Chap 4 Laplace Transform

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Chap 4 Laplace Transform . 中華大學 資訊工程系 Fall 2002. Outline. Basic Concepts Laplace Transform Definition, Theorems, Formula Inverse Laplace Transform Definition, Theorems, Formula Solving Differential Equation Solving Integral Equation. Basic Concepts. 微分方程式. 代數方程式. Laplace Transform.

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Chap 4 Laplace Transform

Fall 2002

Outline
• Basic Concepts
• Laplace Transform
• Definition, Theorems, Formula
• Inverse Laplace Transform
• Definition, Theorems, Formula
• Solving Differential Equation
• Solving Integral Equation
Basic Concepts

Laplace Transform

Differential

Equation f(t)

L{ f(t)} = F(s)

Algebra

Equation F(s)

Inverse Laplace Transform

Solution of

Differential

Equation f(t)

L-1{F(s)} = f(t)

Solution of

Algebra

Equation F(s)

Basic Concepts

Laplace Transform

L{ f(t)} = F(s)

Inverse Laplace Transform

L-1{F(s)} = f(t)

Laplace Transform
• Definition

The Laplace transform of a function f(t) is defined as

• Converges: L{f(t)} exists
• Diverges: L{f(t)} does not exist
Laplace Transform

s=0.125

e-st

s=0.25

s=0.5

s=1

s=2

s=4

s=8

t

Laplace Transform
• Example : Find L{ 1}

Sol:

Laplace Transform
• Example : Find L{ eat }

Sol:

Laplace Transform
• Example 4-2 : Find L{ tt }

Sol:

L{ tt } does not exist

Laplace Transform
• Exercise 4-1 :
• Find
• Find
• Find
• Find
Laplace Transform
• Theorems

Definition of Laplace Transform

Linear Property

Derivatives

Integrals

First Shifting Property

Second Shifting Property

Laplace Transform
• Theorems

Change of Scale Property

Multiplication by tn

Division by t

Unit Impulse Function

Periodic Function

Convolution Theorem

First Shifting Theorem
• If f(t) has the transform F(s) (where s > k), then eatf(t) has the transform F(s-a), (where s-a > k), in formulas,

or, if we take the inverse on both sides

Excises sec 5.1
• #1, #7, #19, #24, #29,#35, #37,#39
Examples
• Example 1:

Let f(t)=t2, Derive L(f) from L(1)

• Example 2:

Derive the Laplace transform of cos wt

Differential Equations, Initial Value Problem
• How to use Laplace transform and Laplace inverse to solve the differential equations with given initial values
Laplace Transform of the Integral of a Function
• Theorem : Integration of f(t)

Let F(s) be the Laplace transform of f(t). If f(t) is piecewise continuous and satisfies an inequality of the form (2), Sec. 5.1 , then

or, if we take the inverse transform on both sides of above form

Laplace Transform
• Unit Step Function (also called Heaviside’s Function)
Second Shifting Theorem; t-shifting
• IF f(t) has the transform F(s), then the “shifted function”

has the transform e-asF(s). That is

Application of Unit Step Functions
• Note
• Find the transform of the function
Example :

Find the inverse Laplace transform f(t) of

Area = 1

Laplace Transform
• Unit Impulse Function (also called Dirac Delta Function)
Laplace Transform
• Example 4-6 : Prove

Proof

Homework
• section 5-2

#4, #7, #9, #18, #19

• Section 5-3

#3, #6, #17, #28, #29

Example

Find the inverse transform of the function

Example1

Using the convolution, find the inverse h(t) of

• Example 2
• Example 3
Laplace Transform
• Example 4-7 : Prove

Proof:

Homeworks
• Section 5-4
• #1,#13
• Section 5-5
• #7, #14, #27
Inverse Laplace Transform
• Definition

The Inverse Laplace Transform of a function F(s) is defined as

Inverse Laplace Transform
• Theorems

Inverse Laplace Transform

Linear Property

Derivatives

Integrals

First Shifting Property

Second Shifting Property

Inverse Laplace Transform
• Theorems

Change of Scale Property

Multiplication by tn

Division by t

Unit Impulse Function

Unit Step Function

Convolution Theorem