中華大學 資訊工程系 Fall 2002 Chap 4 Laplace Transform
Page 2 Outline Basic Concepts Laplace Transform Definition, Theorems, Formula Inverse Laplace Transform Definition, Theorems, Formula Solving Differential Equation Solving Integral Equation
Page 3 Basic Concepts Differential Equation f(t) Solution of Differential Equation f(t) Algebra Equation F(s) Solution of Algebra Equation F(s) Laplace Transform Inverse Laplace Transform L { f(t) } = F(s) L -1 { F(s) } = f(t) 微分方程式代數方程式
Page 4 Basic Concepts Laplace Transform Inverse Laplace Transform L { f(t) } = F(s) L -1 { F(s) } = f(t)
Page 5 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
Page 6 Laplace Transform t e -st s=1 s=2 s=4 s=8 s=0.5 s=0.25 s=0.125
Page 7 Laplace Transform Example : Find L { 1 } Sol:
Page 8 Laplace Transform Example : Find L { e at } Sol:
Page 9 Laplace Transform Example 4-2 : Find L { t t } Sol: L { t t } does not exist
Page 10 Laplace Transform Exercise 4-1 : Find
Page 11 Laplace Transform Theorems Definition of Laplace Transform Linear Property Derivatives Integrals First Shifting Property Second Shifting Property
Page 12 Laplace Transform Theorems Change of Scale Property Multiplication by t n Division by t Unit Impulse Function Periodic Function Convolution Theorem
Page 13 Linearity of Laplace Transform Proof:
Page 14 Application for Linearity of Laplace Transform
Page 15 First Shifting Theorem If f(t) has the transform F(s) (where s > k), then e at f(t) has the transform F(s-a), (where s-a > k), in formulas, or, if we take the inverse on both sides
Page 16 Examples for First Shifting Theorem
Page 17 Excises sec 5.1 #1, #7, #19, #24, #29,#35, #37,#39
Page 18 Laplace of Transform the Derivative of f(t) Prove Proof:
Page 19 Laplace transorm of the derivative of any order n
Page 20 Examples Example 1: Let f(t)=t 2, Derive L(f) from L(1) Example 2: Derive the Laplace transform of cos wt
Page 21 Differential Equations, Initial Value Problem How to use Laplace transform and Laplace inverse to solve the differential equations with given initial values
Page 22 Example : Explanation of the Basic Concept Examples
Page 23 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
Page 24 An Application of Integral Theorem Examples
Page 25 Laplace Transform Unit Step Function (also called Heaviside’s Function)
Page 26
Page 27 Second Shifting Theorem; t-shifting IF f(t) has the transform F(s), then the “shifted function” has the transform e -as F(s). That is
Page 28 The Proof of the T-shifting Theorem Prove Proof:
Page 29 Application of Unit Step Functions Note Find the transform of the function
Page 30 Example : Find the inverse Laplace transform f(t) of
Page 31 Short Impulses. Dirac’s Delta Function Area = 1
Page 32 Laplace Transform Unit Impulse Function (also called Dirac Delta Function) Area = 1
Page 33 Laplace Transform Example 4-6 : Prove Proof
Page 34 Example
Page 35 Homework section 5-2 #4, #7, #9, #18, #19 Section 5-3 #3, #6, #17, #28, #29
Page 36 Differentiation and Integration of Transforms Differentiation of transforms
Page 37 Example
Page 38 Integration of Transform
Page 39 Example Find the inverse transform of the function
Page 40 Convolution. Integration Equation Convolution Properties
Page 41 Example1 Using the convolution, find the inverse h(t) of Example 2 Example 3
Page 42 Laplace Transform Example 4-7 : Prove Proof:
Page 43 Differential Equation
Page 44 Integration Equations Example
Page 45 Homeworks Section 5-4 #1,#13 Section 5-5 #7, #14, #27
Page 46 Laplace Transform Formula
Page 47 Laplace Transform Formula
Page 48 Inverse Laplace Transform Definition The Inverse Laplace Transform of a function F(s) is defined as
Page 49 Inverse Laplace Transform Theorems Inverse Laplace Transform Linear Property Derivatives Integrals First Shifting Property Second Shifting Property
Page 50 Inverse Laplace Transform Theorems Change of Scale Property Multiplication by t n Division by t Unit Impulse Function Unit Step Function Convolution Theorem
Page 51 Inverse Laplace Transform Formula
Page 52 Inverse Laplace Transform Formula
Page 53 Solving Differential Equation