solve this heat conduction equation pde in matlab using implicit euler for time and central differencing for space in matlab. use a jacobi solver in matlab

main.m
% Define parameters
L = 1; % length of domain
N = 101; % number of grid points
dx = L/(N-1); % spatial step size
x = linspace(0, L, N)'; % spatial grid
T = 1; % total time to solve for
dt = 0.01; % temporal step size
Nt = ceil(T/dt); % number of time steps
alpha = 0.1; % thermal diffusivity coefficient
r = alpha*dt/dx^2; % stability parameter
T_init = sin(pi*x); % initial temperature profile

% Initialize temperature matrix
Tn = T_init;
Tnm1 = T_init;

% Set up coefficient matrix and RHS vector
A = diag(1+2*r*ones(N-2, 1)) + diag(-r*ones(N-3, 1), 1) + diag(-r*ones(N-3, 1), -1);
A = sparse(A);
b = zeros(N-2, 1);

% Jacobi iteration
for n = 1:Nt
    % Compute RHS vector
    b(1) = r*Tnm1(1);
    b(N-2) = r*Tnm1(N);
    b(2:N-3) = Tnm1(2:N-3) + r*(Tnm1(1:N-4) - 2*Tnm1(2:N-3) + Tnm1(3:N-2));
    
    % Solve system
    Tn(2:N-1) = jacobi(A, b, Tnm1(2:N-1));
    
    % Update time step
    Tnm1 = Tn;
end

% Plot final temperature profile
plot(x, Tn);
xlabel('x');
ylabel('T');
title('Temperature Profile at t=T');
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main.m
function [x_new] = jacobi(A, b, x_old)
% Performs one iteration of Jacobi method
x_new = (b - A*x_old)./diag(A);
end
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