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Introduction to Laplace Transform: Modeling & Simulation Techniques

Learn about Laplace Transform in modeling and simulation. Discover how transforms simplify problem-solving. Explore applications in solving ODEs and algebraic equations. Understand properties and techniques for inverse Laplace Transform.

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Introduction to Laplace Transform: Modeling & Simulation Techniques

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  1. SE 207: Modeling and SimulationIntroduction 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 Laplace Transform t is a real 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. Exampleunit step

  8. ExampleShifted Step

  9. Integration by parts

  10. ExampleRamp

  11. ExampleExponential Function

  12. Examplesine Function

  13. Examplecosine Function

  14. ExampleRectangle Pulse

  15. Properties of Laplace TransformAddition

  16. Properties of Laplace TransformMultiplication by a constant

  17. Properties of Laplace TransformMultiplication by exponential

  18. Properties of Laplace TransformExamplesMultiplication by exponential

  19. Useful Identities

  20. Examplesin Function

  21. Examplecosine Function Laplace Transform Inverse Laplace Transform

  22. Properties of Laplace TransformMultiplication by time

  23. Properties of Laplace Transform

  24. Properties of Laplace TransformIntegration

  25. Properties of Laplace TransformDelay

  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 SimulationLesson 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

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