Explicit shifted second-kind Chebyshev spectral treatment for fractional Riccati differential equation

  • Waleed M. Abd-Elhameed Department of Mathematics, Faculty of Science, Cairo University
  • Youssri H. Youssri Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt http://orcid.org/0000-0003-0403-8797
Keywords: Chebyshev polynomials of the second kind, spectral methods, linearization formula, hypergeometric functions

Abstract

This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method. A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established. This formula is expressed in terms of a certain terminating hypergeometric function of the type $_4F_{3}(1)$. This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type $_3F_{2}(1)$ which can be summed with the aid of Watson's identity. Five illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.

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Author Biography

Youssri H. Youssri, Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Youssri H. Youssri is an Assistant Professor at Cairo University in Egypt. His major research interests are in the areas of numerical solutions of differential problems with a focus on more fundamental topics like fractional calculus. He has received his master's (2011) and doctorate (2014) degrees in mathematics from Cairo University, Egypt. He has published about 40 refereed papers in different disciplines of numerical analysis, including: orthogonal polynomials, spectral methods, fractional differential equations, wavelets, and others. He seeks opportunities for further collaboration and expansion of his research interests. Youssri is an adjunct faculty member at the Department of Mathematics and Actuarial Science - AUC, Egypt.

Published
2019-12-26
Section
Articles