2P0Lecture02
19
1 Mathematics 2P0 MATH29641, MATH29631 2010-11
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2P0Lecture02
Transcript of 2P0Lecture02
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1Mathematics 2P0
MATH29641, MATH296312010-11
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21.3 Inverse Laplace Transforms
Question:
Once we have found the solution of the transformed problem, how can we transform it back to obtain the solution of the original problem?
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3 For any function f(t), the Laplace
Transform is defined as
1.1 Introduction to Laplace Transforms
( )0
( )stf t e f t dt
=
Recall:
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4 If
Then the process of finding f(t) when only is known, is called taking the Inverse Laplace Transform.
A common notation is
( )f s
1.3 Inverse Laplace Transforms
( )0
( )stf t e f t dt
=
1( ) ( )f t f s =
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51.3 Inverse Laplace Transforms
1.3.1 Inverse Laplace Transforms by Table
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6
The tables of Laplace Transforms can also be used to find Inverse Laplace Transforms. If there is a function of s in the right column, its inverse Laplace Transform will be in the left column.
The linearity properties apply here.
1.3.1 Inverse Laplace Transforms by Table
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7 Example 1.3.1 Find
i)
ii)
iii)
1.3.1 Inverse Laplace Transforms by Table
1 12s
12
5s
1
2
124s
s
+
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81.3 Inverse Laplace Transforms
1.3.2 Inverse Laplace Transforms by Partial Fractions
Review Partial Fractions!
Literature: HELM module 3.6.
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9Partial Fraction Example 1.3.2: Apply the method of partial
fractions to the expression
352107)( 2 ++
+=
ssssf
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10
132352107
2 ++
+=
++
+
sB
sA
sss
)1)(32(352 2 ++=++ ssss
A
Step 1: Factorize
Step 2: Find and such that B
Example 1.3.2
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11
Example 1.3.2 Space for your calculations
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12
13
321
352107
2 ++
+=
++
+
sssss
. we see that
Example 1.3.2
end of example
Space for your calculations
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13
When a rational function is expressed in terms of partial fractions, the resulting functions are in a suitable form for their inverse Laplace Transforms to be identified.
1.3.2 Inverse Laplace Transforms by Partial Fractions
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14
Example 1.3.3 Find the Inverse Laplace Transform of
Solution: in tutorials!
1.3.2 Inverse Laplace Transforms by Partial Fractions
1( )( 1)( 2)
f ss s
=
+
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15
Quadratic Factors
The process for dealing with a quadratic factor of the form
is as follows
1.3.2 Inverse Laplace Transforms by Partial Fractions
2
As Bs qs r
+
+ +
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16
Quadratic Factors
If q = 0, then let so
1.3.2 Inverse Laplace Transforms by Partial Fractions
2
As Bs qs r
+
+ +
2r =-1
2 2 2 2 2 2
cos sin
cos sin
As B s BAs s s
BA t t
BA r t r tr
+ = + + + +
= +
= +
-1-1
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17
Example 1.3.4
Find
1.3.2 Inverse Laplace Transforms by Partial Fractions
-1
2
3 84
ss
+ + Solution: try it
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18
Quadratic Factors 2As B
s qs r+
+ +
2
2 2
If 0, complete the square for to get ( ) .Now, write as ( ) ( )
q s qs rs u v
As B A s u B Au
+ +
+ +
+ + +
1.3.2 Inverse Laplace Transforms by Partial Fractions
-12 2 2 2 2
2 2 2 2
( )( ) ( )
( ) ( )
cos sinut u t
As B A s u B Aus qs r s u v s u v
s u B Au vAs u v v s u v
B AuAe vt e vtv
+ + = + + + + + + +
+ = + + + + +
= +
-1 -1
-1 -1
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19
Example 1.3.5
Find the Inverse Laplace Transform of
Solution: in tutorials.
1.3.2 Inverse Laplace Transforms by Partial Fractions
524)( 2 ++
=
ssssf
Mathematics 2P01.3 Inverse Laplace Transforms Slide 4Slide 5Slide 6 Example 1.3.1Slide 8Partial FractionSlide 10Slide 11Slide 12Slide 13 Example 1.3.3 Quadratic FactorsSlide 16 Example 1.3.4Slide 18 Example 1.3.5
