1. EE3561_Unit 8Al-Dhaifallah14351 EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations Lesson 3: Midpoint and Heun’s Predictor corrector Methods

2. EE3561_Unit 8Al-Dhaifallah14352 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

3. EE3561_Unit 8Al-Dhaifallah14353 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

4. EE3561_Unit 8Al-Dhaifallah14354 Outlines of Lesson 3 Lesson 3: Midpoint and Heun’s Predictor-corrector methods Review Euler Method Heun’s Method Midpoint method

5. EE3561_Unit 8Al-Dhaifallah14355 Euler Method

6. EE3561_Unit 8Al-Dhaifallah14356 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 Introduction

7. EE3561_Unit 8Al-Dhaifallah14357  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 Introduction

8. EE3561_Unit 8Al-Dhaifallah14358 Midpoint Method

9. EE3561_Unit 8Al-Dhaifallah14359 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 y i+1.  Local Truncation error of order O(h 3 )  Comparable to Second order Taylor series method

10. EE3561_Unit 8Al-Dhaifallah143510 Midpoint Method

11. EE3561_Unit 8Al-Dhaifallah143511 Midpoint Method

12. EE3561_Unit 8Al-Dhaifallah143512 Midpoint Method

13. EE3561_Unit 8Al-Dhaifallah143513 Midpoint Method

14. EE3561_Unit 8Al-Dhaifallah143514 Midpoint Method

15. EE3561_Unit 8Al-Dhaifallah143515 Midpoint Method

16. EE3561_Unit 8Al-Dhaifallah143516 Example 1

17. EE3561_Unit 8Al-Dhaifallah143517 Example 1

18. EE3561_Unit 8Al-Dhaifallah143518 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 y i+1.  Local Truncation error of order O(h 3 )  Comparable to Second order Taylor series method

19. EE3561_Unit 8Al-Dhaifallah143519 Heun’s Predictor Corrector

20. EE3561_Unit 8Al-Dhaifallah143520 Heun’s Predictor Corrector Method

21. EE3561_Unit 8Al-Dhaifallah143521 Heun’s Predictor Corrector (Prediction )

22. EE3561_Unit 8Al-Dhaifallah143522 Heun’s Predictor Corrector (Prediction )

23. EE3561_Unit 8Al-Dhaifallah143523 Heun’s Predictor Corrector (Prediction )

24. EE3561_Unit 8Al-Dhaifallah143524 Heun’s Predictor Corrector

25. EE3561_Unit 8Al-Dhaifallah143525 Heun’s Predictor Corrector

26. EE3561_Unit 8Al-Dhaifallah143526 Example 2

27. EE3561_Unit 8Al-Dhaifallah143527 Example 2

28. EE3561_Unit 8Al-Dhaifallah143528 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.

29. EE3561_Unit 8Al-Dhaifallah143529 Comparison Method Local truncation error Global truncation error

30. EE3561_Unit 8Al-Dhaifallah143530 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

31. EE3561_Unit 8Al-Dhaifallah143531 EE 3561 : Computational Methods Topic 8 Solution of Ordinary Differential Equations Lesson 4: Runge-Kutta Methods

32. EE3561_Unit 8Al-Dhaifallah143532 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

33. EE3561_Unit 8Al-Dhaifallah143533 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

34. EE3561_Unit 8Al-Dhaifallah143534 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

35. EE3561_Unit 8Al-Dhaifallah143535 Runge-Kutta Method

36. EE3561_Unit 8Al-Dhaifallah143536 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

37. EE3561_Unit 8Al-Dhaifallah143537 Taylor Series in One Variable Approximation Error

38. EE3561_Unit 8Al-Dhaifallah143538 Taylor Series in One Variable another look

39. EE3561_Unit 8Al-Dhaifallah143539 Definitions

40. EE3561_Unit 8Al-Dhaifallah143540 Taylor Series Expansion

41. EE3561_Unit 8Al-Dhaifallah143541 Taylor Series in Two Variables xx+h y y+k

42. EE3561_Unit 8Al-Dhaifallah143542 Runge-Kutta Method

43. EE3561_Unit 8Al-Dhaifallah143543 Runge-Kutta Method

44. EE3561_Unit 8Al-Dhaifallah143544 Runge-Kutta Method

45. EE3561_Unit 8Al-Dhaifallah143545 Runge-Kutta Method

46. EE3561_Unit 8Al-Dhaifallah143546 Runge-Kutta Method Alternative Formula

47. EE3561_Unit 8Al-Dhaifallah143547 Runge-Kutta Method Alternative Formula

48. EE3561_Unit 8Al-Dhaifallah143548 Runge-Kutta Method Alternative Formulas

49. EE3561_Unit 8Al-Dhaifallah143549 Runge-Kutta Method

50. EE3561_Unit 8Al-Dhaifallah143550 Second order Runge-Kutta Method Example

51. EE3561_Unit 8Al-Dhaifallah143551 Second order Runge-Kutta Method Example

52. EE3561_Unit 8Al-Dhaifallah143552 Second order Runge-Kutta Method Example

53. EE3561_Unit 8Al-Dhaifallah143553

54. EE3561_Unit 8Al-Dhaifallah143554 Summary  Runge Kutta methods generate accurate solution without the need to calculate high order derivatives.  Second order RK have local truncation error of order O(h 3 )  Fourth order RK have local truncation error of order O(h 5 )  N function evaluations are needed in N th order RK method.

55. EE3561_Unit 8Al-Dhaifallah143555 More in this unit Lesson 5: Applications of Runge-Kutta Methods To solve first order differential equations.  Lessons 6: Solving Systems of high order ODE.