Numerical analysis of some partial differential equations with fractal-fractional derivative

Nadiyah Hussain Alharthi; Abdon Atangana; Badr S. Alkahtani; Nadiyah Hussain Alharthi; Abdon Atangana; Badr S. Alkahtani

doi:10.3934/math.2023116

AIMS Mathematics

2023, Volume 8, Issue 1: 2240-2256. doi: 10.3934/math.2023116

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Numerical analysis of some partial differential equations with fractal-fractional derivative

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, P.O. Box 90950, 11623, Riyadh, Saudi Arabia
2.
Institute for Groundwater Studies, Faculty of Natural and Agricultural Science, University of the Free State, 9300 Bloemfontein, South Africa
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
4.
Department of Mathematics, College of Science, King Saud University, P.O. Box 1142, Riyadh 11989, Saudi Arabia

Received: 09 September 2022 Revised: 10 October 2022 Accepted: 14 October 2022 Published: 31 October 2022
MSC : 35-xx

In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed.
- fractal-fractional,
- power-law kernel,
- Mittag-Leffler kernel,
- partial differential equation,
- numerical scheme, heat equation
Citation: Nadiyah Hussain Alharthi, Abdon Atangana, Badr S. Alkahtani. Numerical analysis of some partial differential equations with fractal-fractional derivative[J]. AIMS Mathematics, 2023, 8(1): 2240-2256. doi: 10.3934/math.2023116

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Abstract

In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed.

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