# Chebyshev Collocation Method for 2D Boundary Value Problems

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Consider the 2D boundary value problem given by , with boundary conditions and . You can set the values of and . Using this Demonstration, you can solve the PDE using the Chebyshev collocation method adapted for 2D problems. The solution is shown as either a 3D plot or a contour plot.

Contributed by: Housam Binous, Brian G. Higgins, and Ahmed Bellagi (March 2013)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomial of the first kind, .

The Chebyshev derivative matrix at the quadrature points is an matrix given by

, , for

and

for and ,

where for and .

The discrete Laplacian is given by , where is the identity matrix, is the Kronecker product operator, , and is without the first row and first column.

Reference

[1] L. N. Trefethen, *Spectral Methods in MATLAB*, Philadelphia: SIAM, 2000.

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