A note on the finite element method for the space-fractional advection diffusion equation

Zheng, Yuanyuan; Li, Changpin; Zhao, Zhengang

doi:10.1016/j.camwa.2009.08.071

Cited by 155 publications

(43 citation statements)

References 10 publications

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“…In the past several years, contributions on the research of potentially useful tools for solving FDEs, and various other fractional equations are far too many. For more details, see the work on variational iteration [9], integral transform [10], multigrid [11] and finite element methods [12]. Besides, the Finite Difference Method (FDM) is a very popular method for solving fractional diffusion equations, and nowadays, the initial value problem and boundary value problem for the fractional partial differential equations have been studied extensively [13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Numerical and analytical solutions of new generalized fractional diffusion equation

Agrawal

2013

Computers & Mathematics with Applications

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a b s t r a c tOur current paper is devoted to studying the numerical and analytical solutions for a class of Generalized Fractional Diffusion Equations (GFDEs) with new Generalized TimeFractional Derivative (GTFD). The GTFD we propose here is defined in the Caputo sense. We consider the GFDEs on a bounded domain. The numerical solutions are obtained by using the Finite Difference Method (FDM) of full discretization. The stability of FDM is discussed and the order of convergence is evaluated numerically. Numerical experiments are given, which illustrate that the FDM is simple and effective for solving the GFDEs with different coefficients and source functions. An interesting phenomenon is that we can observe the period-like solution in GFDEs with a particular positive periodic weight function. Using the method of separation of variables, we convert the homogeneous GFDE into two ordinary differential equations, and solve them via the help of solutions of the initial value problem with Caputo derivative. In the analytical solution, we observe that the weight function in the denominator, and scale function mapping the response domain differently. Since the derivative considered in this article is new, many existing results of FDEs are generalized.

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Section: Introductionmentioning

confidence: 99%

Numerical and analytical solutions of new generalized fractional diffusion equation

Agrawal

2013

Computers & Mathematics with Applications

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“…We now restrict the fractional derivative spaces to X ¼ ða; bÞ be a bounded open subinterval of R. By the similar idea of [21][22][23][24][25][26], we have the following definitions and lemmas:…”

Section: Fractional Derivative Spacesmentioning

confidence: 99%

Fractional difference/finite element approximations for the time–space fractional telegraph equation

Zhao

2012

Applied Mathematics and Computation

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“…It is important to solve this equation for a better understanding of advection and diffusion phenomena in a fractional setting, and for this purpose, numerical and approximate analytical methods are usually required. The finite element method was constructed for the space fractional advection-diffusion equation by Zheng et al [8]. Wang and Wang [9] developed a fast characteristic finite difference scheme for space fractional advection-diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

et al. 2018

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In this paper, we investigate a fully implicit finite difference scheme for solving the time fractional advection-diffusion equation. The time fractional derivative is estimated using Caputo's formulation, and the spatial derivatives are discretized using extended cubic B-spline functions. The convergence and stability of the fully implicit scheme are analyzed. Numerical experiments conducted indicate that the scheme is feasible and accurate.

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A note on the finite element method for the space-fractional advection diffusion equation

Cited by 155 publications

References 10 publications

Numerical and analytical solutions of new generalized fractional diffusion equation

Numerical and analytical solutions of new generalized fractional diffusion equation

Fractional difference/finite element approximations for the time–space fractional telegraph equation

A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

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