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

  • Abeer Aldoghaither , Taous-Meriem Laleg-Kirati EMAIL logo and Da-Yan Liu


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.

Funding statement: Research reported in this publication is supported by the King Abdullah University of Science and Technology (KAUST).

A Duhamel’s principle

We refer to [5]. If V(x,t;τ) is a solution to the equation


where L is a linear differential operator involving no time derivatives, then the solution of


has the form


B Properties of the Green’s function Gαθ(,)

We refer to [4]. We have




Using the following scale rule for the Fourier transform,


we get


which is non-negative, since pαθ is a probability density function whose Fourier transform is p^αθ(k)=e-ψθα(k).

Therefore, the inverse Fourier transform of (B.1) is


which is real and normalized (see [4]).


The authors would like to express great appreciation to Professor Manuel Ortigueira and Professor William Rundell for their valuable suggestions and comments.


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Received: 2015-3-31
Revised: 2016-3-27
Accepted: 2016-3-30
Published Online: 2016-5-14
Published in Print: 2017-4-1

© 2017 by De Gruyter

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