Matlab:导数边界值的有限元(Ritz)法

 tic;
 % this method is transform from Ritz method
 %is used for solving two point BVP
 %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('pi^2/2*sin(pi/2*x)');
 %以下是右端向量；
 for i=2:N
     fun1=@(x) f(X(i-1)+h*x).*x+f(X(i)+h*x).*(1-x);
 fi(i-1,1)=h*quad(fun1,0,1);
 end
 funN=@(x) f(X(N-1)+h*x).*x;
 fi(N,1)=h*quad(funN,0,1);
 %以下是stiff矩阵；
 a11=1/h+pi^2*h/12;
 aii=2*a11;
 aij=-1/h+pi^2*h/24;
 A=diag(aii*ones(N,1),0)+diag(aij*ones(N-1,1),1)+diag(aij*ones(N-1,1),-1);
 A(N,N)=a11;
 numerical_solution=A\fi;
 numerical_solution=[0;numerical_solution];
 %以下是真解；
 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
     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;

效果图：