# Chebyshev Collocation Method for Linear and Nonlinear Boundary Value Problems

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Consider two boundary-value problems (BVP), one linear and the other nonlinear.

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

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The linear BVP requires solving a system of linear equations, which is readily done using LinearSolve.

The nonlinear BVP involves a system of nonlinear algebraic equations, which can be conveniently solved using FindRoot.

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by .

The Chebyshev derivative matrix at quadrature points, , is given by:

, , for and for and ,

where for and .

The matrix is used this way: and .

References

[1] P. Moin, *Fundamentals of Engineering Numerical Analysis*, Cambridge, UK: Cambridge University Press, 2001.

[2] S. Biringen and C.-Y. Chow, *An Introduction to Computational Fluid Mechanics by Example*, Hoboken, NJ: John Wiley & Sons, 2011.

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