1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 31 Ordinary Differential Equations

2. Fig 23.1 FORWARD FINITE DIFFERENCE

3. Fig 23.2 BACKWARD FINITE DIFFERENCE

4. Fig 23.3 CENTERED FINITE DIFFERENCE

5. Data with Errors

6. Pendulum W=mg Ordinary Differential Equation

7. ODEs Non Linear Linearization Assume  is small

8. ODEs Second Order Systems of ODEs

9. Application of ODEs in Engineering Problem SOlving

10. ODE

11. ODE - OBJECTIVES Undetermined

12. ODE- Objectives Initial Conditions

13. ODE-Objectives Given Calculate

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

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

16. Euler’s Method Let

17. Example

18. Euler h=0.5

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

20. Sources of Error Propagated Local

21. Euler’s Method

22. Heun’s Method PredictorCorrector 2-Steps

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

24. Heun’s Method Correct Corrector Estimate Let

25. Error in Heun’s Method

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

27. Mid-Point Method Predictor Corrector 2-Steps

28. Mid-Point Method Predictor Predict Let

29. Mid-Point Method Corrector Correct Estimate Let