Employing a Fractional Basis Set to Solve Nonlinear Multidimensional Fractional Differential Equations

Md. Habibur Rahman, Muhammad I. Bhatti () and Nicholas Dimakis
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Md. Habibur Rahman: Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
Muhammad I. Bhatti: Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
Nicholas Dimakis: Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

Mathematics, 2023, vol. 11, issue 22, 1-15

Abstract: Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the Bhatti-fractional polynomials basis set. The method involved approximating the desired solution and treated the resulting equation as a matrix equation. All fractional derivatives are considered in the Caputo sense. The resulting operational matrix was inverted, and the desired solution was obtained. The effectiveness of the method was demonstrated by solving two specific types of nonlinear multidimensional fractional differential equations. The results showed higher accuracy, with absolute errors ranging from 10 − 12 to 10 − 6 when compared with exact solutions. The proposed technique offered computational efficiency that could be implemented in various programming languages. The examples of two partial fractional differential equations were solved using Mathematica symbolic programming language, and the method showed potential for efficient resolution of fractional differential equations.

Keywords: Bernstein polynomial method; fractional differential equations; multidimensional fractional equations; Bhatti-fractional polynomials; operational matrix; computational efficiency; symbolic programming (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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