An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions

Kazmi, Kamran

doi:10.1016/j.apnum.2019.08.019

Cited by 19 publications

(11 citation statements)

References 43 publications

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“…Based on a backward difference formula, Podlubny [ 195 , 196 ] proposed a matrix form to represent discrete analogs of fractional operations and extended this method to the solution of DODEs [ 197 ]. Other mesh-based techniques developed in literature to solve DODEs and multi-terms FDEs include the predictor-corrector method [ 56 , 198 , 199 , 200 , 201 ] and the finite volume method [ 127 , 128 , 202 ].…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…(46) Then the spectral Jacobi-Galerkin scheme (28) in the two-dimensional case can be expressed in the following form…”

Section: Two-dimensional Casementioning

confidence: 99%

“…Numerical methods for solving distributed-order time-fractional diffusion equations are presented in Morgado and Rebelo [24], Abdelkawy et al [25], and Zaky and Machado [26]. Numerical methods for distributedorder space-fractional diffusion equations are provided in Abbaszadeh [27], Kazmi and Khaliq [28], and Fan and Liu [29]. Numerical methods for multi-dimensional distributedorder generalized Schrödinger equations are provided in Bhrawy and Zaky [30].…”

Section: Introductionmentioning

confidence: 99%

Jacobi Spectral Galerkin Method for Distributed-Order Fractional Rayleigh–Stokes Problem for a Generalized Second Grade Fluid

2020

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Distributed-order fractional differential operators provide a powerful tool for mathematical modeling of multiscale multiphysics processes, where the differential orders are distributed over a range of values rather than being just a fixed fraction. In this work, we consider the Rayleigh-Stokes problem for a generalized second-grade fluid which involves the distributed-order fractional derivative in time. We develop a spectral Galerkin method for this model by employing Jacobi polynomials as temporal and spatial basis/test functions. The suggested approach is based on a novel distributed order fractional differentiation matrix for Jacobi polynomials. Numerical results for one-and two-dimensional examples are presented illustrating the performance of the algorithm. The results show that our scheme can achieve the spectral accuracy for the problem under consideration with smooth solution and allows a great flexibility to deal with multi-dimensional temporally-distributed order fractional Rayleigh-Stokes problems as the global behavior of the solution is taken into account.

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“…Thus, to use Equation (1) in practical modeling, it is indispensable to develop efficient numerical methods for solving it in the related numerical simulations. A large number of numerical methods for multi-term or distribution-order fractional partial differential equations have been constructed and analyzed by numerical analysts from all over the world, see References [8][9][10][11][12][13][14][15] and the reference therein, for examples.…”

Section: Introductionmentioning

confidence: 99%

A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity

Huang

Zhang

Arshad

et al. 2021

Numerical Methods Partial

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In this paper, a superlinear convergence scheme for the multi-term and distribution-order fractional wave equation with initial singularity is proposed. The initial singularity of the solution of the multi-term time fractional partial differential equation often generate a singular source, it increases the difficulty to numerically solve the equation. Thus, after discretizing the spatial distribution-order derivative by the midpoint quadrature, an integral transformation is applied to deal with the temporal direction for obtaining a temporal superlinear convergence scheme based on the uniform mesh. Then, the fully discrete scheme is constructed by using Crank-Nicolson technique and L1 approximation in time, and fractional centered difference approximation in space. The convergence and stability of the proposed scheme are rigorously analyzed. Finally, numerical experiments are presented to support the theoretical results of our scheme.

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An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions

Cited by 19 publications

References 43 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

Jacobi Spectral Galerkin Method for Distributed-Order Fractional Rayleigh–Stokes Problem for a Generalized Second Grade Fluid

A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity

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