Second-Order Finite Difference/Spectral Element Formulation for Solving the Fractional Advection-Diffusion Equation

Abbaszadeh, Mostafa; Amjadian, Hanieh

doi:10.1007/s42967-020-00060-y

Cited by 15 publications

(9 citation statements)

References 54 publications

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“…Moreover, hp it‐7500 U CPU 2.90 with 12 GB of RAM is used to analyze the problems understudy while it is noted that the time of computational algorithm can be reduced by using the more efficient computer. Similarly, Figure 8 is plotted to show the algorithm time against the discretization of time‐domain via finite difference scheme (FDM) [36, 37]. It is noted form Figures 7 and 8 that the increase in the convergence control parameter took the same time for fractional and integer order values but time‐discretize parameter or meshing parameter N took less time for integer‐order values while consume more time for fractional derivative.…”

Section: Applications Of Proposed Methodsmentioning

confidence: 99%

A stable computational approach to analyze semi‐relativistic behavior of fractional evolutionary problems

Hamid

Usman

Wang

et al. 2020

Numerical Methods Partial

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The current work is related to the development of a hybrid semi‐spectral computational scheme based on multidimensional Chelyshkov polynomials (CPs). One dimensional traditional CPs have been efficiently converted to multidimensional Chelyshkov polynomials (MDCPs) and used to construct some novel operational matrices of fractional‐order derivatives. A method is developed via obtained polynomials as the Chelyshkov polynomial method (CPM) and further coupled with a finite difference scheme (FDM) and named as a semi‐spectral method (SSM). The method is used to evaluate the results of nonlinear evolutionary problems of fractional order where the space domain is collocated via CPM and the time variable is discretized through FDM. The stability analysis of the proposed scheme is proved to show its mathematical authenticity. Additionally, the reported study is evidence that the proposed methodology is reasonable in terms of accuracy, efficiency and low‐cost to tackle the nonlinear problem in higher dimensions while could be further modified for other class of dynamical problems.

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Section: Applications Of Proposed Methodsmentioning

confidence: 99%

A stable computational approach to analyze semi‐relativistic behavior of fractional evolutionary problems

Hamid

Usman

Wang

et al. 2020

Numerical Methods Partial

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“…Throughout the discussion, we let u, U C−N , U F EG and U M F EG denote the exact, C-N, FEG and MFEG solutions, respectively. In addition, the computational orders of the presented methods are calculated using the formula [47] C…”

Section: Numerical Experimentsmentioning

confidence: 99%

Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods

Salama

Ali

2022

Int. J. Appl. Comput. Math

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In this paper, we shall present the development of two explicit group schemes, namely, fractional explicit group (FEG) and modified fractional explicit group (MFEG) methods for solving the time fractional mobile/immobile equation in two space dimensions. The presented methods are formulated based on two Crank-Nicolson (C-N) finite difference schemes established at two different grid spacings. The stability and convergence of order O(τ 2−α + h 2 ) are rigorously proven using Fourier analysis. Several numerical experiments are conducted to verify the efficiency of the proposed methods. Meanwhile, numerical results show that the FEG and MFEG algorithms are able to reduce the computational times and iterations effectively while preserving good accuracy in comparison to the C-N finite difference method.

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“…A transport equation for confined structures has been used to calculate the ionic currents through various transmembrane proteins in Khodadadian and Heitzinger [24]. Also for more details see [25][26][27][37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

Integrated radial basis functions to simulate modified anomalous sub‐diffusion equation

Dehghan

Ebrahimijahan

Abbaszadeh

2022

Numerical Methods Partial

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This research work presents a newly truly meshless approach based upon the integrated radial basis functions (IRBFs) technique to study the fractional modified anomalous sub-diffusion equation. First, the temporal direction is discretized by a finite difference method with second-order accuracy. The stability analysis and convergence of the proposed time-discrete formulation are investigated, theoretically. Then, the spatial direction is approximated by the IRBFs methodology. In this approach, the largest-order derivative is approximated by a linear combination of RBFs and then the lower-order derivative and also the unknown function are constructed by repeated integration. According to this procedure, the existing derivatives in the main fractional PDE are approximated smoothly. Furthermore, to show the efficiency of the developed numerical procedure, we employed some non-rectangular computational domains to obtain the numerical results. On the other hand, the mentioned model is solved in one-two-, and three dimensional cases. The numerical results confirm the ability and efficiency of the new numerical method for solving time fractional PDEs on complex computational domains.

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Second-Order Finite Difference/Spectral Element Formulation for Solving the Fractional Advection-Diffusion Equation

Cited by 15 publications

References 54 publications

A stable computational approach to analyze semi‐relativistic behavior of fractional evolutionary problems

A stable computational approach to analyze semi‐relativistic behavior of fractional evolutionary problems

Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods

Integrated radial basis functions to simulate modified anomalous sub‐diffusion equation

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