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

## 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.

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 . If V(x,t;τ) is a solution to the equation

{V(x,t:τ)t-LV(x,t:τ)=0,(x,t)D×(0,),V(x;t=τ)=f(x),V|D=0,xD,

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

{v(x,t)t-Lv(x,t)=f(x,t),(x,t)D×(0,),v|D=0,v(x,0)=0,xD,

has the form

v(x,t)=0tV(x,t;τ)dτ.

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

We refer to . We have

(B.1)G^αθ(k,t)=e[iνk-dψθα(k)]t=eiνkte-dψθα(k)t=P^11(k;-νt)P^αθ(k;td)

and

P^αθ(k;c)=e-cψθα(k),c.

Using the following scale rule for the Fourier transform,

f(cx)|a|-1f^(kc),

we get

Pαθ(x;c)=|c|-1pαθ(xc1α),

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

(B.2)Gαθ(x,t)=-+P11(x-k;νt)Pαθ(k;td)dk,

which is real and normalized (see ).

## Acknowledgements

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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