1 tic;
 2 % this method is transform from Galerkin method 
 3 %also call it as finit method
 4 %is used for solving two point BVP which is the first and second term.
 5 %this code was writen by HU.D.dong in February 11th 2017
 6 %MATLAB 7.0
 7 clear;
 8 clc;
 9 N=50;
10 h=1/N;
11 X=0:h:1;
12 f=inline(\'(0.5*pi^2)*sin(0.5*pi.*x)\');
13 %以下是右端向量:
14 for i=2:N
15         fun1=@(x) pi^2/2.*sin(pi/2.*x).*(1-(x-X(i))/h);
16          fun2=@(x) pi^2/2.*sin(pi/2.*x).*((x-X(i-1))/h);
17     f_phi(i-1,1)=quad(fun1,X(i),X(i+1))+quad(fun2,X(i-1),X(i));
18     end
19  funN=@(x) pi^2/2.*sin(pi/2.*x).*(x-X(N))/h;
20     f_phi(N)=quad(funN,X(N),X(N+1));
21     %以下是刚度矩阵：
22  A11=quad(@(x) 2/h+0.25*pi^2*h.*(1-2*x+2*x.^2),0,1);
23  A12=quad(@(x) -1/h+0.25*pi^2*h.*(1-x).*x,0,1);
24  ANN=quad(@(x) 1/h+0.25*pi^2*h*x.^2,0,1);
25  A=diag([A11*ones(1,N-1),ANN],0)+diag(A12*ones(1,N-1),1)+diag(A12*ones(1,N-1),-1);
26  Numerical_solution=A\f_phi;
27  Numerical_solution=[0;Numerical_solution];
28     %Accurate solution on above以下是精确解
29     %%
30     for i=1:length(X)
31    Accurate_solution(i,1)=sin((pi*X(i))/2)/2 - cos((pi*X(i))/2)/2 + exp((pi*X(i))/2)*((exp(-(pi*X(i))/2)*cos((pi*X(i))/2))/2 + (exp(-(pi*X(i))/2)*sin((pi*X(i))/2))/2);
32     end 
33     figure(1);
34     grid on; 
35     subplot(1,2,1);
36      plot(X,Numerical_solution,\'ro-\',X,Accurate_solution,\'b^:\');
37     title(\'Numerical solutions vs Accurate solutions\');
38     legend(\'Numerical_solution\',\'Accurate_solution\');
39     subplot(1,2,2);
40       plot(X,Numerical_solution-Accurate_solution,\'b x\');
41     legend(\'error_solution\');
42     title(\'error\');
43     toc;