Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator

In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite domain with variable coefficients. Fractional Advection- Dispersion equation as a model for transporting heterogeneous subsurface media as one approach to the modeling of the generally non-Fickian behavior of transport. We use a semi-analytical method as Reproducing kernel Method to solve the Space-Time Fractional Advection-Dispersion equation so that we can get better approximate solutions than the methods with which this problem has been solved. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of the reproducing kernel method, which is very time-consuming. So, we introduce the Improved Reproducing Kernel Method, which is a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, the present method increases the accuracy of approximate solutions.

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Homotopy perturbation method for solving the space–time fractional advection–dispersion equation

Advances in Water Resources ◽

10.1016/j.advwatres.2009.09.003 ◽

2009 ◽

Vol 32 (12) ◽

pp. 1711-1716 ◽

Cited By ~ 78

Author(s):

Ahmet Yıldırım ◽

Hüseyin Koçak

Keyword(s):

Perturbation Method ◽

Dispersion Equation ◽

Homotopy Perturbation Method ◽

Space Time ◽

Homotopy Perturbation ◽

Advection Dispersion

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An analytic algorithm for the space–time fractional advection–dispersion equation

Computer Physics Communications ◽

10.1016/j.cpc.2011.01.015 ◽

2011 ◽

Vol 182 (5) ◽

pp. 1134-1144 ◽

Cited By ~ 47

Author(s):

Ram K. Pandey ◽

Om.P. Singh ◽

Vipul K. Baranwal

Keyword(s):

Dispersion Equation ◽

Space Time ◽

Advection Dispersion

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Direct and inverse source problems for a space fractional advection dispersion equation

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2015-0037 ◽

2017 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Abeer Aldoghaither ◽

Taous-Meriem Laleg-Kirati ◽

Da-Yan Liu

Keyword(s):

Inverse Problem ◽

Dispersion Equation ◽

Direct Problem ◽

Source Term ◽

Analytic Solution ◽

Finite Domain ◽

Inverse Source Problem ◽

Direct And Inverse Problems ◽

Advection Dispersion ◽

Inverse Source Problems

Abstract In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.

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