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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 $[0, \pi]$.
The hardest part is to understand how to calculate the $b_n$ coefficients via an odd extension of the initial condition. In many ways, this is somewhat backwards (usually we ask “How do I compute a Fourier series for an odd extension” rather than to associate a Fourier series already given to an odd extension).
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