SOLVING nonlinear reaction diffusion heat equation

Please matlab gurus, I need to solve this pde

How to i expreess this on matlab

ut = uxx + u - eu^2;

0 < x < 1; t > 0; e<<1

u(0,t) =0

u(1,t) =0

u(x,0) =x-x^3

where e is epislon.

e =.01

N = 16 and report the value of u(1=2; 1).

Provide a mesh plot of your numerical solution for

t [0; 1] and x = 0 : :01 : 1.

Answer 1

[Biswajit]

% Given parameters and conditions

e = 0.01; % epsilon

N = 16; % Number of spatial grid points

T = 1; % Total time

dx = 1 / N; % Grid spacing in x

dt = 0.001; % Time step size

x = 0:dx:1; % Spatial grid

t = 0:dt:T; % Time grid

% Initial condition

u0 = x - x.^3;

% Initialize the solution matrix

u = zeros(length(x), length(t));

u(:,1) = u0';

% Implement finite difference method

for n = 1:length(t)-1

for i = 2:length(x)-1

u(i,n+1) = u(i,n) + dt * (u(i+1,n) - 2*u(i,n) + u(i-1,n))/(dx^2) ...

+ dt * (u(i,n) - e*u(i,n)^2);

end

% Boundary conditions

u(1,n+1) = 0;

u(end,n+1) = 0;

end

% Calculate u(1/2, 1)

x_index = find(abs(x - 1/2) < eps);

u_12_1 = u(x_index, end);

disp(['Value of u(1/2, 1): ', num2str(u_12_1)]);

% Mesh plot of numerical solution

[X, T] = meshgrid(x, t);

figure;

mesh(X, T, u');

xlabel('x');

ylabel('t');

zlabel('u(x,t)');

title('Numerical Solution: u(x,t)');

[Biswajit]

Value of u(1/2, 1): 5.3981e-05