Engineering Physics and Mathematics
Application of Chebyshev collocation method for solving two classes of non-classical parabolic PDEs

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Abstract

This article contributes a numerical scheme for finding approximate solutions of one-dimensional parabolic partial differential equations (PDEs) under non-classical boundary conditions. This scheme is based on the direct Chebyshev collocation method that has been frequently used for problems of ordinary differential equations (ODEs). In fact, the approximate solution of the problem in the truncated Chebyshev series form is obtained by this method. By substituting Chebyshev series solution into the considered problems and by using the matrix operations and the collocation points, the suggested scheme reduces the problems into the associated linear algebraic systems. By solving this system of equations, the unknown Chebyshev coefficients can be determined. To show the accuracy and the efficiency of the method, two numerical examples are implemented and the comparisons are given by a new collocation method.

Keywords

Parabolic problems
Non-classical boundary conditions
Chebyshev series
Collocation points
Chebyshev operational matrix of derivative

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Emran Tohidi is currently a Ph.D. degree student at Ferdowsi University of Mashhad, Mashhad, Iran. He received the B.S. and the M.S. degrees in 2009 and 2011, respectively, in applied mathematics from the same university. The subject of his M.S. degree thesis was ”Numerical solution of optimal control problems via Legendre and Chebyshev polynomials.” Up to now, he has at least 34 published papers surrounding the subject of optimal control problems and numerical solutions of ODEs and PDEs specially working by spectral methods.

Peer review under responsibility of Ain Shams University.

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