poissons equation in matlab

The Poisson's equation is a partial differential equation that can be solved numerically using the finite difference method in MATLAB.

To solve the Poisson's equation using the finite difference method, we need to discretize the given domain and approximate the partial differential equation using finite differences.

Here's an example MATLAB code to solve the Poisson's equation using the finite difference method:

main.m
% Define the domain and boundary conditions
L = 1; % length of the domain
N = 100; % number of grid points
x = linspace(0,L,N); % discretized domain
u = zeros(1,N); % initial guess
u(1) = 1; % boundary condition at x=0
u(N) = 0; % boundary condition at x=L

% Define the parameters for the finite difference method
dx = L/(N-1); % grid spacing
alpha = -2/dx^2; % coefficient of u(i)
beta = 1/dx^2; % coefficient of u(i-1) and u(i+1)

% Solve the Poisson's equation using the finite difference method
for iter=1:1000 % maximum 1000 iterations
    unew = u;
    for i=2:N-1 % loop over the interior points
        unew(i) = (beta*u(i-1) + beta*u(i+1) - alpha*u(i))/(-alpha);
    end
    if norm(unew-u,inf) < 1e-6 % check convergence
        break;
    end
    u = unew;
end

% Plot the solution
figure(1); clf;
plot(x,u,'-');
xlabel('x'); ylabel('u');
title('Solution to Poisson Equation using FDM');
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In this code, we define the domain and boundary conditions, and then we discretize the domain using linspace. We initialize the solution u as a vector of zeros, and set the boundary conditions at x=0 and x=L.

Next, we define the parameters for the finite difference method, including the grid spacing dx and the coefficients alpha and beta.

Then, we loop over the interior points and update u using the finite difference method. We check for convergence using the norm function, and continue the iterations until convergence is achieved or we reach the maximum number of iterations.

Finally, we plot the solution u using plot.