Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear

Khaled M. Saad; Manal Alqhtani; Khaled M. Saad; Manal Alqhtani

doi:10.3934/math.2021225

AIMS Mathematics

2021, Volume 6, Issue 4: 3788-3804. doi: 10.3934/math.2021225

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Research article

Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear

Khaled M. Saad ^{1,2
,
,},
Manal Alqhtani ¹

1.
Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Kingdom of Saudi Arabia
2.
Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen

Received: 10 October 2020 Accepted: 18 January 2021 Published: 27 January 2021
MSC : Primary 26A33, 34A08; Secondary 35A20, 35A22

In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, $ 1 < q < 2 $, in the equation on the behavior of numerical solutions of fractal-fractional reaction diffusion equations (FFRDE). Iterative approximations to the solutions of these equations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In key parameter regimes, all these iterative solutions based on a power kernel, an exponential kernel and a generalized Mittag-Leffler kernel are very close. Hence, iterative solutions obtained using one of these kernels are compared with full numerical solutions of the FFRDE and excellent agreement is found. All numerical solutions in this paper were obtained using Mathematica.
Citation: Khaled M. Saad, Manal Alqhtani. Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear[J]. AIMS Mathematics, 2021, 6(4): 3788-3804. doi: 10.3934/math.2021225

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Abstract

In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, $ 1 < q < 2 $, in the equation on the behavior of numerical solutions of fractal-fractional reaction diffusion equations (FFRDE). Iterative approximations to the solutions of these equations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In key parameter regimes, all these iterative solutions based on a power kernel, an exponential kernel and a generalized Mittag-Leffler kernel are very close. Hence, iterative solutions obtained using one of these kernels are compared with full numerical solutions of the FFRDE and excellent agreement is found. All numerical solutions in this paper were obtained using Mathematica.

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