An Explicit Numerical Method for the Fractional Cable Equation

Quintana-Murillo, J.; Yuste, S. B.

doi:10.1155/2011/231920

Cited by 18 publications

(14 citation statements)

References 45 publications

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“…Finally, we list the convergence result. (17), together with (18) and (19) is suitably smooth, and that {u k i,j |0 ≤ i ≤ M 1 , 0 ≤ j ≤ M 2 , 0 ≤ k ≤ N} is the solution of the finite difference scheme (21), together with (22) and (23). Let e k i,j = u(x i , y j , t k ) − u k i,j , then…”

Section: Solvability Stability and Convergence Analysismentioning

confidence: 99%

“…Hu and Zhang proposed two implicit compact difference schemes, where the first scheme was proved to be stable and convergent with order O(τ + h 4 ) by the energy method [9]. In [22], Quintana-Murillo and Yuste constructed an explicit numerical scheme for fractional Cable equation which includes two temporal Riemann-Liouville derivatives, where they showed the stability and convergence conditions by using the Von Neumann method. Zhuang et al [33] considered the one-dimensional time fractional Cable equation by using the Galerkin finite element method, in which the proposed method was based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space.…”

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confidence: 99%

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A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

Ding

2017

IMA Journal of Applied Mathematics

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Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}\; (\text{or}\; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank–Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $\varpi_{\ell}^{(\alpha)}$$(\ell=0,1,\ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|<1. Next, applying the formula to the time fractional Cable equations with Riemann–Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

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Section: Solvability Stability and Convergence Analysismentioning

confidence: 99%

mentioning

confidence: 99%

A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

Ding

2017

IMA Journal of Applied Mathematics

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“…with the following estimate (40) where ℏ is coarse grid step length H or fine grid size h and the norms are defined by w l = 0≤|θ|≤l Ω |D θ w| 2 dx 1 2 with the polynomial's degree l.…”

Section: Error Analysis Based On Two-grid Algorithmmentioning

confidence: 99%

A two-grid finite element approximation for a nonlinear time-fractional Cable equation

Liu

et al. 2016

Nonlinear Dyn

108

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In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Grünwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved; Second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time is proved in detail, which is higher than the L1-approximation result with O(τ 2−α + τ 2−β ). Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU-time is compared to verify our theoretical results.

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“…Let us consider the initial-boundary value problem of the fractional Cable equation which is usually written in the following way (see [11][12][13][14][15]20] and the reference cited therein): In this section, we will use the FWA-FDM to obtain the discretization finite difference formula of the Cable Eq. (3).…”

Section: Approximate Formula For Fractional Derivativementioning

confidence: 99%

Cluster computing for the large scale discrete fractional Cable equation

Sweilam

Moharram

Moniem

2015

Egyptian Informatics Journal

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This paper presents a numerical simulation technique for the fractional Cable equation in large scale domain. Special attention is given to the parallel execution of the fractional weighted average finite difference method (FWA-FDM) on distributed system with explicit message passing, where the fractional derivative is defined in Riemann-Liouville sense. The resultant huge system of equations is studied using precondition conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiments using different number of the Linux PC cluster nodes are studied. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC.Ó 2015 Production and hosting by Elsevier B.V. on behalf

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An Explicit Numerical Method for the Fractional Cable Equation

Cited by 18 publications

References 45 publications

A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

A two-grid finite element approximation for a nonlinear time-fractional Cable equation

Cluster computing for the large scale discrete fractional Cable equation

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