2. HAMPIRAN NUMERIK SOLUSI PERSAMAAN DIFERENSIAL (lanjutan) Pertemuan 12 Matakuliah: METODE NUMERIK I Tahun: 2008

3. Bina Nusantara Runge-Kutta Methods Runge-Kutta methods are very popular because of their good efficiency; and are used in most computer programs for differential equations. They are single-step methods, as the Euler methods.

4. Bina Nusantara Runge-Kutta Methods To convey some idea of how the Runge-Kutta is developed, let’s look at the derivation of the 2 nd order. Two estimates

5. Bina Nusantara Runge-Kutta Methods The initial conditions are: The Taylor series expansion

6. Bina Nusantara Runge-Kutta Methods From the Runge-Kutta The definition of the function Expand the next step

7. Bina Nusantara Runge-Kutta Methods From the Runge-Kutta Compare with the Taylor series 4 Unknowns

8. Bina Nusantara Runge-Kutta Methods The Taylor series coefficients (3 equations/4 unknowns) If you select “a” as Note:These coefficient would result in a modified Euler or Midpoint Method

9. Bina Nusantara Runge-Kutta Method (2 nd Order) Example Consider Exact Solution The initial condition is: The step size is: Use the coefficients

10. Bina Nusantara Runge-Kutta Method (2 nd Order) Example The values are

11. Bina Nusantara Runge-Kutta Method (2 nd Order) Example The values are equivalent of Modified Euler

12. Bina Nusantara Runge-Kutta Method (2 nd Order) Example [b] The values are

13. Bina Nusantara Runge-Kutta Method (2 nd Order) Example [b] The values are

14. Bina Nusantara Runge-Kutta Methods

15. Bina Nusantara The 4 th Order Runge-Kutta The general form of the equations :

16. Bina Nusantara The 4th Order Runge-Kutta This is a fourth order function that solves an initial value problems using a four step program to get an estimate of the Taylor series through the fourth order. This will result in a local error of O(Dh 5 ) and a global error of O(Dh 4 )

17. Bina Nusantara 4 th -order Runge-Kutta Method xixi x i + h/2x i + h f1f1 f2f2 f3f3 f4f4

18. Bina Nusantara Runge-Kutta Method (4 th Order) Example Consider Exact Solution The initial condition is: The step size is :

19. Bina Nusantara The 4 th Order Runge-Kutta The example of a single step:

20. Bina Nusantara Runge-Kutta Method (4 th Order) Example The values for the 4 th order Runge-Kutta method

21. Bina Nusantara Runge-Kutta Method (4 th Order) Example The values are equivalent to those of the exact solution. If we were to go out to x=5. y(5) = -111.4129 (- 111.4132) The error is small relative to the exact solution.

22. Bina Nusantara Runge-Kutta Method (4 th Order) Example A comparison between the 2 nd order and the 4 th order Runge-Kutta methods show a slight difference.

23. Bina Nusantara The 4th Order Runge-Kutta Higher order differential equations can be treated as if they were a set of first-order equations. Runge-Kutta type forward integration solutions can be obtain. A more direct solution can be obtained by repeating the whole process used in first-order cases.

24. Bina Nusantara The 4 th Order Runge-Kutta The general form of the 2 nd order equations :

25. Bina Nusantara The step sizes are: The next step would be :

26. Bina Nusantara

27. Soal Latihan Gunakan Metode Runge-Kutta orde 2 dan orde 4 untuk menyelesaikan PDB berikut: 1. y’= -½ x 2 y, pada 0≤x≤1, y(0)=4, dan h=0,2 2. y’= x + y pada 0≤x≤1, y(0)=0, dan h=0,25 3. y’= y sin 2 (x) pada 0≤x≤3, y(0)=1, dan h=0,1