Numerical simulation of nonlinear Galilei invariant advection–diffusion equations via fractal–fractional operators

Rahimkhani, Parisa; Sabermahani, Sedigheh; Abdeljawad, Thabet

doi:10.22124/jmm.2026.32909.2994

Numerical simulation of nonlinear Galilei invariant advection–diffusion equations via fractal–fractional operators

Articles in Press

Document Type : Research Article

Authors

¹ Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran

² Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran

³ Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833,11586 Riyadh, Saudi Arabia

10.22124/jmm.2026.32909.2994

Abstract

In this paper, a new class of fractal-fractional differential equations called the fractal-fractional Galilei invariant advection–diffusion equation is introduced for the first time. Then, we introduce a method based on the fractional Genocchi functions for solving this problem. To achieve the stated objective, we first derive the fractal-fractional and integer-order integral operators for the fractional Genocchi functions in the Atangana-Riemann-Liouville sense.
Then, the obtained integral operators of the fractional Genocchi functions, along with certain properties of the fractional Genocchi functions, are utilized within a collocation method to reduce the fractal-fractional Galilei invariant advection- diffusion equation to a system of algebraic equations involving the unknown coefficients. Finally, the convergence analysis of the proposed method is examined in the Sobolev space. In addition, illustrative examples are provided to demonstrate the validity and applicability of the proposed method.

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