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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu

IMPACT FACTOR 2017: 1.162

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Volume 20, Issue 2


Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations

W. M. Abd-Elhameed / Y. H. YoussriORCID iD: https://orcid.org/0000-0003-0403-8797
Published Online: 2019-01-29 | DOI: https://doi.org/10.1515/ijnsns-2018-0118


The basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.

Keywords: Chebyshev polynomials of sixth kind; tau method; collocation method; connection formulae; fractional differential equations

JEL Classification: 65M70; 34A08; 33C45; 11B83


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About the article

Received: 2018-05-07

Accepted: 2019-12-01

Published Online: 2019-01-29

Published in Print: 2019-04-26

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 20, Issue 2, Pages 191–203, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2018-0118.

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