SOLVING nonlinear reaction diffusion heat equation
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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.
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Answers (1)
Biswajit
on 21 Nov 2023
% 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)');
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