how to use ode15s to solve diffusion problem in matlab

The ode15s function in MATLAB can be used to solve diffusion problems. To use ode15s, you need to define the system of differential equations that describe your diffusion problem.

Here's an example code that demonstrates how to use ode15s for solving a simple diffusion problem in MATLAB:

main.m
% Define the parameters of the diffusion problem
alpha = 0.1; % diffusion coefficient
L = 1; % length of the domain
tspan = [0, 1]; % time span for simulation

% Define the initial conditions and boundaries
u0 = @(x) sin(pi*x/L); % initial condition
bc = @(t) 0; % boundary condition

% Define the system of differential equations
ode = @(t, u) alpha*diff(u, 2); % using finite difference approximation

% Solve the diffusion problem using ode15s
x = linspace(0, L, 100); % spatial grid
sol = ode15s(ode, tspan, u0(x), [], x);

% Plot the solution
figure;
for i = 1:length(sol.x)
    plot(x, sol.y(:, i));
    hold on;
end
xlabel('x');
ylabel('u(x)');
title('Diffusion problem solution');
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This code defines a simple one-dimensional diffusion problem with a sine-shaped initial condition and zero boundary conditions. The ode function is set up to compute the second derivative of the solution using a finite difference approximation. The ode15s function is called with the defined system of differential equations, time span, initial condition, and additional arguments specifying the spatial grid.

Note that the solution sol returned by ode15s contains the time points in sol.x and the corresponding solution values in sol.y. The solution can be plotted using the plot function.

Make sure to modify and adapt this code to your specific diffusion problem, including defining appropriate initial and boundary conditions and adjusting the values for the diffusion coefficient and length of the domain.