椭圆型方程的五点差分格式的matlab代码

以下是一个简单的 MATLAB 代码实现椭圆型方程的五点差分格式：

% 定义问题参数
a = 1;
b = 2;
alpha = 1;
beta = 2;
f = @(x,y) 0;

% 定义网格
Nx = 20;
Ny = 30;
hx = a/(Nx-1);
hy = b/(Ny-1);
x = linspace(0,a,Nx);
y = linspace(0,b,Ny);

% 初始化解向量
u = zeros(Ny,Nx);

% 设置边界条件
u(1,:) = alpha;
u(Ny,:) = beta;
u(:,1) = alpha;
u(:,Nx) = beta;

% 构造五点差分格式
A = gallery('poisson',Nx-2)/(hx^2);
B = gallery('poisson',Ny-2)/(hy^2);
D = -2*(1/hx^2+1/hy^2)*eye(Nx-2,Ny-2);
F = zeros(Nx-2,Ny-2);
G = zeros(Nx-2,Ny-2);

% 循环求解
for j = 2:Ny-1
    for i = 2:Nx-1
        F(i-1,j-1) = f(x(i),y(j));
        G(i-1,j-1) = -(1/hx^2)*(u(j,i-1)+u(j,i+1))...
                     -(1/hy^2)*(u(j-1,i)+u(j+1,i))...
                     -f(x(i),y(j));
    end
end

% 解线性方程组
U = reshape(u(2:end-1,2:end-1),[],1);
V = reshape(G,[],1);
LHS = kron(A,eye(Ny-2)) + kron(eye(Nx-2),B) + D;
RHS = V - F;
W = LHS\RHS;
U = U + W;
u(2:end-1,2:end-1) = reshape(U,Ny-2,Nx-2);

% 绘图
[X,Y] = meshgrid(x,y);
surf(X,Y,u)

注意，此代码仅适用于简单的椭圆型方程，对于更复杂的问题可能需要更高级的数值方法。