# General code for chebyshev pseudo spectral method

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Bikram Singh el 17 de Jun. de 2023
Respondida: Bikram Singh el 20 de Jun. de 2023
I am studying about the different numerical methods over Matlab. I am studying about Pseudo-spectral collocation method but there is no expample or problem where I can find about coding for this. Is there any possibility or help to get this.
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Kautuk Raj el 18 de Jun. de 2023
Editada: Kautuk Raj el 18 de Jun. de 2023
The pseudo-spectral collocation method is a numerical method for solving differential equations using the spectral representation of the solution. In this method, the solution is represented as a sum of basis functions (e.g., Chebyshev or Fourier polynomials), and the collocation points are used to enforce the equation's constraints. I will give you a simple example of how to implement the Chebyshev collocation method in MATLAB to solve the Poisson equation on the interval [-1, 1]:
Poisson equation: -u''(x) = f(x) with u(-1) = u(1) = 0.
Let f(x) = 10*sin(3*pi*x). The analytical solution is u(x) = (10/(9*pi^2)) * sin(3*pi*x).
This is a MATLAB implementation of the Chebyshev collocation method to solve this problem:
% Problem definition
f = @(x) 10 * sin(3 * pi * x);
analytical_solution = @(x) (10 / (9 * pi^2)) * sin(3 * pi * x);
N = 20; % Number of collocation points
% Chebyshev collocation points
x = cos(pi * (0:N) / N)';
% Chebyshev differentiation matrix
T = chebyshev_differentiation_matrix(N);
% Second-order differentiation matrix
T2 = T^2;
% Boundary conditions
T2 = T2(2:end-1, 2:end-1);
% Evaluate the forcing term at the collocation points
F = f(x);
F = F(2:end-1);
% Solve the linear system
u_inner = T2 \ F;
u = [0; u_inner; 0];
% Plot the numerical and analytical solutions
xx = linspace(-1, 1, 200);
figure;
plot(x, u, 'ro', xx, analytical_solution(xx), 'b-');
legend('Numerical solution', 'Analytical solution');
xlabel('x'); ylabel('u(x)');
title('Chebyshev collocation method for Poisson equation');
In this example, we first define the problem and the number of collocation points N. Then, we calculate the Chebyshev collocation points and the Chebyshev differentiation matrix using a custom helper function chebyshev_differentiation_matrix. We then compute the second-order differentiation matrix and apply the boundary conditions. Finally, we solve the linear system and plot the numerical and analytical solutions.
Here is the chebyshev_differentiation_matrix function:
function D = chebyshev_differentiation_matrix(N)
% Compute the Chebyshev differentiation matrix of order N
x = cos(pi * (0:N) / N)';
c = [2; ones(N-1, 1); 2] .* (-1).^(0:N)';
X = repmat(x, 1, N+1);
dX = X - X';
D = (c * (1./c)')./(dX + eye(N+1));
D = D - diag(sum(D, 2));
end
This is how the output looks like:
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Bikram Singh el 19 de Jun. de 2023
Thank you so much sir for answering. Is it possible to get such example also for Pseudospectral collocation method?
Kautuk Raj el 19 de Jun. de 2023

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Bikram Singh el 20 de Jun. de 2023
I want to put the coupled equations as mentioned in the attached file(file.PNG) in matlab code.
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