Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

Chen, Chang-Ming; Liu, Fawang; Anh, Vo; Turner, Ian

doi:10.1016/j.amc.2010.12.049

Cited by 32 publications

(24 citation statements)

References 32 publications

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“…The exact solution is v(x, t) = e x t 2 . We compare our results with those obtained by Chen et al [43]. For various choices of γ (x, t), N , and M, the maximum absolute errors are listed in Table 1.…”

Section: Numerical Examplementioning

confidence: 67%

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Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection–diffusion equation

et al. 2018

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In this paper, we investigate numerical solution of the variable-order fractional Galilei advection-diffusion equation with a nonlinear source term. The suggested method is based on the shifted Legendre collocation procedure and a matrix form representation of variable-order Caputo fractional derivative. The main advantage of the proposed method is investigating a global approximation for the spatial and temporal discretizations. This method reduces the problem to a system of algebraic equations, which is easier to solve. The validity and effectiveness of the method are illustrated by an easy-to-follow example. IntroductionRecently, kinetic equations with fractional derivatives were recognized as a useful tool for description of anomalous diffusion phenomena. Examples include systems exhibiting underground water pollution, Hamiltonian chaos, disordered medium, dynamics of protein molecules, reactions in complex systems, motions under the influence of optical tweezers, and more; see reviews on fractional kinetics [1][2][3][4]. The kinetic equations with time-fractional derivative are used for description of subdiffusion processes, that is, those for which the mean-squared displacement grows in time slower than linearly [5]. Also, it describes slow relaxation processes that are characterized by stretched exponential or power-law response function [6]. It became clear that further theoretical investigations are required to incorporate adequate tools for description of more realistic random processes, which are described by a set of characteristic exponents and are therefore of multifractional type. An adequate kinetic description of these processes requires the use of generalized fractional kinetics based on the concept of variable-order fractional (V-OF) operators. This calculus was proposed in [7,8] and very recently was introduced in physics [9, 10].The V-OF operators are nonlocal with singular kernels, which makes the V-OF models complicated. Hence, solving V-OF models is also more complicated. Numerical computation of the V-OF operators is the key to understand the behavior and physical meaning of the V-OF models. Lin et al. [11] investigated the stability and convergence of an ex-

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Section: Numerical Examplementioning

confidence: 67%

“…For various choices of γ (x, t), N , and M, the maximum absolute errors are listed in Table 1. Meanwhile, the results of [43] are presented in the first column of this table. Obviously, our method is more accurate than the method proposed in [43].…”

Section: Numerical Examplementioning

confidence: 99%

Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection–diffusion equation

et al. 2018

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“…Moreover, Chen et al [39] developed a new numerical scheme with high spatial accuracy for the variableorder anomalous sub-diffusion equation. They also presented and developed some numerical algorithms to improve temporal accuracy for the variable-order Galilei invariant advection-diffusion equation with a nonlinear source term [40]. More recently, Zhao et al [41] proposed two second-order approximation formulae for the variable-order fractional time derivatives involved in anomalous diffusion and wave propagation.…”

Section: Introductionmentioning

confidence: 98%

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

208

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The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one-and twodimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.Keywords One-dimensional cable equation · Two-dimensional variable-order nonlinear cable equation · Collocation method · Jacobi polynomials · Operational matrix of fractional derivative · Variable-order derivative

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“…Chen et al [30] developed numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. Chen et al [31] also presented numerical simulation for the variable-order Galilei invariant advection-diffusion equation with a nonlinear source term.…”

Section: Introductionmentioning

confidence: 97%

Numerical analysis for a variable-order nonlinear cable equation

Chen

Liu

Burrage

2011

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.

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Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

Cited by 32 publications

References 32 publications

Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection–diffusion equation

Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection–diffusion equation

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Numerical analysis for a variable-order nonlinear cable equation

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