Numerical analysis of multi-term time-fractional nonlinear subdiffusion equations with time delay: What could possibly go wrong?

Zaky, Mahmoud A.; Hendy, Ahmed S.; Alikhanov, Anatoly A.; Пименов, В. Г.

doi:10.1016/j.cnsns.2020.105672

Cited by 27 publications

(16 citation statements)

References 29 publications

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“…Now, we are ready to state and prove a proper discrete fractional Grönwall inequality for multi-term time-fractional difference operators of formula (2). The proof of the next lemma follows the methodology used to prove the main Lemma 2.6 in [26].…”

Section: Discrete Grönwall Inequalitymentioning

confidence: 99%

“…More recently, and as it is not possible to trivially generalize the discrete L 2 − 1 σ Grönwall inequality to deal with multi-term time fractional differential operators of Caputo type. An analysis was done in [26] to construct an appropriate Grönwall inequality for a sequence satisfied…”

Section: Introductionmentioning

confidence: 99%

“…Inspiring by the nontriviality of extending the discrete Grönwall inequality proposed in [11] and the recent work [26], the main purpose of that work is to state and prove a discrete Grönwall inequality to be applied for the analysis of a constructed difference scheme for the multi-term time-fractional reaction-diffusion problem with nonlinear delay and space fractional derivatives of Riemann-Liouville type. It should be noticed that the proof of the discrete Grönwall inequality proposed here can't be deduced from the previously mentioned approaches as we use here Karatay approximation formula [16] for the discretization of time Caputo fractional derivative of multi-term type.…”

Section: Introductionmentioning

confidence: 99%

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A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

Hendy

Zaky

Staelen

2021

Applied Numerical Mathematics

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Section: Discrete Grönwall Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

Hendy

Zaky

Staelen

2021

Applied Numerical Mathematics

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“…There exists a wide variety of numerical methods which deal with space and/or fractional differential equations [24][25][26][27][28][29][30][31]. The coupled space fractional Ginzburg-Landau system was numerically investigated in [32].…”

Section: Introductionmentioning

confidence: 99%

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

2021

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A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

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“…The analytical solution of this problem is only gained for the linear case when the coefficients constants. Recently, the authors of [32][33][34] constructed numerical techniques for this problem with fourth-order fractional diffusion wave equations and some modifications have been considered in [35]. In [36], a Newtonlinearized compact ADI scheme for Riesz space fractional nonlinear reaction-diffusion equations was constructed and analyzed.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

et al. 2021

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The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2−1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time.

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Numerical analysis of multi-term time-fractional nonlinear subdiffusion equations with time delay: What could possibly go wrong?

Cited by 27 publications

References 29 publications

A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

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