1. Dr. Mubashir Alam King Saud University

2. Outline Ordinary Differential Equations (ODE) ODE: An Introduction (8.1) ODE Solution: Euler’s Method (8.2) ODE Solution: Runge-Kutta Method (Order 2) (8.5)

3. Ordinary Differential Equations (ODE)

4. ODE: General Solution

7. ODE: Stability

9. Assume the solution Y(x) is being sought in the interval x 0 ≤ x ≤ b, and for an initial value Y 0 Change the initial value from Y 0 to Y 0 +ε, and lets call the resulting solution Y ε (x), i.e. Then a solution is stable if for small value of ε Thus a small change in the initial solution Y 0 will only lead to small change in the solution Y(x) of the initial value problem.

10. Numerical Methods for ODE

12. Euler’s Method

13. Proof of Euler’s Method Ch#5

14. Geometric Approach

15. Example: 8.2.1 Find the solution: Y`(x) = -Y(x), Y(0)=1 True Solution: Y(x)=e -x Euler Method Solution: y n+1 = y n -hy n, n ≥ 0 Y 0 =1 and x n = nh

16. Example: 8.2.1 hxy h (x)ErrorRelative Error 0.213.2768E-14.02E-20.109 21.0738E-12.80E-20.207 33.5184E-21.46E-20.293 41.1529E-26.79E-30.371 53.7779E-32.96E-30.439

17. Example

19. Runge-Kutta Method of Order 2

20. How to choose constants γ 2 = 1/2, or ¾ or 1

21. Runge-Kutta Method of Order 2

22. Example:8.5.2

23. hxy h (x)Error 0.101.0000000000 20.4912156731.9349E-003 4-1.407898629-2.5475E-003 60.6806967235.8065E-005 80.8413763392.4819E-003 10-1.380966579-2.1261E-003

24. Example: 8.5.2 hxy h (x)Error 0.101.0000000000 0.11.0945004173.37E-004 0.21.1780799106.56E-004 0.31.2499019599.54E-004 0.41.3092476560.00123168 0.51.3555228750.0014852