2
π =οΏ½(17351.1) (2 + π)2
π2
= 215.58
Hence the next largest integer is used
π = 216
To have relative error less than 0.1% compared to exact solution. Therefore using the resultobtained in (9) in part (b) the normalized solution needed is
π¦normalized βΌοΏ½
2π + 2
216οΏ½π=1
1
β1 + sin π₯sin οΏ½ ππ
2 + π(1 + π₯ β cos (π₯))οΏ½
The following is a plot of the above solution adding all the first 216 modes for illustration.
282
5.1. Exam 1 CHAPTER 5. EXAMS
In[42]:= ClearAll[x, n, lam]
mySol[x_, max_] := Sqrt2
Pi + 2 Sum
1
Sqrt[1 + Sin[x]]Sin
n Pi
2 + Pi(1 + x - Cos[x]), {n, 1, max};
In[46]:= p[n_] := Plot[mySol[x, n], {x, 0, Pi}, PlotRange β All, Frame β True,
FrameLabel β {{"y(x)", None}, {"x", Row[{"yn(x) for n =", n}]}}, BaseStyle β 14, GridLines β Automatic,
GridLinesStyle β LightGray, ImageSize β 600, PlotStyle β Red,
FrameTicks β { {Automatic, None}, {{0, Pi / 4, Pi / 2, 3 / 4 Pi, Pi}, None}}, PlotRange β All]
In[47]:= p[216]
Out[47]=
0 Ο
4
Ο
2
3Ο
4Ο
0
20
40
60
80
100
x
y(x)
yn(x) for n =216
283
5.2. Exam 2 CHAPTER 5. EXAMS
5.2 Exam 2
5.2.1 problem 3
2. Consider the 1D heat equation in a semi-infinite domain:
βu
βt= Ξ½
β2u
βx2, x β₯ 0
with boundary conditions: u(0, t) = exp(βiΟt) and u(x, t) bounded as x β β. In orderto construct a real forcing, we need both positive and negative real values of Ο. Considerthat this forcing has been and will be applied for all time. This βpure boundary valueproblemβ could be an idealization of heating the surface of the earth by the sun (periodicforcing). One could then ask, how far beneath the surface of the earth do the periodicfluctuations of the heat propagate?
(a) Consider solutions of the form u(x, t;Ο) = exp(ikx) exp(βiΟt). Find a single expressionfor k as a function of (given) Ο real, sgn(Ο) and Ξ½ real.
Write u(x, t;Ο) as a function of (given) Ο real, sgn(Ο) and Ξ½ real. To obtain the mostgeneral solution by superposition, one would next integrate over all values of Ο, ββ <Ο <β (do not do this).
(b) The basic solution can be written as u(x, t;Ο) = exp(βiΟt) exp(βΟx) exp(iΟ sgn(Ο) x).Find Ο in terms of |Ο| and Ξ½.
(c) Make an estimate for the propagation depth of daily temperature fluctuations.
3. Here we study the competing effects of nonlinearity and diffusion in the context ofBurgerβs equation
βu
βt+ u
βu
βx= Ξ½
β2u
βx2(3a)
which is the simplest model equation for diffusive waves in fluid dynamics. It can be solvedexactly using the Cole-Hopf transformation
u = β2Ξ½ΟxΟ
(3b)
as follows (with 2 steps to achieve the transformation (3b)).
(a) Let u = Οx (where the subscript denotes partial differentiation) and integrate oncewith respect to x.
(b) Let Ο = β2Ξ½ ln(Ο) to get the diffusion equation for Ο.
(c) Solve for Ο with Ο(x, 0) = Ξ¦(x), ββ < x < β. In your integral expression for Ο, usedummy variable Ξ· to facilitate the remaining parts below.
(d) Show that
Ξ¦(x) = exp
[
β1
2Ξ½
β« x
xo
F (Ξ±)dΞ±
]
where u(x, 0) = F (x), with xo arbitrary which we will take to be positive for conveniencebelow (xo > 0).
2
(e) Write your expression for Ο(x, t) in terms of
f(Ξ·, x, t) =
β« Ξ·
xo
F (Ξ±)dΞ±+(xβ Ξ·)2
2t.
(f) Find Οx(x, t) and then use equation (3b) to find u(x, t).
4. (a) Use the Method of Images to solve
βu
βt= Ξ½
β2u
βx2+Q(x, t), 0 β€ x β€ L, t β₯ 0
u(x, 0) = f(x),βu
βx(0, t) = 0, u(L, t) = A
(b) For A = 0, compare your expression for the solution in (a) to the eigenfunction solution.
5. This problem is a simple model for diffraction of light passing through infinitesimallysmall slits separated by a distance 2a.
Solve the diffraction equation
βu
βt=iΞ»
4Ο
β2u
βx2(1)
with initial source u(x, 0) = f(x) = Ξ΄(xβ a) + Ξ΄(x+ a), a > 0.
Show that the solution u(x, t) oscillates wildly, but that the intensity |u(x, t)|2 is well-behaved. The intensity |u(x, t)|2 shows that the diffraction pattern at a distance t consistsof a series of alternating bright and dark fringes with period Ξ»t/(2a).
3
5.2.1.1 Part (a)
ππ’ππ‘
+ π’ππ’ππ₯
= ππ2π’ππ₯2
(1)
284
5.2. Exam 2 CHAPTER 5. EXAMS
Let
π’ = β2πππ₯π₯
=πππ₯
οΏ½β2π lnποΏ½ (1A)
5.2.1.2 Part(b)
Let
π = β2π lnπ (2)
Hence (1A) becomes
π’ =πππ₯π
We now substitute the above back into (1) noting first that
ππ’ππ‘
=πππ‘ππππ₯
Interchanging the order gives
ππ’ππ‘
=πππ₯ππππ‘
=πππ₯ππ‘
Andππ’ππ₯
=πππ₯ππππ₯
= ππ₯π₯
Andπ2π’ππ₯2
= ππ₯π₯π₯
Hence the original PDE (1) now can be written in term of π as the new dependent variableas
πππ₯ππ‘ + ππ₯ οΏ½ππ₯π₯οΏ½ = πππ₯π₯π₯ (3)
But
ππ₯ οΏ½ππ₯π₯οΏ½ =12πππ₯
οΏ½π2π₯οΏ½
285
5.2. Exam 2 CHAPTER 5. EXAMS
Using the above in (3), then (3) becomes
πππ₯ππ‘ +
12πππ₯
οΏ½π2π₯οΏ½ = πππ₯π₯π₯
πππ₯ππ‘ +
12πππ₯
οΏ½π2π₯οΏ½ β π
πππ₯
οΏ½ππ₯π₯οΏ½ = 0
πππ₯ οΏ½
ππ‘ +12π2π₯ β πππ₯π₯οΏ½ = 0
Therefore
ππ‘ +12π2π₯ β πππ₯π₯ = 0 (4)
But from (2) π = β2π lnπ, then using this in (4), we now rewrite (4) in terms of π
πππ‘οΏ½β2π lnποΏ½ + 1
2 οΏ½πππ₯
οΏ½β2π lnποΏ½οΏ½2
β ππ2
ππ₯2οΏ½β2π lnποΏ½ = 0
οΏ½β2πππ‘π οΏ½
+12 οΏ½β2π
ππ₯π οΏ½
2
β ππππ₯ οΏ½
β2πππ₯π οΏ½
= 0
β2πππ‘π+ 2π2 οΏ½
ππ₯π οΏ½
2
+ 2π2πππ₯ οΏ½
ππ₯π οΏ½
= 0
But πππ₯οΏ½ππ₯ποΏ½ = ππ₯π₯
π β π2π₯π2 , hence the above becomes
β2πππ‘π+ 2π2 οΏ½
ππ₯π οΏ½
2
+ 2π2 οΏ½ππ₯π₯π
βπ2π₯π2 οΏ½ = 0
β2πππ‘π+ 2π2 οΏ½
ππ₯π οΏ½
2
+ 2π2ππ₯π₯π
β 2π2π2π₯π2 = 0
β2πππ‘π+ 2π2
ππ₯π₯π
= 0
βππ‘π+ π
ππ₯π₯π
= 0
Since π β 0 identically, then the above simplifies to the heat PDE
ππ‘ = πππ₯π₯ (5)
π (π₯, 0) = Ξ¦ (π₯)ββ < π₯ < β
5.2.1.3 Part (c)
Now we solve (5) for π (π₯, π‘) and then convert the solution back to π’ (π₯, π‘) using the Cole-Hopftransformation. This infinite domain heat PDE has known solution (as π (Β±β, π‘) is boundedwhich is
π (π₯, π‘) = οΏ½β
ββΞ¦οΏ½ποΏ½
1
β4πππ‘exp
βββββββββ οΏ½π₯ β ποΏ½
2
4ππ‘
ββββββββ ππ (6)
286
5.2. Exam 2 CHAPTER 5. EXAMS
5.2.1.4 Part(d)
Now
Ξ¦ (π₯) = π (π₯, 0) (7)
But since π’ (π₯, π‘) = πππ₯οΏ½β2π lnποΏ½, then integrating
οΏ½π₯
π₯0π’ (πΌ, π‘) ππΌ = β2π lnπ
lnπ = β12π οΏ½
π₯
π₯0π’ (πΌ, π‘) ππΌ
π (π₯, π‘) = exp οΏ½β12π οΏ½
π₯
π₯0π’ (πΌ, π‘) ππΌοΏ½
Hence at π‘ = 0 the above becomes
π (π₯, 0) = exp οΏ½β12π οΏ½
π₯
π₯0π’ (πΌ, 0) ππΌοΏ½
= exp οΏ½β12π οΏ½
π₯
π₯0πΉ (πΌ) ππΌοΏ½
Where πΉ (π₯) = π’ (π₯, 0). Hence from the above, comparing it to (6) we see that
Ξ¦ (π₯) = exp οΏ½β12π οΏ½
π₯
π₯0πΉ (πΌ) ππΌοΏ½ (8)
5.2.1.5 Part(e)
From (6), we found π (π₯, π‘) = β«β
ββΞ¦οΏ½ποΏ½ 1
β4πππ‘exp
ββββββοΏ½π₯βποΏ½
2
4ππ‘
βββββ ππ. Plugging (8) into this expression
gives
π (π₯, π‘) =1
β4πππ‘οΏ½
β
ββexp οΏ½
β12π οΏ½
π
π₯0πΉ (πΌ) ππΌοΏ½ exp
βββββββββ οΏ½π₯ β ποΏ½
2
4ππ‘
ββββββββ ππ
=1
β4πππ‘οΏ½
β
ββexp
βββββββββ12π οΏ½
π
π₯0πΉ (πΌ) ππΌ β
οΏ½π₯ β ποΏ½2
4ππ‘
ββββββββ ππ
=1
β4πππ‘οΏ½
β
ββexp
βββββββββ12π
β‘β’β’β’β’β’β’β£οΏ½
π
π₯0πΉ (πΌ) ππΌ +
οΏ½π₯ β ποΏ½2
2π‘
β€β₯β₯β₯β₯β₯β₯β¦
ββββββββ ππ (9)
Let
οΏ½π
π₯0πΉ (πΌ) ππΌ +
οΏ½π₯ β ποΏ½2
2π‘= π οΏ½π, π₯, π‘οΏ½
287
5.2. Exam 2 CHAPTER 5. EXAMS
Hence (9) becomes
π (π₯, π‘) = οΏ½β
ββ
1
β4πππ‘πβποΏ½π,π₯,π‘οΏ½
2π ππ (10)
5.2.1.6 Part(f)
From (10)
ππππ₯
= οΏ½β
ββ
1
β4πππ‘πππ₯
βββββπ
βποΏ½π,π₯,π‘οΏ½2π
βββββ ππ
= οΏ½β
ββ
1
β4πππ‘
βββββπππ₯π οΏ½π, π₯, π‘οΏ½ π
βποΏ½π,π₯,π‘οΏ½2π
βββββ ππ
= οΏ½β
ββ
1
β4πππ‘
ββββββββπππ₯
β‘β’β’β’β’β’β’β£οΏ½
π
π₯0πΉ (πΌ) ππΌ +
οΏ½π₯ β ποΏ½2
2π‘
β€β₯β₯β₯β₯β₯β₯β¦ π
βποΏ½π,π₯,π‘οΏ½2π
ββββββββ ππ
Using Leibniz integral rule the above simplifies to
ππππ₯
= οΏ½β
ββ
1
β4πππ‘
ββββββοΏ½π₯ β ποΏ½π‘
πβποΏ½π,π₯,π‘οΏ½
2π
ββββββ ππ (11)
But
π’ = β2πππ₯π
Hence, using (10) and (11) in the above gives
π’ = β2πβ«β
ββ1
β4πππ‘
βββββοΏ½π₯βποΏ½
π‘ πβποΏ½π,π₯,π‘οΏ½
2π
βββββ ππ
β«β
ββ1
β4πππ‘πβποΏ½π,π₯,π‘οΏ½
2π ππ
Hence the solution is
π’ (π₯, π‘) = β2πβ«β
ββ
οΏ½π₯βποΏ½
π‘ πβποΏ½π,π₯,π‘οΏ½
2π ππ
β«β
ββπβποΏ½π,π₯,π‘οΏ½
2π ππ
Where
π οΏ½π, π₯, π‘οΏ½ = οΏ½π
π₯0πΉ (πΌ) ππΌ +
οΏ½π₯ β ποΏ½2
2π‘
288
5.2. Exam 2 CHAPTER 5. EXAMS
5.2.2 problem 4
(e) Write your expression for Ο(x, t) in terms of
f(Ξ·, x, t) =
β« Ξ·
xo
F (Ξ±)dΞ±+(xβ Ξ·)2
2t.
(f) Find Οx(x, t) and then use equation (3b) to find u(x, t).
4. (a) Use the Method of Images to solve
βu
βt= Ξ½
β2u
βx2+Q(x, t), 0 β€ x β€ L, t β₯ 0
u(x, 0) = f(x),βu
βx(0, t) = 0, u(L, t) = A
(b) For A = 0, compare your expression for the solution in (a) to the eigenfunction solution.
5. This problem is a simple model for diffraction of light passing through infinitesimallysmall slits separated by a distance 2a.
Solve the diffraction equation
βu
βt=iΞ»
4Ο
β2u
βx2(1)
with initial source u(x, 0) = f(x) = Ξ΄(xβ a) + Ξ΄(x+ a), a > 0.
Show that the solution u(x, t) oscillates wildly, but that the intensity |u(x, t)|2 is well-behaved. The intensity |u(x, t)|2 shows that the diffraction pattern at a distance t consistsof a series of alternating bright and dark fringes with period Ξ»t/(2a).
3
5.2.2.1 Part(a)
ππ’ππ‘
= ππ2π’ππ₯2
+ π (π₯, π‘) (1)
0 β€ π₯ β€ πΏπ‘ β₯ 0
Initial conditions are
π’ (π₯, 0) = π (π₯)
Boundary conditions are
ππ’ (0, π‘)ππ₯
= 0
π’ (πΏ, π‘) = π΄
Multiplying both sides of (1) by πΊ (π₯, π‘; π₯0, π‘0) and integrating over the domain gives (wherein the following πΊ is used instead of πΊ (π₯, π‘; π₯0, π‘0) for simplicity).
οΏ½πΏ
π₯=0οΏ½
β
π‘=0πΊπ’π‘ ππ‘ππ₯ = οΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππ’π₯π₯πΊ ππ‘ππ₯ +οΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππΊ ππ‘ππ₯ (1)
For the integral on the LHS, we apply integration by parts once to move the time derivativefrom π’ to πΊ
οΏ½πΏ
π₯=0οΏ½
β
π‘=0πΊπ’π‘ ππ‘ππ₯ = οΏ½
πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
πΏ
π₯=0οΏ½
β
π‘=0πΊπ‘π’ ππ‘ππ₯ (1A)
289
5.2. Exam 2 CHAPTER 5. EXAMS
And the first integral in the RHS of (1) gives, after doing integration by parts two times onit
οΏ½πΏ
π₯=0οΏ½
β
π‘=0ππ’π₯π₯πΊ ππ‘ππ₯ = οΏ½
β
π‘=0[π’π₯πΊ]
πΏπ₯=0 ππ‘ βοΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππ’π₯πΊπ₯ ππ‘ππ₯
= οΏ½πΏ
π‘=0[π’π₯πΊ]
πΏπ₯=0 ππ‘ β οΏ½οΏ½
β
π‘=0[π’πΊπ₯]
πΏπ₯=0 ππ‘ βοΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯οΏ½
= οΏ½β
π‘=0οΏ½[π’π₯πΊ]
πΏπ₯=0 β [π’πΊπ₯]
πΏπ₯=0οΏ½ ππ‘ +οΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯
= οΏ½β
π‘=0[π’π₯πΊ β π’πΊπ₯]
πΏπ₯=0 ππ‘ +οΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯
= βοΏ½β
π‘=0[π’πΊπ₯ β π’π₯πΊ]
πΏπ₯=0 ππ‘ +οΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯ (1B)
Substituting (1A) and (1B) back into (1) results in
οΏ½πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯βοΏ½
πΏ
π₯=0οΏ½
β
π‘=0πΊπ‘π’ ππ‘ππ₯ = οΏ½
β
π‘=0[π’π₯πΊ β π’πΊπ₯]
πΏπ₯=0 ππ‘+οΏ½
πΏ
π₯=0οΏ½
β
π‘=0ππ’πΊπ₯π₯ ππ‘ππ₯+οΏ½
πΏ
π₯=0οΏ½
β
π‘=0πΊπ ππ‘ππ₯
Or
οΏ½πΏ
π₯=0οΏ½
β
π‘=0βπΊπ‘π’ β ππ’πΊπ₯π₯ ππ‘ππ₯ = βοΏ½
πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
β
π‘=0[π’πΊπ₯ β π’π₯πΊ]
πΏπ₯=0 ππ‘ +οΏ½
πΏ
π₯=0οΏ½
β
π‘=0πΊπ ππ‘ππ₯
(2)
We now want to choose πΊ (π₯, π‘; π₯0, π‘0) such that
βπΊπ‘π’ β ππ’πΊπ₯π₯ = πΏ (π₯ β π₯0) πΏ (π‘ β π‘0)βπΊπ‘π’ = ππ’πΊπ₯π₯ + πΏ (π₯ β π₯0) πΏ (π‘ β π‘0) (3)
This way, the LHS of (2) becomes just π’ (π₯0, π‘0). Hence (2) now (after the above choice ofπΊ) reduces to
π’ (π₯0, π‘0) = βοΏ½πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
π‘0
π‘=0[π’πΊπ₯ β π’π₯πΊ]
πΏπ₯=0 ππ‘ +οΏ½
πΏ
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯ (4)
We now need to find the Green function which satisfies (3). But (3) is equivalent to solutionof problem
βπΊπ‘π’ = ππ’πΊπ₯π₯
πΊ (π₯, 0) = πΏ (π₯ β π₯0) πΏ (π‘ β π‘0)ββ < π₯ < β
πΊ (π₯, π‘; π₯0, π‘0) = 0 π‘ > π‘0πΊ (Β±β, π‘; π₯0, π‘0) = 0πΊ (π₯, π‘0; π₯0, π‘0) = πΏ (π₯ β π₯0)
The above problem has a known fundamental solution which we found before, but for theforward heat PDE instead of the reverse heat PDE as it is now. The fundamental solution
290
5.2. Exam 2 CHAPTER 5. EXAMS
to the forward heat PDE is
πΊ (π₯, π‘) =1
β4ππ (π‘ β π‘0)exp
ββββββ (π₯ β π₯0)
2
4π (π‘ β π‘0)
βββββ 0 β€ π‘0 β€ π‘
Therefore, for the reverse heat PDE the above becomes
πΊ (π₯, π‘) =1
β4ππ (π‘0 β π‘)exp
ββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ 0 β€ π‘ β€ π‘0 (5)
We now go back to (4) and try to evaluate all terms in the RHS. Starting with the first termβ«πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯. Since πΊ (π₯,β; π₯0, π‘0) = 0 then the upper limit is zero. But at lower limit π‘ = 0
we are given that π’ (π₯, 0) = π (π₯), hence this term becomes
οΏ½πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯ = οΏ½
πΏ
π₯=0βπ’ (π₯, 0) πΊ (π₯, 0) ππ₯
= οΏ½πΏ
π₯=0βπ (π₯)πΊ (π₯, 0) ππ₯
Now looking at the second term in RHS of (4), we expand it and find
[π’πΊπ₯ β π’π₯πΊ]πΏπ₯=0 = (π’ (πΏ, π‘) πΊπ₯ (πΏ, π‘) β π’π₯ (πΏ, π‘) πΊ (πΏ, π‘)) β (π’ (0, π‘) πΊπ₯ (0, π‘) β π’π₯ (0, π‘) πΊ (0, π‘))
We are told that π’π₯ (0, π‘) = 0 and π’ (πΏ, π‘) = π΄, then the above becomes
[π’πΊπ₯ β π’π₯πΊ]πΏπ₯=0 = π΄πΊπ₯ (πΏ, π‘) β π’π₯ (πΏ, π‘) πΊ (πΏ, π‘) β π’ (0, π‘) πΊπ₯ (0, π‘) (5A)
There are still two terms above we do not know. We do not know π’π₯ (πΏ, π‘) and we also donot know π’ (0, π‘). If we can configure, using method of images, such that πΊ (πΏ, π‘) = 0 andπΊπ₯ (0, π‘) = 0 then we can get rid of these two terms and end up only with [π’πΊπ₯ β π’π₯πΊ]
βπ₯=0 =
π΄πΊπ₯ (πΏ, π‘) which we can evaluate once we know what πΊ (π₯, π‘) is.
This means we need to put images on both sides of the boundaries such that to forceπΊ (πΏ, π‘) = 0 and also πΊπ₯ (0, π‘) = 0.
We see that this result agrees what we always did, which is, If the prescribed boundary
conditions on π’ are such that π’ = π΄, then we want πΊ = 0 there. And if it is ππ’ππ₯ = π΄, then we
want ππΊππ₯ = 0 there. And this is what we conclude here also from the above. In other words,
the boundary conditions on Green functions are always the homogeneous version of theboundary conditions given on π’.
βuβx = 0βGβx = 0
x = 0 x = L
u(L, t) = A
G(L, t) = 0
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5.2. Exam 2 CHAPTER 5. EXAMS
To force πΊπ₯ (0, π‘) = 0, we need to put same sign images on both sides of π₯ = 0. So we endup with this
x0x = 0 x = L
βGβx = 0
G = 0βx0
The above makes πΊπ₯ (0, π‘) = 0 at π₯ = 0. Now we want to make πΊ = 0 at π₯ = πΏ. Then we updatethe above and put a negative image at π₯ = 2πΏ β π₯0 to the right of π₯ = πΏ as follows
x0x = 0 x = L
βGβx = 0
G = 0βx0
2Lβ x0
But now we see that the image at π₯ = βπ₯0 has aοΏ½ected condition of πΊ = 0 at π₯ = πΏ and willmake it not zero as we wanted. So to counter eοΏ½ect this, we have to add another negativeimage at distance π₯ = 2πΏ + π₯0 to cancel the eοΏ½ect of the image at π₯ = βπ₯0. We end up withthis setup
x0x = 0 x = L
βGβx = 0
G = 0βx0
2Lβ x0 2L+ x0
But now we see that the two negative images we added to the right will no longer makeπΊπ₯ (0, π‘) = 0, so we need to counter eοΏ½ect this by adding two negative images to the left sideto keep πΊπ₯ (0, π‘) = 0. So we end up with
292
5.2. Exam 2 CHAPTER 5. EXAMS
x0x = 0 x = L
βGβx = 0
G = 0βx0
2Lβ x0 2L+ x0β2Lβ x0 β2L+ x0
But now we see that by putting these two images on the left, we no longer have πΊ = 0 atπ₯ = πΏ. So to counter eοΏ½ect this, we have to put copies of these 2 images on the right againbut with positive sign, as follows
x0x = 0 x = L
βGβx = 0
G = 0βx0
2Lβ x0 2L+ x0β2Lβ x0 β2L+ x0
4Lβ x0 4L+ x0
But now these two images on the right, no longer keep πΊπ₯ (0, π‘) = 0, so we have to put samesign images to the left, as follows
x0x = 0 x = L
βGβx = 0
G = 0
βx0
2Lβ x0
2L+ x0β2Lβ x0
β2L+ x0
4Lβ x0
4L+ x0
β4L+ x0
β4Lβ x0
And so on. This continues for infinite number of images. Therefore we see from the above,for the positive images, we have the following sum
οΏ½4ππ (π‘0 β π‘)πΊ (π₯, π‘; π₯0, π‘0) = expββββββ (π₯ β π₯0)
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ + π₯0)
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (4πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ
+ expββββββ (π₯ β (β4πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (4πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
+ expββββββ (π₯ β (β4πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ +β―
Or
πΊ (π₯, π‘; π₯0, π‘0) =1
β4ππ (π‘0 β π‘)
βββββ
βοΏ½π=ββ
expββββββ (π₯ β (4ππΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (β4ππΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
βββββ (6)
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5.2. Exam 2 CHAPTER 5. EXAMS
The above takes care of the positive images. For negative images, we have this sum of images
οΏ½4ππ (π‘0 β π‘)πΊ (π₯, π‘; π₯0, π‘0) = expββββββ (π₯ β (2πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (β2πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
+ expββββββ (π₯ β (2πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (β2πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ
+ expββββββ (π₯ β (6πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (β6πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
+ expββββββ (π₯ β (6πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (β6πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ β―
Or
οΏ½4ππ (π‘0 β π‘)πΊ (π₯, π‘; π₯0, π‘0) = ββββββ
βοΏ½π=ββ
expββββββ (π₯ β ((4π β 2) πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β ((4π β 2) πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ
βββββ
(7)
Hence the Green function to use is (6)+(7) which gives
πΊ (π₯, π‘; π₯0, π‘0) =1
β4ππ (π‘0 β π‘)
βββββ
βοΏ½π=ββ
expββββββ (π₯ β (4ππΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β (β4ππΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
βββββ
β1
β4ππ (π‘0 β π‘)
βββββ
βοΏ½π=ββ
expββββββ (π₯ β ((4π β 2) πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ + exp
ββββββ (π₯ β ((4π β 2) πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ
βββββ
(7A)
Using the above Green function, we go back to 5A and finally are able to simplify it
[π’πΊπ₯ β π’π₯πΊ]πΏπ₯=0 = π΄πΊπ₯ (πΏ, π‘) β π’π₯ (πΏ, π‘) πΊ (πΏ, π‘) β π’ (0, π‘) πΊπ₯ (0, π‘)
The above becomes now (with the images in place as above)
[π’πΊπ₯ β π’π₯πΊ]πΏπ₯=0 = π΄
ππΊ (πΏ, π‘; π₯0, π‘0)ππ₯
(8)
Since now we know what πΊ (π₯, π‘; π₯0, π‘0), from (7), we can evaluate its derivative w.r.t. π₯. (brokenup, so it fits on one page)
ππΊ (π₯, π‘; π₯0, π‘0)ππ₯
=1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (π₯ β (4ππΏ β π₯0))2π (π‘0 β π‘)
expββββββ (π₯ β (4ππΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ
+1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (π₯ β (β4ππΏ + π₯0))2π (π‘0 β π‘)
expββββββ (π₯ β (β4ππΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
β1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (π₯ β ((4π β 2) πΏ + π₯0))2π (π‘0 β π‘)
expββββββ (π₯ β ((4π β 2) πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
β1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (π₯ β ((4π β 2) πΏ β π₯0))2π (π‘0 β π‘)
expββββββ (π₯ β ((4π β 2) πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ
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5.2. Exam 2 CHAPTER 5. EXAMS
At π₯ = πΏ, the above derivative becomes
ππΊ (πΏ, π‘; π₯0, π‘0)ππ₯
=1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (πΏ β (4ππΏ β π₯0))2π (π‘0 β π‘)
expββββββ (πΏ β (4ππΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ
+1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (πΏ β (β4ππΏ + π₯0))2π (π‘0 β π‘)
expββββββ (πΏ β (β4ππΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
β1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (πΏ β ((4π β 2) πΏ + π₯0))2π (π‘0 β π‘)
expββββββ (πΏ β ((4π β 2) πΏ + π₯0))
2
4π (π‘0 β π‘)
βββββ
β1
β4ππ (π‘0 β π‘)
βοΏ½π=ββ
β (πΏ β ((4π β 2) πΏ β π₯0))2π (π‘0 β π‘)
expββββββ (πΏ β ((4π β 2) πΏ β π₯0))
2
4π (π‘0 β π‘)
βββββ (9)
From (4), we now collect all terms into the solution
π’ (π₯0, π‘0) = βοΏ½πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯ βοΏ½
π‘0
π‘=0[π’πΊπ₯ β π’π₯πΊ]
πΏπ₯=0 ππ‘ +οΏ½
πΏ
π₯=0οΏ½
π‘0
π‘=0πΊπ ππ‘ππ₯ (4)
We found β«πΏ
π₯=0[π’πΊ]βπ‘=0 ππ₯ = β«πΏ
π₯=0βπ (π₯)πΊ (π₯, 0) ππ₯ and now we know what πΊ is. Hence we can
find πΊ (π₯, 0; π₯0, π‘0). It is, from (7A)
πΊ (π₯, 0; π₯0, π‘0) =1
β4πππ‘0
βββββ
βοΏ½π=ββ
expββββββ (π₯ β (4ππΏ β π₯0))
2
4ππ‘0
βββββ + exp
ββββββ (π₯ β (β4ππΏ + π₯0))
2
4ππ‘0
βββββ
βββββ
β1
β4πππ‘0
βββββ
βοΏ½π=ββ
expββββββ (π₯ β ((4π β 2) πΏ + π₯0))
2
4ππ‘0
βββββ + exp
ββββββ (π₯ β ((4π β 2) πΏ β π₯0))
2
4ππ‘0
βββββ
βββββ
(10)
And we now know what [π’πΊπ₯ β π’π₯πΊ]πΏπ₯=0 is. It is π΄
ππΊ(πΏ,π‘;π₯0,π‘0)ππ₯ . Hence (4) becomes
π’ (π₯0, π‘0) = οΏ½πΏ
π₯=0π (π₯)πΊ (π₯, 0; π₯0, π‘0) ππ₯βοΏ½
π‘0
π‘=0π΄ππΊ (πΏ, π‘; π₯0, π‘0)
ππ₯ππ‘+οΏ½
πΏ
π₯=0οΏ½
π‘0
π‘=0πΊ (π₯, π‘; π₯0, π‘0) π (π₯, π‘) ππ‘ππ₯
Changing the roles of π₯0, π‘0
π’ (π₯, π‘) = οΏ½πΏ
π₯0=0π (π₯0) πΊ (π₯0, π‘0; π₯, 0) ππ₯0 βοΏ½
π‘
π‘=0π΄ππΊ (π₯0, π‘0; πΏ, π‘)
ππ₯0ππ‘0 +οΏ½
πΏ
π₯0=0οΏ½
π‘
π‘0=0πΊ (π₯0, π‘0; π₯, π‘) π (π₯0, π‘0) ππ‘0ππ₯0
(11)
This completes the solution.
Summary
The solution is
π’ (π₯, π‘) = οΏ½πΏ
π₯0=0π (π₯0) πΊ (π₯0, π‘0; π₯, 0) ππ₯0βοΏ½
π‘
π‘=0π΄ππΊ (π₯0, π‘0; πΏ, π‘)
ππ₯0ππ‘0+οΏ½
πΏ
π₯0=0οΏ½
π‘
π‘0=0πΊ (π₯0, π‘0; π₯, π‘) π (π₯0, π‘0) ππ‘0ππ₯0
Where πΊ (π₯0, π‘0; π₯, 0) is given in (10) (after changing roles of parameters):
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5.2. Exam 2 CHAPTER 5. EXAMS
πΊ (π₯0, π‘0; π₯, 0) =1
β4πππ‘
βββββ
βοΏ½π=ββ
expββββββ (π₯0 β (4ππΏ β π₯))
2
4ππ‘
βββββ + exp
ββββββ (π₯0 β (β4ππΏ + π₯))
2
4ππ‘
βββββ
βββββ
β1
β4πππ‘
βββββ
βοΏ½π=ββ
expββββββ (π₯0 β ((4π β 2) πΏ + π₯))
2
4ππ‘
βββββ + exp
ββββββ (π₯0 β ((4π β 2) πΏ β π₯))
2
4ππ‘
βββββ
βββββ
(10A)
and ππΊ(π₯0,π‘0;πΏ,π‘)ππ₯0
is given in (9) (after also changing roles of parameters):
ππΊ (π₯0, π‘0; πΏ, π‘)ππ₯0
=1
β4ππ (π‘ β π‘0)
βοΏ½π=ββ
β (πΏ β (4ππΏ β π₯))2π (π‘ β π‘0)
expββββββ (πΏ β (4ππΏ β π₯))2
4π (π‘ β π‘0)
βββββ
+1
β4ππ (π‘ β π‘0)
βοΏ½π=ββ
β (πΏ β (β4ππΏ + π₯))2π (π‘ β π‘0)
expββββββ (πΏ β (β4ππΏ + π₯))2
4π (π‘ β π‘0)
βββββ
β1
β4ππ (π‘ β π‘0)
βοΏ½π=ββ
β (πΏ β ((4π β 2) πΏ + π₯))2π (π‘ β π‘0)
expββββββ (πΏ β ((4π β 2) πΏ + π₯))2
4π (π‘ β π‘0)
βββββ
β1
β4ππ (π‘ β π‘0)
βοΏ½π=ββ
β (πΏ β ((4π β 2) πΏ β π₯))2π (π‘ β π‘0)
expββββββ (πΏ β ((4π β 2) πΏ β π₯))2
4π (π‘ β π‘0)
βββββ (9A)
and πΊ (π₯0, π‘0; π₯, π‘) is given in (7A), but with roles changed as well to become
πΊ (π₯0, π‘0; π₯, π‘) =1
β4ππ (π‘ β π‘0)
βββββ
βοΏ½π=ββ
expββββββ (π₯0 β (4ππΏ β π₯))
2
4π (π‘ β π‘0)
βββββ + exp
ββββββ (π₯0 β (β4ππΏ + π₯))
2
4π (π‘ β π‘0)
βββββ
βββββ
(7AA)
β1
β4ππ (π‘ β π‘0)
βββββ
βοΏ½π=ββ
expββββββ (π₯0 β ((4π β 2) πΏ + π₯))
2
4π (π‘ β π‘0)
βββββ + exp
ββββββ (π₯0 β ((4π β 2) πΏ β π₯))
2
4π (π‘ β π‘0)
βββββ
βββββ
5.2.2.2 Part(b)
When π΄ = 0, the solution in part (a) becomes
π’ (π₯, π‘) = οΏ½πΏ
π₯0=0π (π₯0) πΊ (π₯0, π‘0; π₯, 0) ππ₯0 +οΏ½
πΏ
π₯0=0οΏ½
π‘
π‘0=0πΊ (π₯0, π‘0; π₯, π‘) π ππ‘0ππ₯0
Where πΊ (π₯0, π‘0; π₯, 0) is given in (10A) in part (a), and πΊ (π₯0, π‘0; π₯, π‘) is given in (7AA) in part(a). Now we find the eigenfunction solution for this problem order to compare it with theabove green function images solution. Since π΄ = 0 then the PDE now becomes
ππ’ππ‘
= ππ2π’ππ₯2
+ π (π₯, π‘) (1)
0 β€ π₯ β€ πΏπ‘ β₯ 0
Initial conditions
π’ (π₯, 0) = π (π₯)
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5.2. Exam 2 CHAPTER 5. EXAMS
Boundary conditions
ππ’ (0, π‘)ππ₯
= 0
π’ (πΏ, π‘) = 0
Since boundary conditions has now become homogenous (thanks for π΄ = 0), we can useseparation of variables to find the eigenfunctions, and then use eigenfunction expansion.Let the solution be
π’π (π₯, π‘) = ππ (π‘) ππ (π₯)
π’ (π₯, π‘) =βοΏ½π=1
ππ (π‘) ππ (π₯) (1A)
Where ππ (π‘) are eigenfunctions for the associated homogeneous PDE ππ’ππ‘ = π
π2π’ππ₯2 which can
be found from separation of variables. To find ππ (π₯), we start by separation of variables.Let π’ (π₯, π‘) = π (π₯) π (π‘) and we plug this solution back to the PDE to obtain
ππβ² = ππβ²β²π1π£πβ²
π=πβ²β²
π= βπ
Hence the spatial ODE is πβ²β²
π = βπ or πβ²β² + ππ = 0 with boundary conditions
πβ² (0) = 0π (πΏ) = 0
case π = 0 The solution is π = π1 + π2π₯. Hence πβ² = π2. Therefore π2 = 0. Hence π = π1 = 0.Trivial solution. So π = 0 is not possible.
case π > 0 The solution is π = π1 cos οΏ½βππ₯οΏ½+π2 sin οΏ½βππ₯οΏ½. and πβ² = ββππ1 sinππ₯+π2βπ cosππ₯.From first B.C. at π₯ = 0 we find 0 = π2βπ, hence π2 = 0 and the solution becomes π =π1 cos οΏ½βππ₯οΏ½. At π₯ = πΏ, we have 0 = π1 cos οΏ½βππΏοΏ½ which leads to βππΏ = π
π2 for π = 1, 3, 5,β― or
οΏ½ππ = οΏ½2π β 12 οΏ½
ππΏ
π = 1, 2, 3,β―
ππ = οΏ½2π β 12
ππΏ οΏ½
2
π = 1, 1, 2, 3,β―
Hence the ππ (π₯) solution is
ππ (π₯) = ππ cos οΏ½οΏ½πππ₯οΏ½ π = 1, 2, 3,β―
The time ODE is now solved using the above eigenvalues. (we really do not need to do thispart, since π΄π (π‘) will be solved for later, and π΄π (π‘) will contain all the time dependent parts,
297
5.2. Exam 2 CHAPTER 5. EXAMS
including those that come from π (π₯, π‘), but for completion, this is done)1π£πβ²
π= βππ
πβ² + π£πππ = 0πππ= βπ£ππππ‘
ln |π| = βπ£πππ‘ + πΆπ = πΆπππ£πππ‘
Hence the solution to the homogenous PDE is
π’π (π₯, π‘) = ππππ= ππ cos οΏ½οΏ½πππ₯οΏ½ ππ£πππ‘
Where constants of integration are merged into one. Therefore
π’ (π₯, π‘) =βοΏ½π=1
ππππ
=βοΏ½π=1
ππ cos οΏ½οΏ½πππ₯οΏ½ ππ£πππ‘ (2)
From the above we see that
ππ (π₯) = cos οΏ½βπππ₯οΏ½
Using this, we now write the solution to ππ’ππ‘ = π
π2π’ππ₯2 + π (π₯, π‘) using eigenfunction expansion
π’ (π₯, π‘) =βοΏ½π=1
π΄π (π‘) ππ (π₯) (3)
Where now π΄π (π‘) will have all the time dependent terms from π (π₯, π‘) as well from the time
solution from the homogenous PDE ππ£οΏ½ 2πβ12
ππΏ οΏ½
2π‘part. We will solve for π΄π (π‘) now.
In this below, we will expand π (π₯, π‘) using these eigenfunctions (we can do this, since theeigenfunctions are basis for the whole solution space which the forcing function is in as well).We plug-in (3) back into the PDE, and since boundary conditions are now homogenous,
then term by term diοΏ½erentiation is justified. The PDE ππ’ππ‘ = π
π2π’ππ₯2 + π (π₯, π‘) now becomes
πππ‘
βοΏ½π=1
π΄π (π‘) ππ (π₯) = ππ2
ππ₯2βοΏ½π=1
π΄π (π‘) ππ (π₯) +βοΏ½π=1
ππ (π‘) ππ (π₯) (4)
Where ββπ=1 ππ (π‘) ππ (π₯) is the eigenfunction expansion of π (π₯, π‘). To find ππ (π‘) we apply
298
5.2. Exam 2 CHAPTER 5. EXAMS
orthogonality as follows
π (π₯, π‘) =βοΏ½π=1
ππ (π‘) ππ (π₯)
οΏ½πΏ
0ππ (π₯)π (π₯, π‘) ππ₯ = οΏ½
πΏ
0οΏ½βοΏ½π=1
ππ (π‘) ππ (π₯)οΏ½ ππ (π₯) ππ₯
οΏ½πΏ
0ππ (π₯)π (π₯, π‘) ππ₯ =
βοΏ½π=1
οΏ½πΏ
0ππ (π‘) ππ (π₯) ππ (π₯) ππ₯
= οΏ½πΏ
0ππ (π‘) π2
π (π₯) ππ₯
= ππ (π‘)οΏ½πΏ
0cos2 οΏ½οΏ½πππ₯οΏ½ ππ₯
But
οΏ½πΏ
0cos2 οΏ½οΏ½πππ₯οΏ½ ππ₯ =
πΏ2
Hence
ππ (π‘) =2πΏ οΏ½
πΏ
0ππ (π₯)π (π₯, π‘) ππ₯ (5)
Now that we found ππ (π‘), we go back to (4) and simplifies it moreβοΏ½π=1
π΄β²π (π‘) ππ (π₯) = π
βοΏ½π=1
π΄π (π‘) πβ²β²π (π₯) +
βοΏ½π=1
ππ (π‘) ππ (π₯)
π΄β²π (π‘) ππ (π₯) = ππ΄π (π‘) πβ²β²
π (π₯) + ππ (π‘) ππ (π₯)
But since ππ (π₯) = cos οΏ½βπππ₯οΏ½ then
πβ²π (π₯) = β οΏ½οΏ½πποΏ½ sin οΏ½οΏ½πππ₯οΏ½
And
πβ²β²π (π₯) = βππ cos οΏ½οΏ½πππ₯οΏ½
= βππππ (π₯)
Hence the above ODE becomes
π΄β²πππ = βππ΄πππππ + ππππ
Canceling the eigenfunction ππ (π₯) (since not zero) gives
π΄β²π (π‘) + ππ΄π (π‘) ππ = ππ (π‘) (6)
We now solve this for π΄π (π‘). Integrating factor is
π = exp οΏ½οΏ½πππππ‘οΏ½
= πππππ‘
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5.2. Exam 2 CHAPTER 5. EXAMS
Hence (6) becomesπππ‘οΏ½ππ΄π (π‘)οΏ½ = πππ (π‘)
πππππ‘π΄π (π‘) = οΏ½π‘
0πππππ ππ (π ) ππ + πΆ
π΄π (π‘) = πβππππ‘οΏ½π‘
0πππππ ππ (π ) ππ + πΆπβππππ‘ (7)
Now that we found π΄π (π‘), then the solution (3) becomes
π’ (π₯, π‘) =βοΏ½π=1
ββββββπ
βοΏ½ 2πβ12ππΏ οΏ½
2ππ‘οΏ½
π‘
0ποΏ½ 2πβ12
ππΏ οΏ½
2ππ ππ (π ) ππ + πΆπ
βοΏ½ 2πβ12ππΏ οΏ½
2ππ‘ββββββ ππ (π₯) (8)
At π‘ = 0 , we are given that π’ (π₯, 0) = π (π₯), hence the above becomes
π (π₯) =βοΏ½π=1
πΆππ (π₯)
To find πΆ, we apply orthogonality again, which gives
οΏ½πΏ
0π (π₯) ππ (π₯) ππ₯ =
βοΏ½π=1
οΏ½πΏ
0πΆππ (π₯) ππ (π₯) ππ₯
= πΆοΏ½πΏ
0π2π (π₯) ππ₯
οΏ½πΏ
0π (π₯) ππ (π₯) ππ₯ =
πΏ2πΆ
πΆ =2πΏ οΏ½
πΏ
0π (π₯) ππ (π₯) ππ₯
Now that we found πΆ, then the solution in (8) is complete. It is
π’ (π₯, π‘) =βοΏ½π=1
π΄π (π‘) ππ (π₯)
=βοΏ½π=1
οΏ½πβππππ‘οΏ½π‘
0πππππ ππ (π ) ππ +
2πΏπβππππ‘οΏ½
πΏ
0π (π₯) ππ (π₯) ππ₯οΏ½ππ (π₯)
Or
π’ (π₯, π‘) =βοΏ½π=1
οΏ½ππ (π₯) πβππππ‘οΏ½π‘
0πππππ ππ (π ) ππ οΏ½
+2πΏ
βοΏ½π=1
οΏ½ππ (π₯) πβππππ‘οΏ½πΏ
0π (π₯) ππ (π₯) ππ₯οΏ½
Summary
The eigenfunction solution is
π’ (π₯, π‘) =βοΏ½π=1
οΏ½ππ (π₯) πβππππ‘οΏ½π‘
0πππππ ππ (π ) ππ οΏ½
+2πΏ
βοΏ½π=1
οΏ½ππ (π₯) πβππππ‘οΏ½πΏ
0π (π₯) ππ (π₯) ππ₯οΏ½
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5.2. Exam 2 CHAPTER 5. EXAMS
Where
ππ (π₯) = cos οΏ½οΏ½πππ₯οΏ½ π = 1, 2, 3,β―
ππ = οΏ½2π β 12
ππΏ οΏ½
2
π = 1, 1, 2, 3,β―
And
ππ (π‘) =2πΏ οΏ½
πΏ
0ππ (π₯)π (π₯, π‘) ππ₯
To compare the eigenfunction expansion solution and the Green function solution, we seethe following mapping of the two solutions
2πΏββπ=1οΏ½ππ(π₯)π
βππππ‘β«πΏ0
π(π₯)ππ(π₯)ππ₯οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
πΏ
0π (π₯0) πΊ (π₯0, π‘0; π₯, 0) ππ₯0 +
ββπ=1οΏ½ππ(π₯)π
βππππ‘β«π‘0πππππ ππ(π )ππ οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
πΏ
0οΏ½
π‘
0πΊ (π₯0, π‘0; π₯, π‘) π (π₯0, π‘0) ππ‘0ππ₯0
Where the top expression is the eigenfunction expansion and the bottom expression is theGreen function solution using method of images. Where πΊ (π₯0, π‘0; π₯, π‘) in above contains theinfinite sums of the images. So the Green function solution contains integrals and insidethese integrals are the infinite sums. While the eigenfunction expansions contains two infinitesums, but inside the sums we see the integrals. So summing over the images seems to beequivalent to the operation of summing over eigenfunctions. These two solutions in thelimit should of course give the same result (unless I made a mistake somewhere). In thisexample, I found method of eigenfunction expansion easier, since getting the images incorrect locations and sign was tricky to get right.
301
5.2. Exam 2 CHAPTER 5. EXAMS
5.2.3 problem 5
(e) Write your expression for Ο(x, t) in terms of
f(Ξ·, x, t) =
β« Ξ·
xo
F (Ξ±)dΞ±+(xβ Ξ·)2
2t.
(f) Find Οx(x, t) and then use equation (3b) to find u(x, t).
4. (a) Use the Method of Images to solve
βu
βt= Ξ½
β2u
βx2+Q(x, t), 0 β€ x β€ L, t β₯ 0
u(x, 0) = f(x),βu
βx(0, t) = 0, u(L, t) = A
(b) For A = 0, compare your expression for the solution in (a) to the eigenfunction solution.
5. This problem is a simple model for diffraction of light passing through infinitesimallysmall slits separated by a distance 2a.
Solve the diffraction equation
βu
βt=iΞ»
4Ο
β2u
βx2(1)
with initial source u(x, 0) = f(x) = Ξ΄(xβ a) + Ξ΄(x+ a), a > 0.
Show that the solution u(x, t) oscillates wildly, but that the intensity |u(x, t)|2 is well-behaved. The intensity |u(x, t)|2 shows that the diffraction pattern at a distance t consistsof a series of alternating bright and dark fringes with period Ξ»t/(2a).
3
ππ’ (π₯, π‘)ππ‘
=ππ4π
π2π’ (π₯, π‘)ππ₯2
(1)
π’ (π₯, 0) = π (π₯) = πΏ (π₯ β π) + πΏ (π₯ + π)
I will Use Fourier transform to solve this, since this is for ββ < π₯ < β and the solution π’ (π₯, π‘)is assumed bounded at Β±β (or goes to zero there), hence π’ (π₯, π‘) is square integrable andtherefore we can assume it has a Fourier transform.
Let π (π, π‘) be the spatial part only Fourier transform of π’ (π₯, π‘). Using the Fourier transformpairs defined as
π (π, π‘) = β± (π’ (π₯, π‘)) = οΏ½β
ββπ’ (π₯, π‘) πβπ2ππ₯πππ₯
π’ (π₯, π‘) = β± β1 (π (π, π‘)) = οΏ½β
ββπ (π, π‘) ππ2ππ₯πππ
Therefore, by Fourier transform properties of derivatives
β±οΏ½ππ’ (π₯, π‘)ππ₯ οΏ½ = (2πππ)π (π, π‘)
β± οΏ½π2π’ (π₯, π‘)ππ₯2 οΏ½ = (2πππ)2π (π, π‘) (2)
And
β±οΏ½ππ’ (π₯, π‘)ππ‘ οΏ½ =
ππ (π, π‘)ππ‘
(3)
Where in (3), we just need to take time derivative of π (π, π‘) since the transform is appliedonly to the space part. Now we take the Fourier transform of the given PDE and using (2,3)
302
5.2. Exam 2 CHAPTER 5. EXAMS
relations we obtain the PDE but now in Fourier space.
ππ (π, π‘)ππ‘
= οΏ½ππ4ποΏ½
(2πππ)2π (π, π‘)
= β οΏ½ππ4ποΏ½
4π2π2π (π, π‘)
= οΏ½βππππ2οΏ½π (π, π‘) (4)
Equation (4) can now be easily solved for π (π, π‘) since it is separable.ππ (π, π‘)π (π, π‘)
= οΏ½βππππ2οΏ½ ππ‘
Integrating
ln |π (π, π‘)| = οΏ½βππππ2οΏ½ π‘ + πΆ
π (π, π‘) = π (π, 0) ποΏ½βππππ2οΏ½π‘
Where π (π, 0) is the Fourier transform of π’ (π₯, 0), the initial conditions, which is π (π₯) and isgiven in the problem. To go back to spatial domain, we now need to do the inverse Fouriertransform. By applying the convolution theorem, we know that multiplication in Fourierdomain is convolution in spatial domain, therefore
β± β1 (π (π, π‘)) = β± β1 (π (π, 0)) β β± β1 οΏ½πβππππ2π‘οΏ½ (5)
But
β± β1 (π (π, π‘)) = π’ (π₯, π‘)β± β1 (π (π, 0)) = π (π₯)
And
β± β1 οΏ½πβππππ2π‘οΏ½ = οΏ½β
ββπβππππ2π‘π2πππ₯πππ (5A)
Hence (5) becomes
π’ (π₯, π‘) = π (π₯) β β± β1 οΏ½πβππππ2π‘οΏ½
Here, I used Mathematica to help me with the above integral (5A) as I could not find it intables so far2. Here is the result
Find inverse Fourier transform, for problem 5, NE 548
In[13]:= InverseFourierTransform[ Exp[-I lam Pi z^2 t], z, x, FourierParameters -> {1, -2 * Pi}]
Out[13]=β
β Ο x2
lam t
2 Ο β lam t
2Trying to do this integral by hand also, but so far having some diοΏ½culty..
303
5.2. Exam 2 CHAPTER 5. EXAMS
Therefore, from Mathematica, we see that
β± β1 οΏ½πβππππ2π‘οΏ½ =ππππ₯2ππ‘
β2πβπβππ‘(6)
But3
βπ =1
β2+ π
1
β2
= cos οΏ½π4οΏ½ + π sin οΏ½π
4οΏ½
= πππ4
Hence (6) becomes
β± β1 οΏ½πβππππ2π‘οΏ½ =1
πππ4β2πππ‘
ππππ₯2ππ‘ (7)
Now we are ready to do the convolution in (5A) since we know everything in the RHS,hence
π’ (π₯, π‘) = π (π₯) β1
πππ4β2πππ‘
ππππ₯2ππ‘
= (πΏ (π₯ β π) + πΏ (π₯ + π)) β1
πππ4β2πππ‘
ππππ₯2ππ‘ (8)
Applying convolution integral on (8), which says that
π (π₯) = π1 (π₯) β π2 (π₯)
= οΏ½β
ββπ1 (π§) π2 (π₯ β π§) ππ§
Therefore (8) becomes
π’ (π₯, π‘) = οΏ½β
ββ(πΏ (π§ β π) + πΏ (π§ + π))
1
πππ4β2πππ‘
πππ(π₯βπ§)2
ππ‘ ππ§
=1
πππ4β2πππ‘
οΏ½β
ββ(πΏ (π§ β π) + πΏ (π§ + π)) ππ
π(π₯βπ§)2
ππ‘ ππ§
=1
πππ4β2πππ‘
βββββοΏ½
β
ββπΏ (π§ β π) ππ
π(π₯βπ§)2
ππ‘ ππ§ +οΏ½β
ββπΏ (π§ + π) ππ
π(π₯βπ§)2
ππ‘ ππ§βββββ
But an integral with delta function inside it, is just the integrand evaluated where the deltaargument become zero which is at π§ = π and π§ = βπ in the above. (This is called the siftingproperty). Hence the above integrals are now easily found and we obtain the solution
π’ (π₯, π‘) =1
πππ4β2πππ‘
βββββexp
βββββππ (π₯ β π)2
ππ‘
βββββ + exp
βββββππ (π₯ + π)2
ππ‘
βββββ
βββββ
3Taking the positive root only.
304
5.2. Exam 2 CHAPTER 5. EXAMS
The above is the solution we need. But we can simplify it more by using Euler relation.
π’ (π₯, π‘) =1
πππ4β2πππ‘
ββββββexp
ββββββππ οΏ½π₯2 + π2 β 2π₯ποΏ½
ππ‘
ββββββ + exp
ββββββππ οΏ½π₯2 + π2 + 2ππ₯οΏ½
ππ‘
ββββββ
ββββββ
=1
πππ4β2πππ‘
ββββββexp
ββββββππ οΏ½π₯2 + π2οΏ½
ππ‘
ββββββ exp οΏ½
βπ2ππ₯πππ‘ οΏ½ + exp
ββββββππ οΏ½π₯2 + π2οΏ½
ππ‘
ββββββ exp οΏ½
π2πππ₯ππ‘ οΏ½
ββββββ
Taking exp οΏ½πποΏ½π₯2+π2οΏ½
ππ‘ οΏ½ as common factor outside results in
π’ (π₯, π‘) =exp οΏ½
πποΏ½π₯2+π2οΏ½
ππ‘ οΏ½
πππ4β2πππ‘
οΏ½exp οΏ½π2πππ₯ππ‘ οΏ½ + exp οΏ½βπ
2ππ₯πππ‘ οΏ½οΏ½
=2 exp οΏ½
πποΏ½π₯2+π2οΏ½
ππ‘ οΏ½
πππ4β2πππ‘
ββββββββββ
exp οΏ½π2πππ₯ππ‘οΏ½ + exp οΏ½βπ2ππ₯πππ‘
οΏ½
2
ββββββββββ
=2 exp οΏ½
πποΏ½π₯2+π2οΏ½
ππ‘ οΏ½
πππ4β2πππ‘
cos οΏ½2πππ₯ππ‘ οΏ½
=β2 exp οΏ½
πποΏ½π₯2+π2οΏ½
ππ‘ β ππ4 οΏ½
βπππ‘cos οΏ½
2πππ₯ππ‘ οΏ½
=β2 exp οΏ½π οΏ½
ποΏ½π₯2+π2οΏ½
ππ‘ β π4 οΏ½οΏ½
βπππ‘cos οΏ½
2πππ₯ππ‘ οΏ½
Hence the final solution is
π’ (π₯, π‘) = οΏ½2
πππ‘ exp οΏ½πποΏ½π₯2+π2οΏ½
ππ‘ β π4 οΏ½ cos οΏ½2πππ
π₯π‘οΏ½ (9)
Hence the real part of the solution is
β (π’ (π₯, π‘)) =οΏ½
2πππ‘
cosββββββπ οΏ½π₯2 + π2οΏ½
ππ‘βπ4
ββββββ cos οΏ½
2πππ
π₯π‘ οΏ½
And the imaginary part of the solution is
β (π’ (π₯, π‘)) =οΏ½
2πππ‘
sinββββββπ οΏ½π₯2 + π2οΏ½
ππ‘βπ4
ββββββ cos οΏ½
2πππ
π₯π‘ οΏ½
The π4 is just a phase shift. Here is a plot of the Real and Imaginary parts of the solution,
using for π = 600 Γ 10β9πππ‘ππ, π = 1000 Γ 10β9πππ‘ππ at π‘ = 1 second
305
5.2. Exam 2 CHAPTER 5. EXAMS
In[15]:= a = 1000 * 10^(-9);
Ξ» = 600 * 10^(-9);
u[x_, t_] :=2
Ο Ξ» t
ExpIΟ x
2+ a2
Ξ» t
-Ο
4 Cos
2 Ο a
Ξ»
x
t
In[59]:= Clear[t];
Plot[Re@u[x, 1], {x, -3000 a, 3000 a}, Frame -> True,
FrameLabel β {{"u(x,t)", None}, {"x", "Real part of solution at t= 1 second"}}, BaseStyle β 12,
PlotStyle β Red, PlotTheme β "Classic"]
Out[60]=
-0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
-400
-200
0
200
400
x
u(x,t)
Real part of solution at t= 1 second
In[61]:= Plot[Im@u[x, 1], {x, -3000 a, 3000 a}, Frame -> True,
FrameLabel β {{"u(x,t)", None}, {"x", "Imaginary part of solution at t= 1 second"}}, BaseStyle β 12,
PlotStyle β Red, PlotTheme β "Classic"]
Out[61]=
-0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
-400
-200
0
200
400
x
u(x,t)
Imaginary part of solution at t= 1 second
We see the rapid oscillations as distance goes away from the origin. This is due to the π₯2
term making the radial frequency value increase quickly with π₯. We now plot the |π’ (π₯, π‘)|2.Looking at solution in (9), and since complex exponential is Β±1, then the amplitude is
governed by οΏ½2
πππ‘ cos οΏ½2ππππ₯π‘οΏ½ part of the solution. Hence
|π’ (π₯, π‘)|2 = 2πππ‘ cos2 οΏ½2πππ
π₯π‘οΏ½
These plots show the intensity at diοΏ½erent time values. We see from these plots, that theintensity is well behaved in that it does not have the same rapid oscillations seen in theπ’ (π₯, π‘) solution plots.
306
5.2. Exam 2 CHAPTER 5. EXAMS
In[154]:= a = 1000 * 10^(-9);
Ξ» = 600 * 10^(-9);
intensity[x_, t_] :=2
Ο Ξ» t
Cos2 Ο a
Ξ»
x
t
2
p =
Plotintensity[x, #], {x, -3000 a, 3000 a}, Frame -> True,
FrameLabel β {"u(x,t)", None}, "x", Row" intensity |u 2 of solution at t =", #, " second",
BaseStyle β 12, PlotStyle β Red, PlotTheme β "Classic", ImageSize β 300 & /@ Range[0.002, 0.012, 0.002];
Out[153]=
-0.003-0.002-0.001 0.000 0.001 0.002 0.003
0
1Γ 108
2Γ 108
3Γ 108
4Γ 108
5Γ 108
x
u(x,t)
intensity |u 2 of solution at t =0.002 second
-0.003-0.002-0.001 0.000 0.001 0.002 0.003
0
5.0Γ 107
1.0Γ 108
1.5Γ 108
2.0Γ 108
2.5Γ 108
xu(x,t)
intensity |u 2 of solution at t =0.004 second
-0.003-0.002-0.001 0.000 0.001 0.002 0.003
0
5.0Γ 107
1.0Γ 108
1.5Γ 108
x
u(x,t)
intensity |u 2 of solution at t =0.006 second
-0.003-0.002-0.001 0.000 0.001 0.002 0.003
0
2.0Γ 1074.0Γ 1076.0Γ 1078.0Γ 1071.0Γ 1081.2Γ 1081.4Γ 108
x
u(x,t)
intensity |u 2 of solution at t =0.008 second
-0.003-0.002-0.001 0.000 0.001 0.002 0.003
0
2Γ 107
4Γ 107
6Γ 107
8Γ 107
1Γ 108
x
u(x,t)
intensity |u 2 of solution at t =0.01 second
-0.003-0.002-0.001 0.000 0.001 0.002 0.003
0
2Γ 107
4Γ 107
6Γ 107
8Γ 107
x
u(x,t)
intensity |u 2 of solution at t =0.012 second
Now Comparing argument to cosine in above to standard form in order to find the period:2πππ
π₯π‘= 2πππ‘
307
5.2. Exam 2 CHAPTER 5. EXAMS
Where π is now in hertz, then when π₯ = π‘, we get by comparing terms that2πππ
1π‘= 2ππ
ππ1π‘= π
But π = 1π where π is the period in seconds. Hence π
π1π‘ =
1π or
π = ππ‘π
So period on intensity is ππ‘π at π₯ = π‘ (why problem statement is saying period is ππ‘
2π?).
308
5.2. Exam 2 CHAPTER 5. EXAMS
5.2.4 problem 2 (optional)
2. Consider the 1D heat equation in a semi-infinite domain:
βu
βt= Ξ½
β2u
βx2, x β₯ 0
with boundary conditions: u(0, t) = exp(βiΟt) and u(x, t) bounded as x β β. In orderto construct a real forcing, we need both positive and negative real values of Ο. Considerthat this forcing has been and will be applied for all time. This βpure boundary valueproblemβ could be an idealization of heating the surface of the earth by the sun (periodicforcing). One could then ask, how far beneath the surface of the earth do the periodicfluctuations of the heat propagate?
(a) Consider solutions of the form u(x, t;Ο) = exp(ikx) exp(βiΟt). Find a single expressionfor k as a function of (given) Ο real, sgn(Ο) and Ξ½ real.
Write u(x, t;Ο) as a function of (given) Ο real, sgn(Ο) and Ξ½ real. To obtain the mostgeneral solution by superposition, one would next integrate over all values of Ο, ββ <Ο <β (do not do this).
(b) The basic solution can be written as u(x, t;Ο) = exp(βiΟt) exp(βΟx) exp(iΟ sgn(Ο) x).Find Ο in terms of |Ο| and Ξ½.
(c) Make an estimate for the propagation depth of daily temperature fluctuations.
3. Here we study the competing effects of nonlinearity and diffusion in the context ofBurgerβs equation
βu
βt+ u
βu
βx= Ξ½
β2u
βx2(3a)
which is the simplest model equation for diffusive waves in fluid dynamics. It can be solvedexactly using the Cole-Hopf transformation
u = β2Ξ½ΟxΟ
(3b)
as follows (with 2 steps to achieve the transformation (3b)).
(a) Let u = Οx (where the subscript denotes partial differentiation) and integrate oncewith respect to x.
(b) Let Ο = β2Ξ½ ln(Ο) to get the diffusion equation for Ο.
(c) Solve for Ο with Ο(x, 0) = Ξ¦(x), ββ < x < β. In your integral expression for Ο, usedummy variable Ξ· to facilitate the remaining parts below.
(d) Show that
Ξ¦(x) = exp
[
β1
2Ξ½
β« x
xo
F (Ξ±)dΞ±
]
where u(x, 0) = F (x), with xo arbitrary which we will take to be positive for conveniencebelow (xo > 0).
2
5.2.4.1 Part (a)
ππ’ππ‘
= ππ2π’ππ₯2
(1)
π’ (0, π‘) = πβπππ‘
And π₯ β₯ 0, π’ (β, π‘) bounded. Let
π’ (π₯, π‘) = ππππ₯πβπππ‘
Henceππ’ (π₯, π‘)ππ‘
= βππππππ₯πβπππ‘
= βπππ’ (π₯, π‘) (2)
Andππ’ (π₯, π‘)ππ₯
= ππππππ₯πβπππ‘
π2π’ (π₯, π‘)ππ₯2
= βπ2ππππ₯πβπππ‘
= βπ2π’ (π₯, π‘) (3)
Substituting (2,3) into (1) gives
βπππ’ (π₯, π‘) = βππ2π’ (π₯, π‘)309
5.2. Exam 2 CHAPTER 5. EXAMS
Since π’π (π₯, π‘) can not be identically zero (trivial solution), then the above simplifies to
βππ = βππ2
Or
π2 = πππ (4)
Writing
π = sgn (π) |π|Where
sgn (π) =
β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ©
+1 π > 00 π = 0β1 π < 0
Then (4) becomes
π2 =π sgn (π) |π|
π
π = Β±βπ βsgn (π)β|π|
βπSince
βπ = π12 = οΏ½ππ
π2 οΏ½
12= ππ
π4
Hence π can be written as
π = Β±ππ
π4 βsgn (π)β|π|
βπcase A. Let start with the positive root hence
π =ππ
π4 βsgn (π)β|π|
βπCase (A1) π < 0 then the above becomes
π =πππ
π4 β|π|
βπ=ππ
π2 ππ
π4 β|π|
βπ=ππ
3π4 β|π|
βπ
= οΏ½cos 34π + π sin 3
4ποΏ½
β|π|
βπ
= οΏ½β1
β2+ π
1
β2οΏ½β|π|
βπ
=ββββββ
1
β2β|π|
βπ+ π
1
β2β|π|
βπ
βββββ
310
5.2. Exam 2 CHAPTER 5. EXAMS
And the solution becomes
π’ (π₯, π‘) = exp (πππ₯) exp (βπππ‘)
= expβββββπββββββ
1
β2β|π|
βπ+ π
1
β2β|π|
βπ
βββββ π₯βββββ exp (βπππ‘)
= expβββββ
ββββββ
1
β2β|π|
βππ β
1
β2β|π|
βπ
βββββ π₯βββββ exp (βπππ‘)
= expββββββ
1
β2β|π|
βππ₯βββββ exp
ββββββ
1
β2β|π|
βπππ₯βββββ exp (βπππ‘)
We are told that π’ (β, π‘) is bounded. So for large π₯ we want the above to be bounded. Thecomplex exponential present in the above expression cause no issue for large π₯ since theyare oscillatory trig functions. We then just need to worry about exp οΏ½β 1
β2β|π|
βππ₯οΏ½ for large π₯.
This term will decay for large π₯ since β 1
β2β|π|
βπis negative (assuming π > 0 always). Hence
positive root hence worked OK when π < 0. Now we check if it works OK also when π > 0
case A2 When π > 0 then π now becomes
π =ππ
π4 βsgn (π)β|π|
βπ
=ππ
π4 βπ
βπ
= οΏ½cos π4+ π sin π
4οΏ½ βπ
βπ
= οΏ½1
β2+ π
1
β2οΏ½ βπ
βπ
=1
β2βπ
βπ+ π
1
β2βπ
βπAnd the solution becomes
π’ (π₯, π‘) = exp (πππ₯) exp (βπππ‘)
= exp οΏ½π οΏ½1
β2βπ
βπ+ π
1
β2βπ
βποΏ½ π₯οΏ½ exp (βπππ‘)
= exp οΏ½οΏ½π1
β2βπ
βπβ
1
β2βπ
βποΏ½ π₯οΏ½ exp (βπππ‘)
= exp οΏ½β1
β2βπ
βππ₯οΏ½ exp οΏ½π
1
β2βπ
βππ₯οΏ½ exp (βπππ‘)
We are told that π’ (β, π‘) is bounded. So for large π₯ we want the above to be bounded. Thecomplex exponential present in the above expression cause no issue for large π₯ since theyare oscillatory trig functions. We then just need to worry about exp οΏ½β 1
β2βπ
βππ₯οΏ½ for large π₯.
311
5.2. Exam 2 CHAPTER 5. EXAMS
This term will decay for large π₯ since β 1
β2βπ
βπis negative (assuming π > 0 always). Hence
positive root hence worked OK when π > 0 as well.
Let check what happens if we use the negative root.
case B. negative root hence
π = βππ
π4 βsgn (π)β|π|
βπCase (A1) π < 0 then the above becomes
π = βπππ
π4 β|π|
βπ= β
πππ2 ππ
π4 β|π|
βπ= β
ππ3π4 β|π|
βπ
= β οΏ½cos 34π + π sin 3
4ποΏ½
β|π|
βπ
= β οΏ½β1
β2+ π
1
β2οΏ½β|π|
βπ
= βββββββ
1
β2β|π|
βπ+ π
1
β2β|π|
βπ
βββββ
=βββββ1
β2β|π|
βπβ π
1
β2β|π|
βπ
βββββ
And the solution becomes
π’ (π₯, π‘) = exp (πππ₯) exp (βπππ‘)
= expβββββπβββββ1
β2β|π|
βπβ π
1
β2β|π|
βπ
βββββ π₯βββββ exp (βπππ‘)
= expβββββ
βββββπ1
β2β|π|
βπ+
1
β2β|π|
βπ
βββββ π₯βββββ exp (βπππ‘)
= expβββββ+
1
β2β|π|
βππ₯βββββ exp
βββββ1
β2β|π|
βπππ₯βββββ exp (βπππ‘)
We are told that π’ (β, π‘) is bounded. So for large π₯ we want the above to be bounded. Thecomplex exponential present in the above expression cause no issue for large π₯ since theyare oscillatory trig functions. We then just need to worry about exp οΏ½+ 1
β2β|π|
βππ₯οΏ½ for large π₯.
This term will blow up for large π₯ since + 1
β2β|π|
βπis positive (assuming π > 0 always). Hence
we reject the case of negative sign on π. And pick
π =ππ
π4 βsgn (π)β|π|
βπ
= οΏ½1
β2+ π
1
β2οΏ½βsgn (π)β|π|
βπ
312
5.2. Exam 2 CHAPTER 5. EXAMS
Therefore the solution is
π’ (π₯, π‘; π) = exp (πππ₯) exp (βπππ‘)
= exp
ββββββββββπ
ββββββββββ
οΏ½ 1
β2+ π 1
β2οΏ½ βsgn (π)β|π|
βπ
ββββββββββ π₯
ββββββββββ exp (βπππ‘)
= exp οΏ½π οΏ½1
β2ποΏ½sgn (π)β|π| + π
1
β2ποΏ½sgn (π)β|π| οΏ½ π₯οΏ½ exp (βπππ‘)
= exp οΏ½π1
β2ποΏ½sgn (π)β|π|π₯ β
1
β2ποΏ½sgn (π)β|π| π₯οΏ½ exp (βπππ‘)
Hence
π’ (π₯, π‘; π) = exp οΏ½ββsgn(π)β|π|β2π
π₯οΏ½ exp οΏ½πβsgn(π)β|π|β2π
π₯οΏ½ exp (βπππ‘)
The general solution π’ (π₯, π‘) is therefore the integral over all π, hence
π’ (π₯, π‘) = οΏ½β
π=ββπ’ (π₯, π‘; π) ππ
= οΏ½β
ββexp
βββββββsgn (π)β|π|
β2ππ₯βββββ exp
βββββπβsgn (π)β|π|
β2ππ₯βββββ exp (βπππ‘) ππ
= οΏ½β
ββexp
βββββββsgn (π)β|π|
β2ππ₯βββββ exp
βββββπβ‘β’β’β’β£βsgn (π)β|π|
β2ππ₯ β ππ‘
β€β₯β₯β₯β¦
βββββ ππ
= οΏ½β
ββexp
βββββββsgn (π)β|π|
β2ππ₯βββββ exp
βββββππ₯
β‘β’β’β’β£βsgn (π)β|π|
β2πβ π
π‘π₯
β€β₯β₯β₯β¦
βββββ ππ
5.2.4.2 Part (b)
From part (a), we found that
π’ (π₯, π‘; π) = exp οΏ½ββsgn(π)β|π|β2π
π₯οΏ½ exp οΏ½πβsgn(π)β|π|β2π
π₯οΏ½ exp (βπππ‘) (1)
comparing the above to expression given in problem which is
π’ (π₯, π‘) = exp (βππ₯) exp (ππ sgn (π) π₯) exp (βπππ‘) (2)
Therefore, by comparing exp (βππ₯) to exp οΏ½ββsgn(π)β|π|β2π
π₯οΏ½ we see that
π = βsgn(π)β|π|β2π
(3)
5.2.4.3 part (c)
Using the solution found in part (a)
π’ (π₯, π‘; π) = exp (βππ₯) exp (ππ sgn (π) π₯) exp (βπππ‘)
313
5.2. Exam 2 CHAPTER 5. EXAMS
To find numerical estimate, assuming π > 0 for now
π’ (π₯, π‘; π) = exp (βππ₯) exp (πππ₯) exp (βπππ‘)= πβππ₯ (cos ππ₯ + π sin ππ₯) (cosππ‘ β π sinππ‘)= πβππ₯ (cos ππ₯ cosππ‘ β π sinππ‘ cos ππ₯ + π cosππ‘ sin ππ₯ + sin ππ₯ sinππ‘)= πβππ₯ (cos (ππ₯) cos (ππ‘) + sin (ππ₯) sin (ππ‘) + π (cos (ππ‘) sin (ππ₯) β sin (ππ‘) cos (ππ₯)))= πβππ₯ (cos (π‘π β π₯π) β π sin (π‘π β π₯π))
Hence will evaluate
Re (π’ (π₯, π‘; π)) = πβππ₯ Re (cos (π‘π β π₯π) β π sin (π‘π β π₯π))= πβππ₯ cos (π‘π β π₯π)
I assume here it is asking for numerical estimate. We only need to determine numericalestimate for π. For π, using the period π = 24 hrs or π = 86 400 seconds, then π = 2π
π is nowfound. Then we need to determine π, which is thermal diοΏ½usivity for earth crust. There doesnot seem to be an agreed on value for this and this value also changed with depth inside theearth crust. The value I found that seem mentioned more is 1.2 Γ 10β6 meter2 per second.Hence
π = οΏ½2π
86 400
οΏ½2 οΏ½1.2 Γ 10β6οΏ½
= 5.505 per meter
Therefore
Re (π’ (π₯, π‘; π)) = πβ5.505π₯ cos (π‘π β 5.505π₯)
Using π = 2π86 400 = 7.29 Γ 10
β5 rad/sec. Now we can use the above to estimate fluctuation ofheat over 24 hrs period. But we need to fix π₯ for each case. Here π₯ = 0 means on the earthsurface and π₯ say 10, means at depth 10 meters and so on as I understand that π₯ is starts at0 at surface or earth and increases as we go lower into the earth crust. Plotting at the abovefor π₯ = 0, 0.5, 1, 1.5, 2 I see that when π₯ > 2 then maximum value of πβ5.505π₯ cos (π‘π β 5.505π₯)is almost zero. This seems to indicate a range of heat reach is about little more than2 meters below the surface of earth.
This is a plot of the fluctuation in temperature at diοΏ½erent π₯ each in separate plot, then latera plot is given that combines them all.
314
5.2. Exam 2 CHAPTER 5. EXAMS
0 20000 40000 60000 80000
-1.0
-0.5
0.0
0.5
1.0
The time in the day (in seconds)
T(x,t)
Flucuation in temperature T at depth x = 0
0 20000 40000 60000 80000
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
The time in the day (in seconds)
T(x,t)
Flucuation in temperature T at depth x = 0.5
0 20000 40000 60000 80000
-0.004
-0.002
0.000
0.002
0.004
The time in the day (in seconds)
T(x,t)
Flucuation in temperature T at depth x = 1
315
5.2. Exam 2 CHAPTER 5. EXAMS
0 20000 40000 60000 80000
-0.0002
-0.0001
0.0000
0.0001
0.0002
The time in the day (in seconds)
T(x,t)
Flucuation in temperature T at depth x = 1.5
0 20000 40000 60000 80000
-0.000015
-0.000010
-5.Γ10-6
0.000000
5.Γ10-6
0.000010
0.000015
The time in the day (in seconds)
T(x,t)
Flucuation in temperature T at depth x = 2
0 20000 40000 60000 80000
-1.Γ10-6
-5.Γ10-7
0
5.Γ10-7
1.Γ10-6
The time in the day (in seconds)
T(x,t)
Flucuation in temperature T at depth x = 2.5
This plot better show the diοΏ½erence per depth, as it combines all the plots into one.
316
5.2. Exam 2 CHAPTER 5. EXAMS
0 20000 40000 60000 80000
-1.0
-0.5
0.0
0.5
1.0
The time in the day (in seconds)
T(x,t)
Flucuation in temperature T at different depth x
u(0, t)
u(0.5, t)
u(1, t)
u(1.5, t)
317