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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

Chap 4 Laplace Transform

中華大學 資訊工程系

Fall 2002

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

微分方程式

代數方程式

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 concepts1
Basic Concepts

Laplace Transform

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

Inverse Laplace Transform

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

laplace transform
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 transform1
Laplace Transform

s=0.125

e-st

s=0.25

s=0.5

s=1

s=2

s=4

s=8

t

laplace transform2
Laplace Transform
  • Example : Find L{ 1}

Sol:

laplace transform3
Laplace Transform
  • Example : Find L{ eat }

Sol:

laplace transform4
Laplace Transform
  • Example 4-2 : Find L{ tt }

Sol:

L{ tt } does not exist

laplace transform5
Laplace Transform
  • Exercise 4-1 :
    • Find
    • Find
    • Find
    • Find
laplace transform6
Laplace Transform
  • Theorems

Definition of Laplace Transform

Linear Property

Derivatives

Integrals

First Shifting Property

Second Shifting Property

laplace transform7
Laplace Transform
  • Theorems

Change of Scale Property

Multiplication by tn

Division by t

Unit Impulse Function

Periodic Function

Convolution Theorem

first shifting 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
Excises sec 5.1
  • #1, #7, #19, #24, #29,#35, #37,#39
examples
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
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
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 transform8
Laplace Transform
  • Unit Step Function (also called Heaviside’s Function)
second shifting theorem t shifting
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
Application of Unit Step Functions
  • Note
  • Find the transform of the function
slide30
Example :

Find the inverse Laplace transform f(t) of

laplace transform9

Area = 1

Laplace Transform
  • Unit Impulse Function (also called Dirac Delta Function)
laplace transform10
Laplace Transform
  • Example 4-6 : Prove

Proof

homework
Homework
  • section 5-2

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

  • Section 5-3

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

slide39
Example

Find the inverse transform of the function

slide41
Example1

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

  • Example 2
  • Example 3
laplace transform11
Laplace Transform
  • Example 4-7 : Prove

Proof:

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

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

inverse laplace transform1
Inverse Laplace Transform
  • Theorems

Inverse Laplace Transform

Linear Property

Derivatives

Integrals

First Shifting Property

Second Shifting Property

inverse laplace transform2
Inverse Laplace Transform
  • Theorems

Change of Scale Property

Multiplication by tn

Division by t

Unit Impulse Function

Unit Step Function

Convolution Theorem