Time fractional advection-dispersion equation

Liu, Fawang; Anh, Vo; Turner, Ian; Zhuang, Pinghui

doi:10.1007/bf02936089

Cited by 214 publications

(112 citation statements)

References 16 publications

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“…This topic has received a great 1 Email address :3425linran@sohu.com 2 Corresponding author. Email address :fwliu@xmu.edu.cn,f.liu@qut.edu.au deal of attention in the last decade [7,15,[22][23][24], especially in the fields of viscoelastic materials [1,10,11,27], electrochemical processes [8], dielectric polarization [28], colored noise [29], anomalous diffusion, signal processing [21], control theory [24], advection and dispersion of solutes in natural porous or fractured media [2,3] and chaos [20]. Djrbasjan et al [6] considered the Cauchy problem with multi-term fractional derivatives, and proved the Cauchy problem has a unique solution.…”

Section: Introductionmentioning

confidence: 99%

Fractional high order methods for the nonlinear fractional ordinary differential equation

Lin

Liu

2007

Nonlinear Analysis: Theory, Methods & Applications

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In this paper, we consider the nonlinear fractional order ordinary differential equa-Fractional order linear multiple step methods are introduced. The high order (2-6) approximations of the fractional order ordinary differential equation with initial value are proposed. The consistence, convergence and stability of the fractional high order methods are proved. Finally, some numerical examples are provided to show that the fractional high order methods for solving the fractional order nonlinear ordinary differential equation are computationally efficient solution methods.

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Section: Introductionmentioning

confidence: 99%

Fractional high order methods for the nonlinear fractional ordinary differential equation

Lin

Liu

2007

Nonlinear Analysis: Theory, Methods & Applications

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“…Sousa [23] proposed an approximation of the Caputo fractional derivative of order 1 < α 2. Liu et al [18] developed a numerical method for solving the time fractional advectiondispersion equation. Zhang and Han [30] proposed a quasi-wavelet method for time-dependent fractional partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of time-fractional hyperbolic partial differential equations

Arshed¹

2017

J. Math. Computer Sci.

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In this study, a numerical scheme is developed for the solution of time-fractional hyperbolic partial differential equation. In the proposed scheme, cubic B-spline collocation is used for space discretization and time discretization is obtained by using central difference formula. Caputo fractional derivative is used for time-fractional derivative. The stability and convergence of the developed scheme, have also been proved. The numerical examples support the theoretical results.

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“…In the case of the fractional advection-diffusion equation, as a rule, different numerical methods have been used to find the solution: the implicit difference method based on the shifted Grünwald approximation [9] and the explicit difference method [10], the fractional variational method [11], the finite volume method [12], etc. In [13,14] the solution to one-dimensional time-fractional advection-diffusion equation was obtained in terms of the H-function.…”

Section: Introductionmentioning

confidence: 99%