Matlab-10:Ritz-Galerkin方法求解二阶常微分方程

一、代数多项式法：

 tic;
 clear
 clc
 % N=input('please key in the value of ''N''');
 N=10;
 M=100;
 h=1/M;
 X=0:h:1;
 accurate_fun=inline('x.^2 - (2*exp(x))/(exp(1) + 1) - (2*exp(-x)*exp(1))/(exp(1) + 1) + 2');
  f=inline('x.^2-x');
  phi=inline('x.*(1-x).*x.^(i-1)','i','x');
  diff_phi=inline('i*x.^(i-1)-(i+1)*x.^i','i','x');
  for j=1:N
      for i=1:N
 A(i,j)=quad(@(x)phi(i,x).*phi(j,x)+diff_phi(i,x).*diff_phi(j,x),0,1);
      end
      b(j,1)=quad(@(x) phi(j,x).*f(x),0,1);
  end
  C=A\b;
 syms x;
  Un=0;
 for i=1:N
 Un=Un+C(i)*phi(i,x);
 end
 Un=Un+x;
  numerical= double(vpa(subs(Un,'x',X)));
  accurate=accurate_fun(X);
  subplot(1,2,1)
  plot(X,numerical,'r -',X,accurate,'b >');
   title('numerical VS accurate');
  legend('numerical','accurate');
  grid on;
   subplot(1,2,2);
   plot(X,numerical-accurate,'g');
   title('error');
   grid on;
  toc;

二、三角函数法：

 tic;
 clear
 clc
 % N=input('please key in the value of ''N''');
 N=10;
 M=100;
 h=1/M;
 X=0:h:1;
 accurate_fun=inline('x.^2 - (2*exp(x))/(exp(1) + 1) - (2*exp(-x)*exp(1))/(exp(1) + 1) + 2');
  f=inline('x.^2-x');
  phi=inline('sin(i*pi*x)','i','x');
  diff_phi=inline('i*pi*cos(i*pi*x)','i','x');
  for j=1:N
      for i=1:N
 A(i,j)=quad(@(x)phi(i,x).*phi(j,x)+diff_phi(i,x).*diff_phi(j,x),0,1);
      end
      b(j,1)=quad(@(x) phi(j,x).*f(x),0,1);
  end
  C=A\b;
  syms x;
  Wn=0;
 for i=1:N
 Wn=Wn+C(i)*phi(i,x);
 end
 Un=Wn+x;
 numerical=vpa(subs(Un,'x',X));
 accurate=accurate_fun(X);
 subplot(1,2,1)
 plot(X,numerical,'r -',X,accurate,'g ^');
 title('Ritz Galerkin method');
 legend('numerical solution','accurate solution');
 grid on;
 subplot(1,2,2)
 plot(X,numerical-accurate,'b');
 title('error');
 grid on;
 toc;