Numerical solution of the time fractional reaction–diffusion equation with a moving boundary

Zheng, Min; Liu, Fawang; Liu, Qingxia; Burrage, Kevin; Simpson, Matthew J.

doi:10.1016/j.jcp.2017.03.006

Cited by 28 publications

(12 citation statements)

References 33 publications

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“…The difference equation of ECBS method for the TFRD model at mth time level, can be stated as (27) and the boundary conditions are mentioned below:…”

Section: Convergence Analysismentioning

confidence: 99%

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A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

et al. 2020

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The time–fractional reaction–diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo–Fabrizio operator is applied for the discretization of time levels while the extended cubic B-spline (ECBS) function is applied for the space direction. The ECBS function preserves geometrical invariability, convex hull and symmetry property. Unconditional stability and convergence analysis are also proved. The projected numerical method is tested on two numerical examples. The theoretical and numerical results demonstrate that the order of convergence of 2 in time and space directions.

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“…The difference equation of ECBS method for the TFRD model at mth time level, can be stated as (27) and the boundary conditions are mentioned below:…”

Section: Convergence Analysismentioning

confidence: 99%

“…Ersoy and Dag [26] solved the FRDM using the exponential cubic B-spline technique. Zheng et al [27] presented the numerical algorithm of FRDM with a moving boundary using FDM and spectral approximation. Owelabi and Dutta [28] considered the Laplace and the Fourier transform to solve FRDM numerically.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

et al. 2020

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“…Using this fact, Seki et al 25 and Yuste et al 26 derived the fractional reaction sub‐diffusion equation. In the last few years, many papers appeared to study the fractional reaction sub‐diffusion equation, for example, the literature 3,27–29 …”

Section: Introductionmentioning

confidence: 99%

A simple numerical method for two‐dimensional nonlinear fractional anomalous sub‐diffusion equations

Sweilam

Ahmed

Adel

2020

Math Methods in App Sciences

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Recently, many numerical techniques were presented to solve the fractional anomalous sub‐diffusion equations, and the results were excellent. In this paper, we study a simple numerical technique to solve two important types of fractional anomalous sub‐diffusion equations that appear strongly in chemical reactions and spiny neuronal dendrites, which are the two‐dimensional fractional Cable equation and the two‐dimensional fractional reaction sub‐diffusion equation. The proposed technique is a simple one which is an extension of the weighted average finite difference technique. The stability analysis of the proposed method is studied by means of John von Neumann stability analysis technique. An accurate stability criterion which is valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced. Four numerical examples are presented (two for the Cable equation and two for the reaction sub‐diffusion equation) to demonstrate the effectiveness and the accuracy of the presented method.

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“…As we know, fractional differential equations are highly effective mathematical tools to describe the complex behaviors and phenomena of memory processes [1,10,15,35], it also can effectively characterize the ubiquitous power-law phenomena [36]. Many theoretical analysis and numerical methods are developed for fractional differential equations, see the literatures [9,10,12,[16][17][18][19][20][21][22][25][26][27][28][29][30][31][32][33][34][35] and the references therein. On the other hand, stochastic perturbations are coming from many natural sources in the practically physical system, they can not be ignored and the presence of noises might give rise to some statistical features and important phenomena, then the stochastic differential equations are produced, which are more realistic mathematical model of the real-world situations [8].…”

Section: Introductionmentioning

confidence: 99%

A Galerkin finite element method for time-fractional stochastic heat equation

Zou

2018

Computers & Mathematics with Applications

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In this study, a Galerkin finite element method is presented for time-fractional stochastic heat equation driven by multiplicative noise, which arises from the consideration of heat transport in porous media with thermal memory with random effects. The spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained. Numerical techniques are developed by the standard Galerkin finite element method in spatial direction, and Gorenflo-Mainardi-Moretti-Paradisi scheme is applied in temporal direction. The convergence error estimates for both semi-discrete and fully discrete schemes are established. Finally, numerical example is provided to verify the theoretical results.

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Numerical solution of the time fractional reaction–diffusion equation with a moving boundary

Cited by 28 publications

References 33 publications

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

A simple numerical method for two‐dimensional nonlinear fractional anomalous sub‐diffusion equations

A Galerkin finite element method for time-fractional stochastic heat equation

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