Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation

Bu, Weiping; Zhao, Yanxing; Shen, Chen

doi:10.1016/j.amc.2021.125985

Cited by 4 publications

(2 citation statements)

References 37 publications

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“…The main aim of Feischl et al (2017) was to analyze a multilevel diagonal additive Schwarz preconditioner for the adaptive coupling of FEM and BEM for a linear 2D Laplace transmission problem. Ren et al (2022) proposed a stochastic age-structured HIV/AIDS model with nonlinear incidence rates.A numerical approximation combining finite difference method in time and finite element method in space is proposed in Bu et al (2021) to solve the two-dimensional time-space fractional Bloch-Torrey equation on rectangular computational domain. A numerical simulation is proposed in Xu et al (2021) based on the finite difference scheme and an unstructured-mesh Galerkin FEM based on the matrix transfer technique (MTT) for solving multi-term time and space fractional Bloch-Torrey equations.…”

Section: Introductionmentioning

confidence: 99%

“…Ren et al. (2022) proposed a stochastic age-structured HIV/AIDS model with nonlinear incidence rates.A numerical approximation combining finite difference method in time and finite element method in space is proposed in Bu et al. (2021) to solve the two-dimensional time-space fractional Bloch–Torrey equation on rectangular computational domain.…”

Section: Introductionmentioning

confidence: 99%

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A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

Abbaszadeh,

Bagheri Salec,

Kamel Abd Al-Khafaji

2023

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PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

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