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

Abeer Aldoghaither; Taous-Meriem Laleg-Kirati; Da-Yan Liu

doi:10.1515/jiip-2015-0037

Artikel

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

Abeer Aldoghaither , Taous-Meriem Laleg-Kirati und Da-Yan Liu

Veröffentlicht/Copyright: 14. Mai 2016

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Journal of Inverse and Ill-posed Problems Band 25 Heft 2

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.

Keywords: Space fractional advection dispersion equation; inverse source problem; ill-posedness; Tikhonov

MSC 2010: 26A33; 34K29; 35R11; 65F22

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

{∂V(x,t:τ)∂⁡t-LV(x,t:τ)=0,(x,t)∈D×(0,∞),V(x;t=τ)=f⁢(x),V|∂⁡D=0,x∈D,

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

{∂⁡v⁢(x,t)∂⁡t-L⁢v⁢(x,t)=f⁢(x,t),(x,t)∈D×(0,∞),v|∂⁡D=0,v⁢(x,0)=0,x∈D,

has the form

v⁢(x,t)=∫0tV⁢(x,t;τ)⁢dτ.

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

We refer to [4]. We have

(B.1)G^αθ⁢(k,t)=e[i⁢ν⁢k-d⁢ψθα⁢(k)]⁢t=ei⁢ν⁢k⁢t⁢e-d⁢ψθα⁢(k)⁢t=P^11⁢(k;-ν⁢t)⁢P^αθ⁢(k;t⁢d)

and

P^αθ⁢(k;c)=e-c⁢ψθα⁢(k),c∈ℝ.

Using the following scale rule for the Fourier transform,

f(cx)↔ℱ|a|-1f^(kc),

we get

Pαθ⁢(x;c)=|c|-1⁢pαθ⁢(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;ν⁢t)⁢Pαθ⁢(k;t⁢d)⁢dk,

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

Acknowledgements

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

References

[1] Andrle M., Ben Belgacem F. and El Badia A., Identification of moving pointwise sources in an advection-dispersion-reaction equation, Inverse Problems 27 (2011), Article ID 025007. 10.1088/0266-5611/27/2/025007Suche in Google Scholar

[2] Chi G., Li G. and Jia X., Numerical inversions of a source term in the FADE with a Dirichlet boundary condition using final observations, Comput. Math. Appl. 62 (2011), 1619–1626. 10.1016/j.camwa.2011.02.029Suche in Google Scholar

[3] Furdui O., Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, Springer, New York, 2013. 10.1007/978-1-4614-6762-5Suche in Google Scholar

[4] Huang F. and Liu F., The fundamental solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Comput. 18 (2005), 339–350. 10.1007/BF02936577Suche in Google Scholar

[5] Jeffrey A., Applied Partial Differential Equations. An Introduction, Academic Press, San Diego, 2003. Suche in Google Scholar

[6] Kirsch A., An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 2011. 10.1007/978-1-4419-8474-6Suche in Google Scholar

[7] Meerschaert M. and Tadjeran C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), 65–77. 10.1016/j.cam.2004.01.033Suche in Google Scholar

[8] Meerschaert M. and Tadjeran C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), 80–90. 10.1016/j.apnum.2005.02.008Suche in Google Scholar

[9] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999. Suche in Google Scholar

[10] Qian Z., Optimal modified method for a fractional-diffusion inverse heat conduction problem, Inverse Probl. Sci. Eng. 18 (2010), 521–533. 10.1080/17415971003624348Suche in Google Scholar

[11] Salim T. and El-Kahlout A., Analytical solution of time-fractional advection dispersion equation, Appl. Appl. Math. 4 (2009), 176–188. Suche in Google Scholar

[12] Schumer R., Benson D., Meerschaert M. and Wheatcraft S., Eulerian derivation of the fractional advection-dispersion equation, J. Contaminant Hydrol. 48 (2001), 69–88. 10.1016/S0169-7722(00)00170-4Suche in Google Scholar

[13] Schumer R., Meerschaert M. and Baeumer B., Fractional advection-dispersion equations for modeling transport at the Earth surface, J. Geophys. Res. 114 (2009), 10.1029/2008JF001246. 10.1029/2008JF001246Suche in Google Scholar

[14] Wei H., Chen W., Sun H. and Li X., A coupled method for inverse source problem of spatial fractional anomalous diffusion equations, Inverse Probl. Sci. Eng. 18 (2010), 945–956. 10.1080/17415977.2010.492515Suche in Google Scholar

[15] Xiong X., Zhou Q. and Hon Y. C., An inverse problem for fractional diffusion equation in 2-dimensional case: Stability analysis and regularization, J. Math. Anal. Appl. 393 (2012), 185–199. 10.1016/j.jmaa.2012.03.013Suche in Google Scholar

[16] Zhang H., Liu F. and Anh V., Numerical approximation of Levy–Feller diffusion equation and its probability interpretation, J. Comput. Appl. Math. 206 (2007), 1098–1115. 10.1016/j.cam.2006.09.017Suche in Google Scholar

[17] Zheng G. H. and Wei T., Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation, J. Comput. Appl. Math. 233 (2010), 2631–2640. 10.1016/j.cam.2009.11.009Suche in Google Scholar

[18] Zheng G. H. and Wei T., Two regularization methods for solving a Riesz–Feller space-fractional backward diffusion problem, Inverse Problems 26 (2010), Article ID 115017. 10.1088/0266-5611/26/11/115017Suche in Google Scholar

Received: 2015-3-31

Revised: 2016-3-27

Accepted: 2016-3-30

Published Online: 2016-5-14

Published in Print: 2017-4-1

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/jiip-2015-0037

Schlagwörter für diesen Artikel

Space fractional advection dispersion equation; inverse source problem; ill-posedness; Tikhonov