Finite difference approximations for the fractional advection–diffusion equation

Su, Lijuan; Wang, Wenqia; Yang, Zhichao

doi:10.1016/j.physleta.2009.10.004

Cited by 57 publications

(41 citation statements)

References 18 publications

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“…In the riche area of numerical methods of FPDEs, little work has been done by spectral methods when compared to finite difference and finite element methods. This partially motivates our interest in such methods [39][40][41].…”

Section: Introductionmentioning

confidence: 90%

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Doha

Bhrawy²,

Ezz-Eldien

2013

Open Physics

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Abstract:In this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the timedependent fractional diffusion equations.PACS (2008)

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Section: Introductionmentioning

confidence: 90%

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Doha

Bhrawy²,

Ezz-Eldien

2013

Open Physics

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“…For example, some modern of epidemics or Avian influenza can spread around the world in a few weeks and seem to follow a non-Gaussian, scale-free dynamics [1]. In hydrology system, such equations have been shown to govern pulse propagation along orthogonal polarization axes in Lévy motion and in wavelength-division-multiplexed systems [2,23]. The concept of the fractional equations can define as a model of beam propagation or water wave interactions.…”

Section: Introductionmentioning

confidence: 99%

“…As for the numerical methods for this type of problems, the finite difference method has been considered by several authors in various settings, see e.g. [23].…”

Section: Introductionmentioning

confidence: 99%

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Convergence analysis and approximation solution for the coupled fractional convection-diffusion equations

Rostamy¹,

Mottaghi²

2016

J. Math. Computer Sci.

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By using maximum principle approach, the existence, uniqueness and stability of a coupled fractional partial differential equations is studied. A new fractional characteristic finite difference scheme is given for solving the coupled system. This method is based on shifted Grünwald approximation and Diethelm's algorithm. We obtain the optimal convergence rate for this scheme and drive the stability estimates. The results are justified by implementing an example of the fractional order time and space dependent in concept of the complex Lévy motion. Also, the numerical results are examined for disinfection and sterilization of tetanus.

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“…Some difference schemes for the space-fractional heat equations are presented in [1,31,32]. It is easy to apply the Crank-Nicholson difference schemes for the space fractional heat equations.…”

Section: Introductionmentioning

confidence: 99%

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

Karatay

Kale

Bayramoglu

2013

fcaa

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In this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2−α + h 2 ), conditionally. Numerical experiments are carried out to support the theoretical claims.MSC 2010 : 65M12, 65M06

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Finite difference approximations for the fractional advection–diffusion equation

Cited by 57 publications

References 18 publications

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Convergence analysis and approximation solution for the coupled fractional convection-diffusion equations

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

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