1. SE 207: Modeling and Simulation Introduction to Laplace Transform
Dr. Samir Al-Amer
Term 072

2. Why do we use them
We use transforms to transform the problem into a one that is easier to solve then use the inverse transform to obtain the solution to the original problem

3. Laplace Transform L L-1 t is a real variable s is complex variable
f(t) is a real function
Time Domain
s is complex variable
F(s) is a complex valued function
Frequency Domain
L-1
Inverse
Laplace Transform

4. Use of Laplace Transform in solving ODE
Differential Equation
Algebraic Equation
Laplace
Transform
Solution of the
Algebraic Equation
Solution of the
Differential Equation
Inverse
Laplace
transform

5. Definition of Laplace Transform
Sufficient conditions for existence of the Laplace transform

6. Examples of functions of exponential order

7. Example unit step

8. Example Shifted Step

9. Integration by parts

10. Example Ramp

11. Example Exponential Function

12. Example sine Function

13. Example cosine Function

14. Example Rectangle Pulse

15. Properties of Laplace Transform Addition

16. Properties of Laplace Transform Multiplication by a constant

17. Properties of Laplace Transform Multiplication by exponential

18. Properties of Laplace Transform Examples Multiplication by exponential

19. Useful Identities

20. Example sin Function

21. Example cosine Function
Laplace Transform
Inverse Laplace Transform

22. Properties of Laplace Transform Multiplication by time

23. Properties of Laplace Transform

24. Properties of Laplace Transform Integration

25. Properties of Laplace Transform Delay

26. Properties of Laplace Transform
Slope =A L

27. Properties of Laplace Transform4
Slope =A _ _
Slope =A
A L L L
Slope =A = L

28. Summary

29. SE 207: Modeling and Simulation Lesson 3: Inverse Laplace Transform
Dr. Samir Al-Amer
Term 072

30. Properties of Laplace Transform

31. Solving Linear ODE using Laplace Transform

32. Inverse Laplace Transform

33. Notation

34. Notation

35. Notation

36. Examples

37. Partial Fraction Expansion

38. Partial Fraction Expansion

39. Partial Fraction Expansion

40. Example

41. Example

42. Alternative Way of Obtaining Ai

43. Repeated poles

44. Repeated poles

45. Repeated poles

46. Repeated poles

47. Common Error

48. Complex Poles

49. Complex Poles

50. What do we do if F(s) is not strictly proper

51. Solving for the Response

52. Final value theorem

53. Final value theorem

54. Step function A

55. impulse function

56. impulse function

57. Initial Value& Final Value Theorems

58. Initial Value Theorem

59. Final Value Theorems

60. SE 207: Modeling and Simulation Lesson 4: Additional properties of Laplace transform and solution of ODE
Dr. Samir Al-Amer
Term 072

61. Outlines What to do if we have proper function? Time delay
Inversion of some irrational functions
Examples

62. Step function A

63. impulse function

64. impulse function
You can consider the unit impulse as the limiting case for a rectangle pulse with unit area as the width of the pulse approaches zero
Area=1

65. impulse function

66. Sample property of impulse function

67. Time delay
g(t) G(s)
f(t) F(s)

68. What do we do if F(s) is not strictly proper

69. What do we do if F(s) is not strictly proper

70. Example
− − −

71. Example

72. Solving for the Response