Introduction Course Contents Numerical Methods

NUMMET HECE Computational Numerical Methods

Engr. Melvin Kong CabatuanDe La Salle University

Manila, Philippines

May 2014

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods

Self Introduction

Engr. Melvin K. Cabatuan, ECEMasters of Engineering, NAIST (Japan)

Thesis: Cognitive Radio (Wireless Communication)ECE Reviewer/Mentor (Since 2005)

2nd Place, Nov. 2004 ECE Board ExamTest Engineering Cadet, ON SemiconductorsDOST Academic Excellence Awardee 2004

Mathematician of the Year 2003DOST Scholar (1999-2004)

Panasonic Scholar, Japan (2007-2010)Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods

1 Introduction

2 Course ContentsEvaluation CriteriaPre-requisiteReferences

3 Numerical MethodsMathematical ModelingProblem SolvingExample

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part I

1 Mathematical Modeling & EngineeringProblem Solving

2 Approximation and Round-off Errors3 Truncation Errors and the Taylor Series4 Roots of Equations

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part I

1 Mathematical Modeling & EngineeringProblem Solving

2 Approximation and Round-off Errors3 Truncation Errors and the Taylor Series4 Roots of Equations

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part I

1 Mathematical Modeling & EngineeringProblem Solving

2 Approximation and Round-off Errors3 Truncation Errors and the Taylor Series4 Roots of Equations

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part I

1 Mathematical Modeling & EngineeringProblem Solving

2 Approximation and Round-off Errors3 Truncation Errors and the Taylor Series4 Roots of Equations

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part II

1 Linear Algebraic Equations

2 Curve Fitting

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part II

1 Linear Algebraic Equations

2 Curve Fitting

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part III

1 Numerical Integration & Differentiationwith Engineering Applications

2 Ordinary Differential Equations &Engineering Applications

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part III

1 Numerical Integration & Differentiationwith Engineering Applications

2 Ordinary Differential Equations &Engineering Applications

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part III

1 Numerical Integration & Differentiationwith Engineering Applications

2 Ordinary Differential Equations &Engineering Applications

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Course Contents - Part III

1 Numerical Integration & Differentiationwith Engineering Applications

2 Ordinary Differential Equations &Engineering Applications

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Evaluation Criteria

Quiz Average: 45%Final Exam: 40%Project: 10%Teacher‘s Evaluation: 5%

Total: 100%PASSING GRADE: 65%

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Pre-requisite

1 LBYEC12 (Hard)2 CONTSIG (Soft)3 Mathematical Background4 C++ or MATLAB/SCILAB

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Pre-requisite

1 LBYEC12 (Hard)2 CONTSIG (Soft)3 Mathematical Background4 C++ or MATLAB/SCILAB

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Pre-requisite

1 LBYEC12 (Hard)2 CONTSIG (Soft)3 Mathematical Background4 C++ or MATLAB/SCILAB

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

Pre-requisite

1 LBYEC12 (Hard)2 CONTSIG (Soft)3 Mathematical Background4 C++ or MATLAB/SCILAB

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

References

1 Canale, R., & Chapra, S. (2009). NumericalMethods for Engineers (6 ed.), New York,McGraw-Hill

2 Fausett, L.V. (2008). Applied NumericalAnalysis using Matlab. USA: PearsonPrentice Hall.

3 Online Resources

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

References

1 Canale, R., & Chapra, S. (2009). NumericalMethods for Engineers (6 ed.), New York,McGraw-Hill

2 Fausett, L.V. (2008). Applied NumericalAnalysis using Matlab. USA: PearsonPrentice Hall.

3 Online Resources

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References

References

1 Canale, R., & Chapra, S. (2009). NumericalMethods for Engineers (6 ed.), New York,McGraw-Hill

2 Fausett, L.V. (2008). Applied NumericalAnalysis using Matlab. USA: PearsonPrentice Hall.

3 Online Resources

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Why study numerical methods?

Numerical Methods expand the types of problems youcan address, i.e. handling large systems of equations,nonlinearities, and complicated geometries.

Numerical Methods allow you to use "canned" softwarewith insight.

Numerical Methods enable you to design your ownprograms to solve problems without having to buy orcommission expensive software.

Numerical Methods are an efficient vehicle for learningto use computers and also reinforce your understanding inmathematics.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Why study numerical methods?

Numerical Methods expand the types of problems youcan address, i.e. handling large systems of equations,nonlinearities, and complicated geometries.

Numerical Methods allow you to use "canned" softwarewith insight.

Numerical Methods enable you to design your ownprograms to solve problems without having to buy orcommission expensive software.

Numerical Methods are an efficient vehicle for learningto use computers and also reinforce your understanding inmathematics.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Why study numerical methods?

Numerical Methods expand the types of problems youcan address, i.e. handling large systems of equations,nonlinearities, and complicated geometries.

Numerical Methods allow you to use "canned" softwarewith insight.

Numerical Methods enable you to design your ownprograms to solve problems without having to buy orcommission expensive software.

Numerical Methods are an efficient vehicle for learningto use computers and also reinforce your understanding inmathematics.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Why study numerical methods?

Numerical Methods expand the types of problems youcan address, i.e. handling large systems of equations,nonlinearities, and complicated geometries.

Numerical Methods allow you to use "canned" softwarewith insight.

Numerical Methods enable you to design your ownprograms to solve problems without having to buy orcommission expensive software.

Numerical Methods are an efficient vehicle for learningto use computers and also reinforce your understanding inmathematics.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Why study numerical methods?

Numerical Methods expand the types of problems youcan address, i.e. handling large systems of equations,nonlinearities, and complicated geometries.

Numerical Methods allow you to use "canned" softwarewith insight.

Numerical Methods enable you to design your ownprograms to solve problems without having to buy orcommission expensive software.

Numerical Methods are an efficient vehicle for learningto use computers and also reinforce your understanding inmathematics.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Numerical Methods/ Analysis

PurposeTo find approximate solutions tocomplicated mathematical problems usingarithmetic operations.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Numerical Methods/ Analysis

Insight} Numerical methods solve hard problemsby doing lots of easy steps. ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Numerical Methods/ Analysis

Insight} Computers are great tools, but w/ofundamental understanding ofengineering problems, they will beuseless! ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Mathematical Modeling

Traditional vs. Modern Approach

FormulationFundamental lawsexplained briefly.

SolutionElaborate and complicated

method.

InterpretationIn-depth analysis limitedtime-consuming solution.

FormulationIn-depth exposition of the

problem.

SolutionEasy-to-use computer

method.

InterpretationMore time for in-depthanalysis due to ease of

calculation.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Mathematical Modeling

Traditional vs. Modern Approach

FormulationFundamental lawsexplained briefly.

SolutionElaborate and complicated

method.

InterpretationIn-depth analysis limitedtime-consuming solution.

FormulationIn-depth exposition of the

problem.

SolutionEasy-to-use computer

method.

InterpretationMore time for in-depthanalysis due to ease of

calculation.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Mathematical Modeling

Traditional vs. Modern Approach

FormulationFundamental lawsexplained briefly.

SolutionElaborate and complicated

method.

InterpretationIn-depth analysis limitedtime-consuming solution.

FormulationIn-depth exposition of the

problem.

SolutionEasy-to-use computer

method.

InterpretationMore time for in-depthanalysis due to ease of

calculation.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Mathematical Modeling

Traditional vs. Modern Approach

FormulationFundamental lawsexplained briefly.

SolutionElaborate and complicated

method.

InterpretationIn-depth analysis limitedtime-consuming solution.

FormulationIn-depth exposition of the

problem.

SolutionEasy-to-use computer

method.

InterpretationMore time for in-depthanalysis due to ease of

calculation.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Mathematical Modeling

Traditional vs. Modern Approach

FormulationFundamental lawsexplained briefly.

SolutionElaborate and complicated

method.

InterpretationIn-depth analysis limitedtime-consuming solution.

FormulationIn-depth exposition of the

problem.

SolutionEasy-to-use computer

method.

InterpretationMore time for in-depthanalysis due to ease of

calculation.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Mathematical Modeling

Traditional vs. Modern Approach

FormulationFundamental lawsexplained briefly.

SolutionElaborate and complicated

method.

InterpretationIn-depth analysis limitedtime-consuming solution.

FormulationIn-depth exposition of the

problem.

SolutionEasy-to-use computer

method.

InterpretationMore time for in-depthanalysis due to ease of

calculation.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Mathematical Modeling

Traditional vs. Modern Approach

FormulationFundamental lawsexplained briefly.

SolutionElaborate and complicated

method.

InterpretationIn-depth analysis limitedtime-consuming solution.

FormulationIn-depth exposition of the

problem.

SolutionEasy-to-use computer

method.

InterpretationMore time for in-depthanalysis due to ease of

calculation.Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Engineering Problem Solving

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Engineering Problem Solving

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Engineering Problem Solving

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Engineering Problem Solving

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Example: Bungee-jumping

Predict the velocity of a jumper as afunction of time during the free-fall part ofthe jump.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Example: Bungee-jumping

Predict the velocity of a jumper as afunction of time during the free-fall part ofthe jump.

F = FD + FUF = Net force acting on the bodyFD = Force due to gravity = mg

FU = Force due to air resistance = −cv(c = drag coefficient)

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Example: Bungee-jumping

Predict the velocity of a jumper as afunction of time during the free-fall part ofthe jump.

dvdt = g − c

mv} This is a first order ordinary linear differential equation. ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analytical Solution

If the jumper is initially at rest (v = 0 att = 0), dv/dt can be solved to give theresult:

v(t) = gmc

(1− e−(c/m)t

)

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analytical Solution

v(t) = gmc

(1− e−(c/m)t

)g = 9.8 m/s2 , c = 12.5 kg/s, m = 68.1 kg

t (sec.) V (m/s)0 02 16.404 27.778 41.10

10 44.8712 47.49∞ 53.39

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Numerical Solution

dvdt = g − c

mv

dvdt∼=

∆v∆t = v(ti+1)− v(ti)

ti+1 − ti........

dvdt = lim

∆t→0

∆v∆t

v(ti+1)− v(ti)ti+1 − ti

= g − cmv(ti)

v(ti+1) = v(ti) + [g − cmv(ti)](ti+1 − ti)

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Numerical Solution

v(ti+1) = v(ti) + [g − cmv(ti)](ti+1 − ti)

@ ∆t = 2 sect (sec.) V (m/s)

0 02 19.604 32.008 44.82

10 47.9712 49.96∞ 53.39

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analytical vs. Numerical Solution

AnalyticalSolution

t (sec.) V (m/s)0 02 16.404 27.778 41.10

10 44.8712 47.49∞ 53.39

Numerical@ ∆t = 2 sect (sec.) V (m/s)

0 02 19.604 32.008 44.82

10 47.9712 49.96∞ 53.39

Numerical@ ∆t = 0.01 sect (sec.) V (m/s)

0 02 16.414 27.838 41.13

10 44.9012 47.51∞ 53.39

} Minimize the error by using smaller step size, ∆t. ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analytical vs. Numerical Solution

AnalyticalSolution

t (sec.) V (m/s)0 02 16.404 27.778 41.10

10 44.8712 47.49∞ 53.39

Numerical@ ∆t = 2 sect (sec.) V (m/s)

0 02 19.604 32.008 44.82

10 47.9712 49.96∞ 53.39

Numerical@ ∆t = 0.01 sect (sec.) V (m/s)

0 02 16.414 27.838 41.13

10 44.9012 47.51∞ 53.39

} Minimize the error by using smaller step size, ∆t. ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analytical vs. Numerical Solution

AnalyticalSolution

t (sec.) V (m/s)0 02 16.404 27.778 41.10

10 44.8712 47.49∞ 53.39

Numerical@ ∆t = 2 sect (sec.) V (m/s)

0 02 19.604 32.008 44.82

10 47.9712 49.96∞ 53.39

Numerical@ ∆t = 0.01 sect (sec.) V (m/s)

0 02 16.414 27.838 41.13

10 44.9012 47.51∞ 53.39

} Minimize the error by using smaller step size, ∆t. ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analytical vs. Numerical Solution

AnalyticalSolution

t (sec.) V (m/s)0 02 16.404 27.778 41.10

10 44.8712 47.49∞ 53.39

Numerical@ ∆t = 2 sect (sec.) V (m/s)

0 02 19.604 32.008 44.82

10 47.9712 49.96∞ 53.39

Numerical@ ∆t = 0.01 sect (sec.) V (m/s)

0 02 16.414 27.838 41.13

10 44.9012 47.51∞ 53.39

} Minimize the error by using smaller step size, ∆t. ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analytical vs. Numerical Solution

AnalyticalSolution

t (sec.) V (m/s)0 02 16.404 27.778 41.10

10 44.8712 47.49∞ 53.39

Numerical@ ∆t = 2 sect (sec.) V (m/s)

0 02 19.604 32.008 44.82

10 47.9712 49.96∞ 53.39

Numerical@ ∆t = 0.01 sect (sec.) V (m/s)

0 02 16.414 27.838 41.13

10 44.9012 47.51∞ 53.39

} Minimize the error by using smaller step size, ∆t. ~

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Analogy

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Key Insights

Numerical Methods find approximate solutions tocomplicated problems using arithmetic operations.

} Solving hard problems with lots of easy steps. ~

Computers are great tools, but w/o fundamentalunderstanding of engineering problems, they will be useless!

You can minimize the error in numerical solutions by usingsmaller step size, ∆t.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Key Insights

Numerical Methods find approximate solutions tocomplicated problems using arithmetic operations.

} Solving hard problems with lots of easy steps. ~

Computers are great tools, but w/o fundamentalunderstanding of engineering problems, they will be useless!

You can minimize the error in numerical solutions by usingsmaller step size, ∆t.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Key Insights

Numerical Methods find approximate solutions tocomplicated problems using arithmetic operations.

} Solving hard problems with lots of easy steps. ~

Computers are great tools, but w/o fundamentalunderstanding of engineering problems, they will be useless!

You can minimize the error in numerical solutions by usingsmaller step size, ∆t.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Key Insights

Numerical Methods find approximate solutions tocomplicated problems using arithmetic operations.

} Solving hard problems with lots of easy steps. ~

Computers are great tools, but w/o fundamentalunderstanding of engineering problems, they will be useless!

You can minimize the error in numerical solutions by usingsmaller step size, ∆t.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

Key Insights

Numerical Methods find approximate solutions tocomplicated problems using arithmetic operations.

} Solving hard problems with lots of easy steps. ~

Computers are great tools, but w/o fundamentalunderstanding of engineering problems, they will be useless!

You can minimize the error in numerical solutions by usingsmaller step size, ∆t.

Engr. Melvin Kong Cabatuan N UMMET H

Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example

END

} Thank you for your attention ~

Engr. Melvin Kong Cabatuan N UMMET H