A numerical scheme to solve a class of two-dimensional nonlinear time-fractional diffusion equations of distributed orde
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ORIGINAL ARTICLE
A numerical scheme to solve a class of two‑dimensional nonlinear time‑fractional diffusion equations of distributed order A. Babaei1 · H. Jafari1,2,3 · S. Banihashemi1 Received: 11 August 2020 / Accepted: 23 September 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract This article is devoted to obtain the numerical solution for a class of nonlinear two-dimensional distributed-order timefractional diffusion equations. We discretize the problem by using a finite difference scheme in the time direction. Then, we solve the discretized nonlinear problem by a collocation approach based on the Legendre polynomials. The numerical algorithm is fully described and convergence analysis of the scheme is evaluated. Finally, few numerical implementations are presented to highlight the flexibility and the convergence rate of this method. Keywords Distributed-order fractional derivative · Nonlinear multi-dimensional equation · Finite difference scheme · Legendre polynomials · Convergence analysis
1 Introduction During the recent decades, fractional calculus has been received the attention of many researchers in various fields of science and engineering, such as physics, biology, finance, and fluid mechanics [1–9]. In a large number of natural phenomena, the next state of a system depends on its current and all previous states. Fractional operators preserve these hereditary and memory properties of real problems [10–14]. Nonlocality and the extra degree of freedom introduced by the fractional order are the other main features of these operators. In recent years, some new fractional-order operators with nonlocal and non-singular kernel are proposed and applied in several practical problems [15–19]. Fractional-order differential equations (FDEs) are the special case of distributed-order fractional differential equations (DFDEs) [20, 21]. Thus, models based on the DFDEs became a growing research area due to their potential for * A. Babaei [email protected] 1
Department of Applied Mathematics, University of Mazandaran, Babolsar, Iran
2
Department of Mathematical Sciences, University of South Africa, UNISA0003, Pretoria, South Africa
3
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan
describing complex phenomena and several problems in different fields have been modeled via these types of equations [22–30]. In physics, a diffusion process is characterized by the mean square displacement of a moving particle. In some cases, the growth of mean square displacement is logarithmic and the diffusion process is named ultraslow. The motion of a polyampholyte hooked around an obstacle in polymer physics and particle’s motion in a quenched random force field are some examples of this type anomalous diffusion that their models are defined as DFDEs [27]. Also, the author in [23] used DFDE to describe dielectric induction phenomena. Due to the importance of finding suitable solutions for DFDEs, there has been significant in
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