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EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations

EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations. Lesson 3: Midpoint and Heun’s Predictor corrector Methods. Lessons in Topic 8. Lesson 1: Introduction to ODE Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method

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EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations

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  1. EE 3561 : Computational MethodsUnit 8Solution of Ordinary Differential Equations Lesson 3: Midpoint and Heun’s Predictor corrector Methods Al-Dhaifallah1435

  2. Lessons in Topic 8 • Lesson 1: Introduction to ODE • Lesson 2: Taylor series methods • Lesson 3: Midpoint and Heun’s method • Lessons 4-5: Runge-Kutta methods • Lesson 6: Solving systems of ODE Al-Dhaifallah1435

  3. Learning Objectives of Lesson 3 • To be able to solve first order differential equation using Midpoint Method • To be able to solve first order differential equation using Heun’s Predictor Corrector method Al-Dhaifallah1435

  4. Outlines of Lesson 3 • Lesson 3: Midpoint and Heun’s • Predictor-corrector methods • Review Euler Method • Heun’s Method • Midpoint method Al-Dhaifallah1435

  5. Euler Method Al-Dhaifallah1435

  6. Introduction • We have seen Taylor series method • Euler method is simple but not accurate • Higher order Taylor series methods are accurate • but require calculating higher order derivatives • analytically Al-Dhaifallah1435

  7. Introduction • The methods proposed in this lesson have the general form • For the case of Euler • Different forms of will be used for the midpoint and Heun’s methods Al-Dhaifallah1435

  8. Midpoint Method Al-Dhaifallah1435

  9. Motivation • The midpoint can be summarized as • Euler method is used to estimate the solution at the midpoint. • The value of the rate function f(x,y) at the mid point is calculated • This value is used to estimate yi+1. • Local Truncation error of order O(h3) • Comparable to Second order Taylor series method Al-Dhaifallah1435

  10. Midpoint Method Al-Dhaifallah1435

  11. Midpoint Method Al-Dhaifallah1435

  12. Midpoint Method Al-Dhaifallah1435

  13. Midpoint Method Al-Dhaifallah1435

  14. Midpoint Method Al-Dhaifallah1435

  15. Midpoint Method Al-Dhaifallah1435

  16. Example 1 Al-Dhaifallah1435

  17. Example 1 Al-Dhaifallah1435

  18. Summary • The midpoint can be summarized as • Euler method is used to estimate the solution at the midpoint. • The value of the rate function f(x,y) at the mid point is calculated • This value is used to estimate yi+1. • Local Truncation error of order O(h3) • Comparable to Second order Taylor series method Al-Dhaifallah1435

  19. Heun’s Predictor Corrector Al-Dhaifallah1435

  20. Heun’s Predictor Corrector Method Al-Dhaifallah1435

  21. Heun’s Predictor Corrector(Prediction) Al-Dhaifallah1435

  22. Heun’s Predictor Corrector(Prediction) Al-Dhaifallah1435

  23. Heun’s Predictor Corrector(Prediction) Al-Dhaifallah1435

  24. Heun’s Predictor Corrector Al-Dhaifallah1435

  25. Heun’s Predictor Corrector Al-Dhaifallah1435

  26. Example 2 Al-Dhaifallah1435

  27. Example 2 Al-Dhaifallah1435

  28. Summary • Euler, Midpoint and Heun’s methods are similar in the following sense: • Different methods use different estimates of the slope • Both Midpoint and Heun’s methods are comparable in accuracy to second order Taylor series method. Al-Dhaifallah1435

  29. Comparison Al-Dhaifallah1435

  30. More in this Unit • Lessons 4-5: Runge-Kutta Methods • Lesson 6: Systems of High order ODE • Lesson 7: Multi-step methods • Lessons 8-9: Boundary Value Problems Al-Dhaifallah1435

  31. EE 3561 : Computational MethodsTopic 8Solution of Ordinary Differential Equations Lesson 4: Runge-Kutta Methods Al-Dhaifallah1435

  32. Lessons in Topic 8 • Lesson 1: Introduction to ODE • Lesson 2: Taylor series methods • Lesson 3: Midpoint and Heun’s method • Lessons 4-5: Runge-Kutta methods • Lesson 6: Solving systems of ODE Al-Dhaifallah1435

  33. Learning Objectives of Lesson 4 • To understand the motivation for using Runge Kutta method and basic idea used in deriving them. • To Familiarize with Taylor series for functions of two variables • Use Runge Kutta of order 2 to solve ODE Al-Dhaifallah1435

  34. Motivation • We seek accurate methods to solve ODE that does not require calculating high order derivatives. • The approach is to suggest a formula involving unknown coefficients then determine these coefficients to match as many terms of the Taylor series expansion Al-Dhaifallah1435

  35. Runge-Kutta Method Al-Dhaifallah1435

  36. Lecture Taylor Series in Two Variables The Taylor Series discussed in Chapter 4 is extended to the 2-independent variable case. This is used to prove RK formula Al-Dhaifallah1435

  37. Taylor Series in One Variable Approximation Error Al-Dhaifallah1435

  38. Taylor Series in One Variableanother look Al-Dhaifallah1435

  39. Definitions Al-Dhaifallah1435

  40. Taylor Series Expansion Al-Dhaifallah1435

  41. Taylor Series in Two Variables y+k y x x+h Al-Dhaifallah1435

  42. Runge-Kutta Method Al-Dhaifallah1435

  43. Runge-Kutta Method Al-Dhaifallah1435

  44. Runge-Kutta Method Al-Dhaifallah1435

  45. Runge-Kutta Method Al-Dhaifallah1435

  46. Runge-Kutta MethodAlternative Formula Al-Dhaifallah1435

  47. Runge-Kutta MethodAlternative Formula Al-Dhaifallah1435

  48. Runge-Kutta MethodAlternative Formulas Al-Dhaifallah1435

  49. Runge-Kutta Method Al-Dhaifallah1435

  50. Second order Runge-Kutta Method Example Al-Dhaifallah1435

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