Exact Solutions, Convergence, and Stability Analysis of Fractional Diffusion Models with Nonlocal Interactions

Document Type : Review Article

Authors

Department of Mathematics, Babol Noshirvani University of Technology, Shariati Ave., Babol, Iran

Abstract

In this paper, a fractional integro-differential model involving the Caputo time-fractional derivative and the Riesz space-fractional operator is proposed and analyzed. The model incorporates both nonlinear reaction terms and nonlocal integral interactions, allowing an accurate description of anomalous diffusion processes with memory and spatial long-range effects. By applying the Fourier transform with respect to the spatial variables and the Laplace transform with respect to time, the governing equation is transformed into an algebraic equation in the transform domain, leading to an explicit representation of the solution in terms of Mittag--Leffler functions. The existence, convergence, and stability of the mild solution are established by means of an iterative scheme combined with fixed-point arguments and a fractional Gronwall inequality. It is shown that the approximate solutions converge uniformly to the unique mild solution and that the solution depends continuously on the initial data. To illustrate the theoretical results, three representative examples are presented, including a pure fractional diffusion model, a reaction--diffusion model, and a multi-mode system with nonzero integral kernels. The obtained exact solutions demonstrate the significant influence of the fractional orders on the temporal decay rate and spatial behavior of the solution. The proposed framework provides a mathematically rigorous and physically meaningful tool for modeling and analyzing fractional-order transport phenomena arising in engineering and industrial applications such as heat conduction in heterogeneous materials, diffusion in porous media, and dynamic processes in complex systems.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 11, Issue 1
June 2026
Pages 186-203

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History

  • Receive Date: 26 February 2026
  • Revise Date: 08 April 2026
  • Accept Date: 18 April 2026
  • Publish Date: 07 June 2026

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