ib_euler.m

偏微分方程matlab m文件

function [u,x,t] = IB_Euler(A,xf,T,it0,bx0,bxf,M,N)
%解方程 A1 u_xx = u_t ， 0 <= x <= xf, 0 <= t <= T
%初值: u(x,0) = it0(x)
% 边界条件: u(0,t) = bx0(t), u(xf,t) = bxf(t)
% M ： x 轴的等分段数
% N ： t 轴的等分段数
dx = xf/M; x = [0:M]*dx;
dt = T/N; t = [0:N]'*dt;
for i = 1:M + 1
    u(i,1) = it0(x(i));
end
for j = 1:N + 1
    u([1 M + 1],j) = [bx0(t(j)); bxf(t(j))]; 
end
r = A*dt/dx/dx;
r2 = 1 + 2*r;
for i = 1:M - 1
    P(i,i) = r2; %构造9.2.9的矩阵
    if i > 1
        P(i - 1,i) = -r; P(i,i - 1) = -r; 
    end
end
for j = 2:N + 1
    b = [r*u(1,j); zeros(M - 3,1); r*u(M + 1,j)] + u(2:M,j - 1); %Eq.(9.2.9)
    u(2:M,j) = linsolve(P,b);
end
u=u';