求解波动方程数值解的matlab程序隐式格式2010.doc

求解波动方程数值解的matlab程序 隐式格式2010-04-19 13:45function varargout=liu(varargin)
a=1;T=1;a=1;b=0.5;h=1/20;k=1/40;
f=inline('0','x','t');
fx1=inline('exp(x)');
fx2=inline('exp(x)');
ft1=inline('exp(t)');
ft2=inline('exp(1+t)');
[X,Y,Z]=chfenmethed(f,fx1,fx2,ft1,ft2,a,T,h,k);
mesh(X,Y,Z);
shading flat;
xlabel('X','FontSize',14);
ylabel('t','FontSize',14);
zlabel('error','FontSize',14);
title('误差图');
function [X,T,Z]=chfenmethed(f,fx1,fx2,ft1,ft2,a,T,h,k)
%求解下问题
%u_tt-a^2*u_xx=f(x,t) 0<x<1,0<t<T
%u(0,t)=ft1,u(1,t)=ft2,
%u(x,0)=fx1(x)
%u_t(x,0)=fx2(x)
%h离散x方向的步长
%k离散t方向的步长
x=0:h:a;
t=0:k:T;
m=length(x);
n=length(t);
s=a*k/h;
[X,T]=meshgrid(x,t);
Z=zeros(n,m);
U=zeros(n,m);
for i=2:m-1
U(1,i)=feval(fx1,x(i));
U(2,i)=U(1,i)+k*feval(fx2,x(i))+k^2/2*(a^2/h^2* ...
(feval(fx1,x(i+1))-2*feval(fx1,x(i))+feval(fx1,x(i-1))+feval(f,x(i),0)));
Z(2,i)=abs(U(2,i)-f0(x(i),t(2)));
end
for j=1:n
U(j,1)=feval(ft1,t(j));
U(j,m)=feval(ft2,t(j));
end
A=-0.5*s^2*ones(1,m-2);
C=A;
B=(1+s^2)*ones(1,m-2);
UU=zeros(1,m-2);
f1=UU;
for i=3:n
for j=2:m-1
UU(j-1)=f0(x(j),t(i));
f1(j-1)=0.5*s^2*U(i-2,j-1)-(1+s^2)*U(i-2,j)...
+0.5*s^2*U(i-2,j+1)+2*U(i-1,j)...
+k^2*feval(f,x(j),t(i-1));
end
f1(1)=f1(1)+0.5*s^2*U(i,1);
f1(end)=f1(end)+0.5*s^2*U(i,m);
U(i,2:m-1)=zgf(A,B,C,f1);
Z(i,2:m-1)=abs(U(i,2:m-1)-UU);
end
function x=zgf(A,B,C,f)
%解 [b1 c1
% a2 b2 c2
% . . . *x=f
%
% an bn]
n=length(B);
B1=zeros(1,n-1);
Y=zeros(1,n);
x1=zeros(1,n);
B1(1)=C(1)/B(1);
for i=2:n-1
B1(i)=C(i)/(B(i)-A(i)*B1(i-1));
end
Y(1)=f(1)/B(1);
for i=2:n
Y(i)=(f(i)-A(i)*Y(i-1))/(B(i)-A(i)*B1(i-1));
end
x1(n)=Y(n);
for i=n-1:-1:1
x1(i)=Y(i)-B1(i)*x1(i+1);
end
x=x1;
function z=f0(x,t)
%精确解函数
z=exp(x+t);