Finite difference method for Riesz space fractional diffusion equations with delay and a nonlinear source term

Yang, Shiliang

doi:10.22436/jnsa.011.01.03

Cited by 5 publications

(4 citation statements)

References 26 publications

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“…In this section, we recall the classical CNFD scheme for the equations (1.1), which is presented in [35]. Let N and M be two positive integers, τ = T/N be the time step-size and h = L/M be the spatial step-size.…”

Section: The Classical Cnfd Scheme For the Riesz Space Fodes With A Nmentioning

confidence: 99%

“…However, they usually have no analytic solution so that they mainly depend on numerical solutions (see, e.g., [12,14,16,26]). In recently, a classical CNFD scheme for the equations (1.1) has developed in [35], but it includes many degrees of freedom (i.e., unknowns). Thus, due to the truncated error amassing in the calculating process, it would appear floating point overflow after computing some steps so that it can't gain desired results.…”

Section: Introductionmentioning

confidence: 99%

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A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay

Cao¹,

Luo²

2018

J. Nonlinear Sci. Appl.

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This article mainly studies the order-reduction of the classical Crank-Nicolson finite difference (CNFD) scheme for the Riesz space fractional order differential equations (FODEs) with a nonlinear source function and delay on a bounded domain. For this reason, the classical CNFD scheme for the Riesz space FODE and the existence, stability, and convergence of the classical CNFD solutions are first recalled. And then, a reduced-order extrapolating CNFD (ROECNFD) scheme containing very few degrees of freedom but holding the fully second-order accuracy for the Riesz space FODEs is established by means of proper orthogonal decomposition and the existence, stability, and convergence of the ROECNFD solutions are discussed. Finally, some numerical experiments are presented to illustrate that the ROECNFD scheme is far superior to the classical CNFD one and to verify the correctness of theoretical results. This indicates that the ROECNFD scheme is very effective for solving the Riesz space FODEs with a nonlinear source function and delay.Keywords: Crank-Nicolson finite difference scheme, Riesz space fractional order differential equation, existence and stability as well as convergence, reduced-order extrapolating Crank-Nicolson finite difference scheme, proper orthogonal decomposition.

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Section: The Classical Cnfd Scheme For the Riesz Space Fodes With A Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay

Cao¹,

Luo²

2018

J. Nonlinear Sci. Appl.

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“…This equation was considered as the reaction‐diffusion equation with delay in 1D . For the delay field with fractional derivatives in 1D, we take K α =0 and the resulting equation is known as the Riesz space fractional diffusion equation with delay and nonlinear source term in 1D, see Reference . For the field without delay with fractional derivatives in 2D, we take K α =0 and this equation is known as the Riesz space fractional Fisher' equation in 2D, see Reference .…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of difference methods for two‐dimensional Riesz space fractional advection‐dispersion equations with delay

Heris

Javidi

Ahmad

2020

Comp and Math Methods

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In this article, the Riesz space fractional advection‐dispersion equations with delay in two‐dimensional (RFADED in 2D) are considered. The Riesz space fractional derivative is approximated with the aid of backward differential formulas method of second order and shifted Grünwald difference operators. We develop the Crank‐Nicolson scheme using the finite difference method for the RFADED in 2D and show that it is conditionally stable and convergent with the accuracy order O(κ2+h2+k2). Finally, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical method.

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“…For the delay field with fractional derivatives, we take K α = 0 in Eq. (1) and the resulting equation is known as the Riesz space fractional diffusion equation with delay and nonlinear source term, see Yang [Yang (2018)].…”

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay

Heris¹,

Javidi²,

Ahmad³

2019

Computer Modeling in Engineering &Amp; Sciences

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In this paper, we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay (RFADED). We utilize the fractional backward differential formulas method of second order (FBDF2) and weighted shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED. Firstly, the FBDF2 and the shifted Grünwald methods are introduced. Secondly, based on the FBDF2 method and the WSGD operators, the finite difference method is applied to the problem. We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(κ + h 2) and O(κ 2 + h 2) respectively. Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions. Finally, some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.

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Finite difference method for Riesz space fractional diffusion equations with delay and a nonlinear source term

Cited by 5 publications

References 26 publications

A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay

A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay

Stability and convergence of difference methods for two‐dimensional Riesz space fractional advection‐dispersion equations with delay

Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay

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