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| //This lecture is about terminology but it is important not to be bogged down by terminology. You will practice by doing!// | //This lecture is about terminology but it is important not to be bogged down by terminology. You will practice by doing!// | ||
| + | |||
| + | < | ||
| + | <iframe width=" | ||
| + | </ | ||
| + | |||
| + | ===== An example ===== | ||
| + | |||
| We're now going to return to partial differential equations (which motivated Fourier series). We will mainly study **boundary value problems** (BVPs). These come with **initial conditions** (ICs) and **boundary conditions** (BCs). Here is an example: | We're now going to return to partial differential equations (which motivated Fourier series). We will mainly study **boundary value problems** (BVPs). These come with **initial conditions** (ICs) and **boundary conditions** (BCs). Here is an example: | ||
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| ==== The 1D heat equation with zero Dirichlet conditions ==== | ==== The 1D heat equation with zero Dirichlet conditions ==== | ||
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| + | Now let's return to the study of the heat equation with zero Dirichlet conditions stated at the top of this note. In the video, we will go through the procedure from start to end in solving the BVP. In the end, the solution will be given by a Fourier sine series, | ||
| + | |||
| + | $$ | ||
| + | u(x, t) \sim \sum_1^\infty \left[b_n \sin\left(\frac{n\pi x}{L}\right)\right]. | ||
| + | $$ | ||
| + | |||
| + | where $b_n$ will be the Fourier sine coefficients of the odd $2\pi$ extension of $f(x)$ on $[0, \pi]$. | ||
| + | |||
| + | < | ||
| + | % 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/ | ||
| + | |||
| + | % 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([0, pi], [1, 1], ' | ||
| + | ylim([-0.2, | ||
| + | xlim([0, x0]); % Set the x limits | ||
| + | xlabel(' | ||
| + | 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, ' | ||
| + | else | ||
| + | set(p, ' | ||
| + | end | ||
| + | drawnow | ||
| + | title([' | ||
| + | pause(0.1); | ||
| + | |||
| + | if j == 1 | ||
| + | pause | ||
| + | end | ||
| + | end | ||
| + | </ | ||