How can I code the time fractional differential equation with delay using the Crank Nicholson method by using MATLAB?
Solving time fractional differential equations with delay using the Crank-Nicolson method involves discretizing the equation in both time and space. The Crank-Nicolson method is a numerical scheme that combines implicit and explicit methods to approximate the solution. Below is an example MATLAB code to solve a time fractional differential equation with delay using the Crank-Nicolson method. Note that this is a basic template, and you may need to adapt it to your specific equation and boundary/initial conditions.
% Parameters alpha = 0.5; % Fractional order (0 < alpha < 1) T = 1; % Total simulation time dt = 0.01; % Time step N = T / dt; % Number of time steps % Space domain L = 10; % Length of the spatial domain dx = 0.1; % Spatial step x = 0:dx:L; % Spatial grid % Initial condition u0 = sin(pi * x / L); % Crank-Nicolson method A = sparse(N + 1, N + 1); rhs = zeros(N + 1, 1); for n = 1:N % Discretized fractional derivative with delay if n > 1 u_delayed = u(:, n - 1); else u_delayed = u0; end D_alpha = fracderiv(x, u(:, n), alpha) + fracderiv(x, u_delayed, alpha); % Crank-Nicolson discretization A(n, n) = 1; A(n, n + 1) = -0.25 * dt * D_alpha; A(n + 1, n) = 0.25 * dt * D_alpha; A(n + 1, n + 1) = 1; % Right-hand side rhs(n) = u(:, n); rhs(n + 1) = u(:, n + 1); % Solve the linear system u(:, n + 1) = A \ rhs; end % Plot the results figure; surf(x, (1:N + 1) * dt, u'); xlabel('Space (x)'); ylabel('Time (t)'); zlabel('u(x, t)'); title('Numerical Solution using Crank-Nicolson Method'); % Fractional derivative function function D_alpha = fracderiv(x, u, alpha) N = length(x); h = x(2) - x(1); D_alpha = zeros(N, 1); for i = 2:N-1 D_alpha(i) = (u(i+1) - u(i-1)) / (2 * h) + ... alpha / (h * gamma(2 - alpha)) * (u(i+1) - 2 * u(i) + u(i-1)); end % Forward and backward differences at boundaries D_alpha(1) = (u(2) - u(1)) / h; D_alpha(N) = (u(N) - u(N-1)) / h; end
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