Exploring fractional Advection-Dispersion equations with computational methods: Caputo operator and Mohand techniques

Azzh Saad Alshehry; Humaira Yasmin; Ali M. Mahnashi; Azzh Saad Alshehry; Humaira Yasmin; Ali M. Mahnashi

doi:10.3934/math.2025012

AIMS Mathematics

2025, Volume 10, Issue 1: 234-269. doi: 10.3934/math.2025012

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Exploring fractional Advection-Dispersion equations with computational methods: Caputo operator and Mohand techniques

1.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O.Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia

Received: 26 August 2024 Revised: 10 December 2024 Accepted: 24 December 2024 Published: 03 January 2025
MSC : 34G20, 35A20, 35A22, 35R11

This study presented a comprehensive analysis of nonlinear fractional systems governed by the advection-dispersion equations (ADE), utilizing the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM). By incorporating the Caputo fractional derivative, we enhanced the modeling capability for fractional-order differential equations, accounting for nonlocal effects and memory in the systems dynamics. We demonstrated that both MTIM and MRPSM were effective for solving fractional ADEs, providing accurate numerical solutions that were validated against exact results. The steady-state solutions, complemented by graphical representations, highlighted the behavior of the system for varying fractional orders and showcased the flexibility and robustness of the methods. These findings contributed significantly to the field of computational physics, offering powerful tools for tackling complex fractional-order systems and advancing research in related fields.
Citation: Azzh Saad Alshehry, Humaira Yasmin, Ali M. Mahnashi. Exploring fractional Advection-Dispersion equations with computational methods: Caputo operator and Mohand techniques[J]. AIMS Mathematics, 2025, 10(1): 234-269. doi: 10.3934/math.2025012

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Abstract

This study presented a comprehensive analysis of nonlinear fractional systems governed by the advection-dispersion equations (ADE), utilizing the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM). By incorporating the Caputo fractional derivative, we enhanced the modeling capability for fractional-order differential equations, accounting for nonlocal effects and memory in the systems dynamics. We demonstrated that both MTIM and MRPSM were effective for solving fractional ADEs, providing accurate numerical solutions that were validated against exact results. The steady-state solutions, complemented by graphical representations, highlighted the behavior of the system for varying fractional orders and showcased the flexibility and robustness of the methods. These findings contributed significantly to the field of computational physics, offering powerful tools for tackling complex fractional-order systems and advancing research in related fields.

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