Chap 4 Laplace Transform

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

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

3. 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)

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

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

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

8. Laplace Transform • Example : Find L{ eat } Sol:

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

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

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

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

13. Linearity of Laplace Transform Proof:

14. Application for Linearity of Laplace Transform

15. 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

16. Examples for First Shifting Theorem

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

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

19. Laplace transorm of the derivative of any order n

20. Examples • Example 1: Let f(t)=t2, Derive L(f) from L(1) • Example 2: Derive the Laplace transform of cos wt

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

22. Example : Explanation of the Basic Concept • Examples

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. An Application of Integral Theorem • Examples

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

27. 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

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

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

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

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

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

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

34. Example

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

36. Differentiation and Integration of Transforms • Differentiation of transforms

37. Example

38. Integration of Transform

39. Example Find the inverse transform of the function

40. Convolution. Integration Equation • Convolution • Properties

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

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

43. Differential Equation

44. Integration Equations • Example

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

46. Laplace Transform • Formula

47. Laplace Transform • Formula

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

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

50. Inverse Laplace Transform • Theorems Change of Scale Property Multiplication by tn Division by t Unit Impulse Function Unit Step Function Convolution Theorem