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vpde_lecture25 [2020/03/31 08:21]
trinh created 		vpde_lecture25 [2020/04/04 21:33] (current)
trinh
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 This lecture will do apply separation of variables and Fourier series in order to solve for the wave equation on a finite interval.  This lecture will do apply separation of variables and Fourier series in order to solve for the wave equation on a finite interval. 
 +
 +<html>
 +<iframe width="560" height="315" src="https://www.youtube.com/embed/VMz3R6d2ZZ8" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 +</html>
  
 ===== Definition 16.1 (1D wave equation with homogeneous Dirichlet BCs ===== ===== Definition 16.1 (1D wave equation with homogeneous Dirichlet BCs =====
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 which models the deflection of a string of length $L$ given initial displacement and velocities.  which models the deflection of a string of length $L$ given initial displacement and velocities. 
  
-===== Separation of variables procedure =====+===== Sec. 16.2: Separation of variables procedure =====
  
 In the video, we'll cover the separation of variables procedure, which follows Sec. 16.2 of the notes. The procedure is virtually identical to the case of the heat equation, with the exception that the equation for two components,  In the video, we'll cover the separation of variables procedure, which follows Sec. 16.2 of the notes. The procedure is virtually identical to the case of the heat equation, with the exception that the equation for two components, 
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 $$ $$
  
 +Once you sum this over all the positive integers, then you will obtain the general solution in terms of the Fourier series. To this solution, we must satisfy the initial displacement and velocities. 
 +
 +===== 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 $c = 1$. 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>
 +
 +==== Implementation of the Fourier series ====
 +
 +Now we need to look to solve for the coefficients of our series by applying the boundary conditions. We have that 
 +$$
 +u(x, t) = \sum_{n=0}^\infty \sin\left(\frac{n\pi x}{L}\right) \left[ A_n \cos\left(\frac{n\pi ct}{L}\right) + B_n \sin\left(\frac{n\pi ct}{L}\right)\right]
 +$$
 +
 +Imposing the initial displacement, we have 
 +$$
 +u_0(x) = \sum_{n=0}^\infty A_n \sin\left(\frac{n\pi x}{L}\right), 
 +$$
 +
 +which we recognise as the sine series for the odd $2L$-periodic extension of the function $u_0(x)$ originally defined on $[0, L]$. So the coefficients are (see theorem 12.7)
 +$$
 +A_n = \frac{2}{L} \int_0^L u_0(x) \sin\left(\frac{n\pi x}{L}\right). 
 +$$
 +
 +Imposing the initial velocity, we have
 +$$
 +v_0(x) = \sum_{n=1}^\infty \left(\frac{n\pi c}{L}\right)B_n \sin\left(\frac{n\pi x}{L}\right)
 +$$
 +
 +Again we recognise this as the sine series, so we now need to equate
 +$$
 +\left(\frac{n\pi c}{L}\right)B_n = \frac{2}{L} \int_0^L v_0(x) \sin\left(\frac{n\pi x}{L}\right). 
 $$ $$
  
 +//(The video gets very close to the end of this; we managed to get the $A_n$ coefficients and need to address the $B_n$ coefficients in lecture 26)//
Last modified: 2020/03/31 08:21