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--- vpde_lecture25 [2020/03/31 08:25]
trinh
+++ vpde_lecture25 [2020/03/31 08:49]
trinh
@@ Line 35: / Line 35: @@
 ===== Example 16.2 (Imagining the modes) =====
-I want to help you imagine what the modes, $u_n$, look like. For simplicity, take the length $L = \pi$. Also, we may take $c = 1$.
+I want to help you imagine what the modes, $u_n$, look like. For simplicity, take the length $L = \pi$. Also, we may take $c = 1$. The individual modes we want to examine are:
+$$
+u_n(x, t) = \sin\left(nx\right) \left[ A_n \cos(nct) + B_n \sin(nct)\right]
+$$
+<Code:Matlab linenums:1 |Demonstration of the modes>
+% Code for MA20223 30 Mar 2020
+clear
+close all
+% Length and time
+L = pi; T = 2*pi/(c*n);
+% Function
+n = 2; c = 1;
+un = @(x,t) sin(n*x/L).*cos(n*c*t/L);
+% Make vectors for space and time
+x = linspace(0, pi, 50); t = linspace(0, 2*T, 50);
+% Create a mesh of x vs. t
+[X,T] = meshgrid(x,t);
+% Matrix of U values to imagine the surface
+U = sin(n*X).*cos(n*T);
+figure(1); subplot(1,2,1); xlabel('x'); ylabel('u');
+subplot(1,2,2);
+% Plot the surface and make it pretty
+s = surf(X,T,U); set(s, 'EdgeColor', 'none', 'FaceColor', 'interp');
+view([-48, 17]); xlabel('x'); ylabel('t'); zlabel('u');
+hold on
+% Plot an animation in time
+for j = 1:length(t)
+    tt = t(j); uu = un(x,tt);
+    subplot(1,2,1);
+    plot(x, uu);   title(['t = ', num2str(tt)]);
+    ylim([-1,1]); xlim([0, pi]);
+    subplot(1,2,2);
+    if j == 1
+        p = plot3(x, tt*ones(size(x)), un(x, t(j)), 'LineWidth', 3);
+        pause;
+    else
+        set(p, 'XData', x, 'YData', tt*ones(size(x)), 'ZData', un(x,t(j)));
+    end
+    drawnow
+    shg
+end
+</Code>

Trinh @ Bath

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