1. 中華大學 資訊工程系 Fall 2002 Chap 4 Laplace Transform

2. Page 2 Outline Basic Concepts Laplace Transform Definition, Theorems, Formula Inverse Laplace Transform Definition, Theorems, Formula Solving Differential Equation Solving Integral Equation

3. 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

4. Page 4 Basic Concepts Laplace Transform Inverse Laplace Transform L { f(t) } = F(s) L -1 { F(s) } = f(t)

5. 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

6. Page 6 Laplace Transform t e -st s=1 s=2 s=4 s=8 s=0.5 s=0.25 s=0.125

7. Page 7 Laplace Transform Example : Find L { 1 } Sol:

8. Page 8 Laplace Transform Example : Find L { e at } Sol:

9. Page 9 Laplace Transform Example 4-2 : Find L { t t } Sol:  L { t t } does not exist

10. Page 10 Laplace Transform Exercise 4-1 : Find

11. Page 11 Laplace Transform Theorems Definition of Laplace Transform Linear Property Derivatives Integrals First Shifting Property Second Shifting Property

12. Page 12 Laplace Transform Theorems Change of Scale Property Multiplication by t n Division by t Unit Impulse Function Periodic Function Convolution Theorem

13. Page 13 Linearity of Laplace Transform Proof:

14. Page 14 Application for Linearity of Laplace Transform

15. 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

16. Page 16 Examples for First Shifting Theorem

17. Page 17 Excises sec 5.1 #1, #7, #19, #24, #29,#35, #37,#39

18. Page 18 Laplace of Transform the Derivative of f(t) Prove Proof:

19. Page 19 Laplace transorm of the derivative of any order n

20. 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

21. Page 21 Differential Equations, Initial Value Problem How to use Laplace transform and Laplace inverse to solve the differential equations with given initial values

22. Page 22 Example : Explanation of the Basic Concept Examples

23. 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

24. Page 24 An Application of Integral Theorem Examples

25. Page 25 Laplace Transform Unit Step Function (also called Heaviside’s Function)

26. Page 26

27. 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

28. Page 28 The Proof of the T-shifting Theorem Prove Proof:

29. Page 29 Application of Unit Step Functions Note Find the transform of the function

30. Page 30 Example : Find the inverse Laplace transform f(t) of

31. Page 31 Short Impulses. Dirac’s Delta Function Area = 1

32. Page 32 Laplace Transform Unit Impulse Function (also called Dirac Delta Function) Area = 1

33. Page 33 Laplace Transform Example 4-6 : Prove Proof

34. Page 34 Example

35. Page 35 Homework section 5-2 #4, #7, #9, #18, #19 Section 5-3 #3, #6, #17, #28, #29

36. Page 36 Differentiation and Integration of Transforms Differentiation of transforms

37. Page 37 Example

38. Page 38 Integration of Transform

39. Page 39 Example Find the inverse transform of the function

40. Page 40 Convolution. Integration Equation Convolution Properties

41. Page 41 Example1 Using the convolution, find the inverse h(t) of Example 2 Example 3

42. Page 42 Laplace Transform Example 4-7 : Prove Proof:

43. Page 43 Differential Equation

44. Page 44 Integration Equations Example

45. Page 45 Homeworks Section 5-4 #1,#13 Section 5-5 #7, #14, #27

46. Page 46 Laplace Transform Formula

47. Page 47 Laplace Transform Formula

48. Page 48 Inverse Laplace Transform Definition The Inverse Laplace Transform of a function F(s) is defined as

49. Page 49 Inverse Laplace Transform Theorems Inverse Laplace Transform Linear Property Derivatives Integrals First Shifting Property Second Shifting Property

50. 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

51. Page 51 Inverse Laplace Transform Formula

52. Page 52 Inverse Laplace Transform Formula

53. Page 53 Solving Differential Equation