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- | //Need to write this.// | + | ====== Example 15.4: Solution of the inhomogeneous Dirichlet problem ====== |
+ | |||
+ | This lecture starts off with Example 15.4 from the notes, where we look to solve | ||
+ | $$ | ||
+ | \begin{gathered} | ||
+ | u_t = u_{xx} \\ | ||
+ | u(0, t) = 2, \qquad u(\pi, t) = 1 \\ | ||
+ | u(x, 0) = 0. | ||
+ | \end{gathered} | ||
+ | $$ | ||
+ | |||
+ | We show that the solution is given by | ||
+ | $$ | ||
+ | u(x, t) = U(x) + \sum_{n=1}^\infty B_n \sin(nx) \mathrm{e}^{-n^2 t} | ||
+ | $$ | ||
+ | where we have found the steady-state solution | ||
+ | $$ | ||
+ | U(x) = 2 - \frac{x}{\pi}, | ||
+ | $$ | ||
+ | as well as the coefficients | ||
+ | $$ | ||
+ | B_n = \frac{2}{n\pi}[(-1)^n - 2]. | ||
+ | $$ | ||
+ | |||
+ | We also illustrated the solution using the code: | ||
+ | |||
+ | < | ||
+ | %% Plot the Fourier series for Example 15.4 | ||
+ | % Written for MA20223 Vectors & PDEs 2019-20 | ||
+ | |||
+ | clear % Clear all variables | ||
+ | close all % Close all windows | ||
+ | N = 100; % How many Fourier modes to include? | ||
+ | |||
+ | R = @(n, t, x) 2/(n*pi)*((-1)^n - 2)*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); | ||
+ | plot(x, 2 - x/pi, ' | ||
+ | ylim([-0.2, | ||
+ | xlim([0, x0]); % Set the x limits | ||
+ | xlabel(' | ||
+ | hold on | ||
+ | |||
+ | for j = 1: | ||
+ | tj = t(j); | ||
+ | |||
+ | u = (2 - x/pi); | ||
+ | for n = 1:N | ||
+ | u = u + R(n, tj, x); | ||
+ | end | ||
+ | if j == 1 | ||
+ | p = plot(x, u, ' | ||
+ | else | ||
+ | set(p, ' | ||
+ | end | ||
+ | drawnow | ||
+ | title([' | ||
+ | pause(0.1); | ||
+ | |||
+ | if j == 1 | ||
+ | pause | ||
+ | end | ||
+ | end | ||
+ | </ | ||
+ | |||
+ | ===== Problem set 8 Q1 ===== | ||
+ | |||
+ | The next thing we studied was PS8, Q1, which is the solution of the homogeneous Neumann problem for the heat equation. | ||
+ | |||
+ | < | ||
+ | %% Plot the Fourier series for a made-up modification of PS8 Q1. | ||
+ | % 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: | ||
+ | R = @(n, t, x) 4*(-1)^n/ | ||
+ | |||
+ | % 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); | ||
+ | plot(x, x.^2, ' | ||
+ | ylim([-0.2, | ||
+ | xlim([0, x0]); % Set the x limits | ||
+ | xlabel(' | ||
+ | hold on | ||
+ | |||
+ | for j = 1: | ||
+ | tj = t(j); | ||
+ | |||
+ | u = 1/2*(2*pi^2/ | ||
+ | for n = 1:N | ||
+ | u = u + R(n, tj, x); | ||
+ | end | ||
+ | if j == 1 | ||
+ | p = plot(x, x.^2, ' | ||
+ | else | ||
+ | set(p, ' | ||
+ | end | ||
+ | drawnow | ||
+ | title([' | ||
+ | pause(0.1); | ||
+ | |||
+ | if j == 1 | ||
+ | pause | ||
+ | end | ||
+ | end | ||
+ | </Code> |