ArticleOpen Access

THE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS

M. HOSSEININIA

Faculty of Mathematics, Yazd University, Yazd, Iran

E-mail Address: m.hoseininia@stu.yazd.ac.ir

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M. H. HEYDARI

Department of Mathematics, Shiraz University of Technology, Shiraz, Iran

E-mail Address: heydari@sutech.ac.ir

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, and

Z. AVAZZADEH

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

E-mail Address: zakiehavazzadeh@tdtu.edu.vn

Corresponding author.

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https://doi.org/10.1142/S0218348X20400423Cited by:17 (Source: Crossref)

This article is part of the issue:

Special Issue on Fractal and Fractional with Applications to Nature
Guest Editors: Abdon Atangana, Emile Franc Goufo Ndongmo, José Francisco Gómez Aguilar, Muhammad Altaf Khan and Ilknur Koca

Abstract

This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the $θ$ $𝜃$ -weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.

Keywords:

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Vol. 28, No. 08

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History

Received 21 December 2019

Accepted 12 February 2020

Published: July 10, 2020

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This is an Open Access article in the “Special Issue on Fractal and Fractional with Applications to Nature” published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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THE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS

Abstract

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