M athematical M odeling

1. MathematicalModeling Team B 김도현 서수빈 이혜경

2. Estimate the Solution with Euler s Method • ‘’

3. Context Preliminary Problem Summary Further • Exact solution • MATLAB Code • Result

4. Preliminary. ODE solving method -seperable -integrate factor

5. Preliminary. Eulers Method ‘’

6. Euler s Method • ‘’

7. Problem 1. Use Euler s method with step size h = 0.1 to approximate the solution to the IVP ‘’ at the points x = 1.1, 1.2, 1.3, 1.4 and 1.5. Compare your numerical result with the exact solution.

8. Problem 1. Find Exact Solution

9. Problem 1. Find Exact Solution

10. Problem 1. Find Exact Solution

11. Problem 1. Find Exact Solution

12. Problem 1. Find Exact Solution

13. Problem 1. Find Exact Solution

14. Problem 1. Coding i=1; x(1,1)=1; y(1,1)=4; slope(1,1)=x(1,1).*(y(1,1).^(1/2)); whilei<=5 i=i+1; x(1,i)=x(1,i-1)+0.1; y(1,i)=slope(1,i-1).*0.1+y(1,i-1); slope(1,i)=x(1,i).*(y(1,i).^(1/2)); end z=((x.^2+7).^2)/16; plot(x,y,'r'); hold on plot(x,z,'b'); error=abs(y-z)

15. Problem 1. Result Exact solution Estimated solution

16. Problem 2. Use Euler s method to find approximates to the solution of the IVP ‘’ at x = 1, taking 1, 2, 4, 8, and 16 steps. Compare your numerical result with the exact solution.

17. Problem 2. Find Exact Solution

18. Problem 2. Find Exact Solution

19. Problem 2. Find Exact Solution

20. Problem 2. Find Exact Solution

21. Problem 2. Find Exact Solution

22. Problem 2. Coding i=1; y(1,1)=1; n=input('How many steps ?'); x=0:1/n:1; whilei<=n y(1,i+1)=y(1,i)/n+y(1,i); i=i+1; end s=0:1/1000:1; z=exp(s); plot(x,y,'r'); hold on plot(s,z,'b'); hold off

23. Problem 2. Result n=1 Exact solution Estimated solution Error = 0.7183

24. Problem 2. Result n=2 Exact solution Estimated solution Error = 0.4684

25. Problem 2. Result n=4 Exact solution Estimated solution Error = 0.2769

26. Problem 2. Result n=8 Exact solution Estimated solution Error = 0.1525

27. Problem 2. Result n=16 Exact solution Estimated solution Error = 0.0804

28. Problem 2. Result n=1024 Exact solution Estimated solution Error = 0.0013

29. Problem 2. Result n=1 n=2 n=4 n=8 n=16

30. Problem 3. ‘’ Use Euler s method with step size h = 0.1 to approximate to the solution of the IVP with x0 = 0, and compute the first 10 iterations of this scheme, and compare the result with the exact solution.

31. Problem 3. Find Exact Solution

32. Problem 3. Find Exact Solution

33. Problem 3. Find Exact Solution

34. Problem 3. Find Exact Solution

35. Problem 3. Find Exact Solution

36. Problem 3. Find Exact Solution

37. Problem 3. Find Exact Solution

38. Problem 3. Find Exact Solution

39. Problem 3. Coding i=1; y(1,1)=1; x=0:0.1:1; whilei<=10 y(1,i+1)=(y(1,i)+sin(x(1,i))).*0.1+y(1,i); i=i+1; end s=0:1/1000:1; z=3*exp(s)/2-sin(s)/2-cos(s)/2; plot(x,y,'r'); hold on plot(s,z,'b'); error=abs(3*exp(x)/2-sin(x)/2-cos(x)/2-y)

40. Problem 3. Result Exact solution Estimated solution Error = 0.2371

41. Summary - error 1) Same mesh size, different x value 2) Same x value, different mesh size 3) error=0.2371 with mesh h=0.1 at x=1

42. Summary - compare whenΔx=1 1) mesh=0.1 (10 iteration) error=0.1983 2) mesh=0.125 (8 steps) error=0.1525 3) mesh h=0.1 error=0.2371

43. Further... Taylor Series Method Improved Euler s Method Runge-Kutta Method ‘’ Chebyshev Node Choosing