solve heat conduction equation pde in matlab using implicit euler for time and central differencing for space in matlab in matlab

main.m
% Define problem parameters
L = 1;      % Length of rod
T = 1;      % Final time
n = 10;     % Number of grid points
m = 100;    % Number of time steps
a = 0.1;    % Thermal diffusivity
h = L/n;

% Discretize the problem domain
x = linspace(0, L, n+1)';
t = linspace(0, T, m+1)';
u = zeros(n+1, m+1);

% Set initial condition
u(:,1) = sin(pi*x/L);

% Set boundary conditions
u(1,:) = 0;
u(n+1,:) = 0;

% Create tridiagonal matrix
r = a/h^2;
A = diag(1+2*r*ones(n-1,1)) + diag(-r*ones(n-2,1),1) + diag(-r*ones(n-2,1),-1);

% Set time step size
dt = T/m;

% Update solution at each time step
for k=2:m+1
    % Right-hand side of equation
    b = u(2:n,k-1) + r*(u(3:n+1,k-1)-2*u(2:n,k-1)+u(1:n-1,k-1));
    b(1) = b(1) + r*u(1,k);
    b(n-1) = b(n-1) + r*u(n+1,k);
    
    % Solve tridiagonal system of equations
    u(2:n,k) = tridiag(A,b);
end

% Plot the solution
surf(t,x,u')
xlabel('Time')
ylabel('Position')
zlabel('Temperature')
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