A computational approach for the space-time fractional advection–diffusion equation arising in contaminant transport through porous media

Aghdam, Y. Esmaeelzade; Mesgrani, H.; Javidi, Mohammad; Nikan, O.

doi:10.1007/s00366-020-01021-y

Cited by 34 publications

(8 citation statements)

References 29 publications

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“…The appropriate mathematical instances are the diffusion equations with space fractional derivatives in which the classical second-order derivative in the space is basically replaced by the Caputo fractional derivative of order 1 < ≤ 2. The physical explanation and numerical methods for studying space fractional diffusion equations (SFDEs) have been investigated mostly with some general ideas [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

Tuấn¹,

Aghdam

Jafari

et al. 2020

Numerical Methods Partial

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Fractional diffusion equations include a consistent and efficient explanation of transport phenomena that manifest abnormal diffusion, that cannot be often represented by second‐order diffusion equations. In this article, a two‐dimensional space fractional diffusion equation (SFDE‐2D) with nonhomogeneous and homogeneous boundary conditions is considered in Caputo derivative sense. An instant and nevertheless accurate scheme is obtained by the finite‐difference discretization to get the semidiscrete in temporal derivative with convergence order Ofalse(δτ2false). Moreover, space fractional derivative can be approximated based on the Chebyshev polynomials of second kind which are powerful methods for basing the operational matrix. The convergence and stability of the proposed scheme are discussed theoretically in detail. Finally, two numerical problems with an exact solution are given that numerical results show the effectiveness of the new techniques. These schemes can be simply extended to three spatial dimensions, which will be the subject of our subsequent research.

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Section: Introductionmentioning

confidence: 99%

A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

Tuấn¹,

Aghdam

Jafari

et al. 2020

Numerical Methods Partial

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“…There are many ways to solve fractional differential equations such as finite difference methods [5,6], finite element methods [7], finite volume methods [8], spectral methods [9,10], and meshless methods [11]. In recently, many authors applied the various methods for solving these equations [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

The impact of the Chebyshev collocation method on solutions of the time-fractional Black–Scholes

2020

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This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the $$(2-\alpha)$$ ( 2 - α ) -order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.

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“…The derivatives and integrals of arbitrary order are very convenient for describing properties of many real-world physical systems, and the new fractional models are more satisfying than former integer-order ones (Kilbas et al 2006;Podlubny 1999;Samko et al 1993). In this sense, with the growing developments in the various fields of science and engineering (Debnath 2003;Ansari 2016, 2017;Eshaghi et al 2019Eshaghi et al , 2020Esmaeelzade et al 2020;Golbabai et al 2019;Mainardi 1994Mainardi , 1997Metzler et al 1995;Nikan et al 2020a, b), the concepts of stability analysis of the fractional differential systems have attracted increasing interest for many researchers. For example, some authors studied the stability of fractional order nonlinear systems with the Caputo derivative by using the Lyapunov direct method with the concept of the Mittag-Leffler stability (Li and Chen 2010;Liu et al 2016;Zhang et al 2011).…”

Section: Introductionmentioning

confidence: 99%

Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems

2020

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In this work, we investigate the asymptotic stability analysis for two classes of nonlinear fractional systems with the regularized Prabhakar derivative. The stability analysis of the neutral and integro-differential nonlinear fractional systems are studied by assessing the eigenvalues of associated matrix and applying conditions on the nonlinear part of these types of systems. We use a numerical method to solve the fractional differential equations with the regularized Prabhakar fractional derivative. We further present the numerical simulations on several test cases to examine and reveal the complex dynamics of the analytical obtained results.

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A computational approach for the space-time fractional advection–diffusion equation arising in contaminant transport through porous media

Cited by 34 publications

References 29 publications

A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

The impact of the Chebyshev collocation method on solutions of the time-fractional Black–Scholes

Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems

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