1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 32 Ordinary Differential Equations

2. Pendulum W=mg Ordinary Differential Equation

3. ODEs Non Linear Linearization Assume  is small

4. ODEs Second Order Systems of ODEs

5. ODE

6. ODE - OBJECTIVES Undetermined

7. ODE- Objectives Initial Conditions

8. ODE-Objectives Given Calculate

9. Runge-Kutta Methods New Value = Old Value + Slope X Step Size

10. Runge Kutta Methods Definition of  yields different Runge-Kutta Methods

11. Euler’s Method Let

12. Sources of Error Truncation: Caused by discretization Local Truncation Propagated Truncation Roundoff: Limited number of significant digits

13. Sources of Error Propagated Local

14. Euler’s Method

15. Heun’s Method PredictorCorrector 2-Steps

16. Heun’s Method Predict Predictor-Corrector Solution in 2 steps Let

17. Heun’s Method Correct Corrector Estimate Let

18. Error in Heun’s Method

19. The Mid-Point Method Remember: Definition of  yields different Runge-Kutta Methods

20. Mid-Point Method Predictor Corrector 2-Steps

21. Mid-Point Method Predictor Predict Let

22. Mid-Point Method Corrector Correct Estimate Let

24. Runge Kutta – 2 nd Order

25. Runge Kutta – 3rd Order

26. Runge Kutta – 4th Order

27. Boundary Value Problems

28. Fig 23.1 FORWARD FINITE DIFFERENCE

29. Fig 23.2 BACKWARD FINITE DIFFERENCE

30. Fig 23.3 CENTERED FINITE DIFFERENCE

31. xoxo Boundary Value Problems x1x1 x2x2 x3x3 x n-1 xnxn...

32. Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

33. Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

34. Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

35. Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

36. Boundary Value Problems Collect Equations: BOUNDARY CONDITIONS

37. Example x1x1 x2x2 x3x3 x4x4