Numerical simulation and error analysis of distributed-order time-fractional equations with nonlinear Riesz diffusion

Derakhshan, Mohammadhossein; Ordokhani, Yadollah; Nemati, Somayeh

doi:10.22124/jmm.2026.33154.3019

Numerical simulation and error analysis of distributed-order time-fractional equations with nonlinear Riesz diffusion

Articles in Press

Document Type : Research Article

Authors

¹ Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

² Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

10.22124/jmm.2026.33154.3019

Abstract

Fractional partial differential equations (FPDEs) have emerged as powerful modeling tools for describing complex systems exhibiting memory, hereditary effects, and anomalous transport phenomena, which cannot be captured by classical integer-order models [5, 16, 38]. In recent years, fractional and distributed-order differential equations have gained significant attention in applied mathematics and engineering due to their superior ability to describe memory effects, hereditary behaviors, and anomalous transport phenomena observed in complex physical systems.
Unlike classical integer-order models, fractional models incorporate nonlocality in both time and space, which is crucial for capturing the dynamics of heterogeneous media, porous materials, viscoelastic substances, and biological tissues [7, 29, 31, 34]. Among these, distributed-order derivatives provide an even more flexible framework by incorporating a continuum of fractional orders, allowing the model to account for multi-scale memory effects and variable relaxation dynamics. This added generality makes distributed-order models more accurate in describing real-world industrial systems, where processes exhibit time-varying and history-dependent behavior that
cannot be adequately captured using single-order derivatives. Moreover, the combination of nonlinear dynamics with distributed-order operators introduces a richer mathematical structure and presents significant analytical and computational challenges.

Keywords