Trinh @ Bath

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vpde_lecture25 [2020/03/31 08:26]
trinh 		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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 u_n(x, t) = \sin\left(nx\right) \left[ A_n \cos(nct) + B_n \sin(nct)\right] 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 08:26