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
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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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