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# Untitled

a guest Apr 23rd, 2019 75 Never
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1. d = 70;
2. M = d-1;
3. N = d-1;
4.
5. %solve Laplace equation
6. % nabla^2 u(x,y) = 0
7.
8. % Specification of rectangular area
9. x0 = 0; xf = 4; % minimal and maximal value of x-coordinate
10. Mx = d; % number of intervals in x direction
11. y0 = 0; yf = 4; % minimal and maximal value of y-coordinate
12. My = d; % number of intervals in y direction
13.
14. bx0 = inline('exp(y) - cos(y)','y'); % Boundary condition for left edge
15. bxf = inline('exp(y)*cos(4) - exp(4)*cos(y)','y'); % Boundary condition for right edge
16. by0 = inline('cos(x) - exp(x)','x'); % Boundary condition for bottom edge
17. byf = inline('exp(4)*cos(x) - exp(x)*cos(4)','x'); % Boundary condition for top edge
18.
19. % a vector for defining rectangular area
20. D = [x0 xf y0 yf];
21.
22. %Parameters for Jacobi Method
23. MaxIter = 1000; % Maximal number of iteration
24. tol = 1e-3; % tolerance of Jacobi method
25.
26. % Solving Laplace equation via function sollap
27. [Ua,x,y] = sollap(bx0,bxf,by0,byf,D,Mx,My,tol,MaxIter);
28.
29. Un = zeros(d+1);
30.
31. for i = 1:length(x)
32.     for j = 1:length(y)
33.         Un(i,j) = -((exp(y(j)) * cos(x(i))) - (exp(x(i)) * cos(y(j))));
34.     end
35. end
36.
37. Ud = Un - Ua;
38.
39. Ud2 = Ud.^2;
40.
41. totals = zeros(1, length(Ud2));
42.
43. for i = 1:length(Ud2)
44.     totals(i) = sum(Ud2(i,:));
45. end
46.
47. USum = sum(totals);
48.
49. error = sqrt((1/(M*N)) * USum);
50.
51.
52.
53.
54. In new script thing....
55. d=[5 10 15 20 25 30 35 40 45 50 55 60 65 70]
56. Error= [0.2459297 0.057170249 0.019832251 0.008179454 0.016384573 0.028631307 0.041942391 0.05724528 0.073729247 0.091975748 0.111647881 0.176497771 0.294126947 0.440691868]
57.
58.
59. In new script....
60. zz=zeros(1,70);
61.
62.
63. for i=1:1:70
64.     zz(i) = ((3.424e-07*i^4) -(5.145e-05*i^3) + (0.002808*i^2) -(0.06226*i) + 0.4775)
65. end
66.
67. figure(1)
68.
69. plot(zz,':b')
70.
71. hold on
72. box on
73. plot(d,Error, 'or')
74. xlabel('Grid Size, d')
75. ylabel('Error')
76. title('Error between the Analytical and Numerical solution')
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