This is an old revision of the document!
We will complete our determination of the Fourier series solution of the heat equation as presented in the previous lecture. This was the problem of: $$ \begin{gathered} u_t = \kappa u_{xx}, \quad x\in[0, \pi], t \geq 0 \\ u(0, t) = 0 = u(\pi, t) \\ u(x, 0) = f(x) = 1. \end{gathered} $$
$$ u(x, t) \sim \sum_1^\infty \left[b_n \sin\left(\frac{n\pi x}{L}\right)\right]. $$
where $b_n$ will be the Fourier sine coefficients of the odd $2\pi$ extension of $f(x)$ on $[0, \pi]$.
Solution of 1D heat equation with zero Dirichlet
% Written for MA20223 Vectors & PDEs clear % Clear all variables close all % Close all windows N = 20; % How many Fourier modes to include? % Define an in-line function that takes in three inputs: % Input 1: n value [scalar] % Input 2: x value [vector] % Input 3: t value [scalar] R = @(n, t, x) -2/(n*pi)*((-1)^n - 1)*exp(-n^2*t)*sin(n*x); % Create a mesh of points between two limits x0 = pi; x = linspace(0, x0, 1000); % Create a mesh of points in time t = linspace(0, 5, 200); figure(1); % Open the figure plot([0, pi], [1, 1], 'b', 'LineWidth', 2); % Plot the base function ylim([-0.2,1.2]); % Set the y limits xlim([0, x0]); % Set the x limits xlabel('x'); ylabel('u(x,t)'); hold on for j = 1:length(t) tj = t(j); u = 0; for n = 1:N u = u + R(n, tj, x); end % Plotting commands if j == 1 p = plot(x, u, 'r'); else set(p, 'YData', u); end drawnow title(['t = ', num2str(tj)]); pause(0.1); if j == 1 pause end end