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 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
We refer to . We have
Using the following scale rule for the Fourier transform,
which is non-negative, since is a probability density function whose Fourier transform is .
Therefore, the inverse Fourier transform of (B.1) is
which is real and normalized (see ).
The authors would like to express great appreciation to Professor Manuel Ortigueira and Professor William Rundell for their valuable suggestions and comments.
 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. 10.1088/0266-5611/27/2/025007Search in Google Scholar
 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.2011.02.029Search in Google Scholar
 Furdui O., Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, Springer, New York, 2013. 10.1007/978-1-4614-6762-5Search in Google Scholar
 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/BF02936577Search in Google Scholar
 Jeffrey A., Applied Partial Differential Equations. An Introduction, Academic Press, San Diego, 2003. Search in Google Scholar
 Kirsch A., An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 2011. 10.1007/978-1-4419-8474-6Search in Google Scholar
 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.2004.01.033Search in Google Scholar
 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.2005.02.008Search in Google Scholar
 Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999. Search in Google Scholar
 Qian Z., Optimal modified method for a fractional-diffusion inverse heat conduction problem, Inverse Probl. Sci. Eng. 18 (2010), 521–533. 10.1080/17415971003624348Search in Google Scholar
 Salim T. and El-Kahlout A., Analytical solution of time-fractional advection dispersion equation, Appl. Appl. Math. 4 (2009), 176–188. Search in Google Scholar
 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)00170-4Search in Google Scholar
 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. 10.1029/2008JF001246Search in Google Scholar
 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. 10.1080/17415977.2010.492515Search in Google Scholar
 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.2012.03.013Search in Google Scholar
 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.2006.09.017Search in Google Scholar
 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.2009.11.009Search in Google Scholar
 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. 10.1088/0266-5611/26/11/115017Search in Google Scholar
© 2017 by De Gruyter