chebyshev collocation method to solve an eigenvalue problem for coupled equation using collocation points

i want to find the critical values to an eigenvalue problem using chebyshev collocation method and need to plot the critical curves , neutral curves as well. Also i need to print those values as a grid. How can i get this

[Torsten]

How can i get this

By investing quite a lot of time.

Or making a google search.

Or looking into the File Exchange.

https://uk.mathworks.com/matlabcentral/fileexchange/48862-chebyshev-collocation-code-for-solving-two-phase-orr-sommerfeld-eigenvalue-problem

[Kavitha G N]

thank you

Answer 1

[Ayush Anand]

Hello Kavitha,

I understand that you want to find the critical values to an eigenvalue problem using Chebyshev collocation method.

In this case, assuming that you have the eigenvalue problem setup in the form of a differential equation or a matrix equation, please follow the steps below.

1. You can use the “chebpts” function to choose the Chebyshev collocation points, by defining the interval and number of points you want in the parameters.
2. Next, solve the differential/matrix equation defined in the problem at each of the collocation points; this will result in a system of equations that you can solve for the eigenvalues.
3. You can use “eig” or “eigs” function in MATLAB to solve the collocation system for the eigenvalues.
4. For plotting the critical and neutral curves, set a range for the parameter you want to vary, and repeat step 3. Then classify the eigenvalues obtained as critical or neutral depending on your criteria. Store these in a list and plot them against the parameter range specified by you initially.
5. You can plot the eigenvalues as a grid using the “table” command.

You can refer to the answer below to check an example of how to implement the Chebyshev collocation method:

General code for chebyshev pseudo spectral method - Help Center Answers - MATLAB & Simulink (mathworks.com)

I hope this helps.