tic;

 % this method is transform from Galerkin method

 %also call it as finit method

 %is used for solving two point BVP which is the first and second term.

 %this code was writen by HU.D.dong in February 11th 2017

 %MATLAB 7.0

 clear;

 clc;

 N=50;

 h=1/N;

 X=0:h:1;

 f=inline('(0.5*pi^2)*sin(0.5*pi.*x)');

 %以下是右端向量:

 for i=2:N

         fun1=@(x) pi^2/2.*sin(pi/2.*x).*(1-(x-X(i))/h);

          fun2=@(x) pi^2/2.*sin(pi/2.*x).*((x-X(i-1))/h);

     f_phi(i-1,1)=quad(fun1,X(i),X(i+1))+quad(fun2,X(i-1),X(i));

     end

  funN=@(x) pi^2/2.*sin(pi/2.*x).*(x-X(N))/h;

     f_phi(N)=quad(funN,X(N),X(N+1));

     %以下是刚度矩阵：

  A11=quad(@(x) 2/h+0.25*pi^2*h.*(1-2*x+2*x.^2),0,1);

  A12=quad(@(x) -1/h+0.25*pi^2*h.*(1-x).*x,0,1);

  ANN=quad(@(x) 1/h+0.25*pi^2*h*x.^2,0,1);

  A=diag([A11*ones(1,N-1),ANN],0)+diag(A12*ones(1,N-1),1)+diag(A12*ones(1,N-1),-1);

  Numerical_solution=A\f_phi;

  Numerical_solution=[0;Numerical_solution];

     %Accurate solution on above以下是精确解

     %%

     for i=1:length(X)

    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);

     end

     figure(1);

     grid on;

     subplot(1,2,1);

      plot(X,Numerical_solution,'ro-',X,Accurate_solution,'b^:');

     title('Numerical solutions vs Accurate solutions');

     legend('Numerical_solution','Accurate_solution');

     subplot(1,2,2);

       plot(X,Numerical_solution-Accurate_solution,'b x');

     legend('error_solution');

     title('error');

     toc;