Parameters optimization for the fractional advection--dispersion equation using Crank--Nicolson and particle swarm optimization methods

Irandoust-Pakchin, Safar; Zaki, Golamreza; Derakhshan, Mohammadhossein; Abdi-Mazraeh, Somaiyeh

doi:10.22124/jmm.2026.32871.2989

Parameters optimization for the fractional advection--dispersion equation using Crank--Nicolson and particle swarm optimization methods

Articles in Press

Document Type : Research Article

Authors

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

10.22124/jmm.2026.32871.2989

Abstract

In this paper, we consider the time--fractional advection--dispersion equation with the Caputo fractional derivative of order $0<\alpha\le1$, dispersion coefficient $\lambda>0$, and average fluid velocity $\mu\ge0$. For the direct problem, the initial and boundary conditions are prescribed, and a numerical method based on the Crank--Nicolson scheme is employed.
For the inverse problem, the goal is to identify the optimal values of $\lambda$, $\mu$, and $\alpha$. To achieve this, the particle swarm optimization method is used to minimize the fitting error so that the numerical solution matches the observed data at the selected time levels.
Several numerical examples are presented to demonstrate the efficiency of the proposed approach and to validate the theoretical analysis.

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