Fastfourier - dt =.001 Defining an appropriate time step u0...

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Fast Fourier Transform Sudarshan Balakrishnan August 23, 2011 The purpose of this write up is to notice the similarity between the exact solution of the linearly dispersive wave equation and the solution computed using the fast fourier transform. The commented code is as follows: ‘1 %This program computes the solution to the linearly dispersive wave equation using the fast fourier transform. N = 1000; %Number of grid points. h = 2*pi/N; %Defining the size of each grid. x = h*(1:N); %Defining the variable x as an array. t = .05*pi; %Defining the variable t as a rational multiple of \pi.
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Unformatted text preview: dt = .001; % Defining an appropriate time step. u0 = zeros(1,N); %Defining the initial data u0(N/2+1:N)= ones(1,N/2); % Defining the initial data k=(1i*[0:N/2-1 0 -N/2+1:-1]); k3=k.^3; u=ifft(exp(k3*t).*fft(u0));%Solution to the Linearly dispersive wave equation. plot(u)%Command to plot the solution At time t = . 05 * π , using the above code we get the following figure: 1 Using the exact solution, we get: At an irrational multiple of π , we notice a fractal pattern to the graph: and the exact solution produces the same plot 2...
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