An inverse problem for a one-dimensional time-fractional diffusion problem

We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented.

[1]  Victor Isakov,et al.  Some inverse problems for the diffusion equation , 1999 .

[2]  William Rundell,et al.  The determination of a coefficient in a parabolic equation from input sources , 1994 .

[3]  I. Podlubny.  Fractional differential equations , 1998 .

[4]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[5]  Benny Y. C. Hon,et al.  Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem , 2011, Adv. Comput. Math..

[6]  Xiang Xu,et al.  Carleman estimate for a fractional diffusion equation with half order and application , 2011 .

[7]  Ying Zhang,et al.  Inverse source problem for a fractional diffusion equation , 2011 .

[8]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[9]  Masahiro Yamamoto,et al.  Tikhonov Regularization by a Reproducing Kernel Hilbert Space for the Cauchy Problem for an Elliptic Equation , 2008, SIAM J. Sci. Comput..

[10]  Masahiro Yamamoto,et al.  Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems , 2011 .

[11]  Guanhua Huang,et al.  Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk. , 2006, Journal of contaminant hydrology.

[12]  Wei Jiang,et al.  Approximate solution of the fractional advection-dispersion equation , 2010, Comput. Phys. Commun..

[13]  Masahiro Yamamoto,et al.  Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation , 2009 .

[14]  W. Rundell.  An inverse problem for a parabolic partial differential equation , 1983 .

[15]  Naomichi Hatano,et al.  Dispersive transport of ions in column experiments: An explanation of long‐tailed profiles , 1998 .

[16]  T. Wei,et al.  A New Regularization Method for the Time Fractional Inverse Advection-Dispersion Problem , 2011, SIAM J. Numer. Anal..

[17]  T. Wei,et al.  Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation , 2010, J. Comput. Appl. Math..

[18]  Yury F. Luchko.  Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation , 2010, Comput. Math. Appl..

[19]  William Rundell,et al.  An inverse Sturm-Liouville problem with a fractional derivative , 2012, J. Comput. Phys..

[20]  Ralf Metzler,et al.  Boundary value problems for fractional diffusion equations , 2000 .

[21]  L. Gelhar,et al.  Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis , 1992 .

[22]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[23]  M. Khader.  On the numerical solutions for the fractional diffusion equation , 2011 .

[24]  Massimiliano Giona,et al.  Fractional diffusion equation for transport phenomena in random media , 1992 .

[25]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[26]  H. M. Selim,et al.  Application of the Fractional Advection‐Dispersion Equation in Porous Media , 2003 .

[27]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[28]  Anatoly N. Kochubei,et al.  Cauchy problem for fractional diffusion equations , 2003 .

[29]  Khosrow Chadan,et al.  An Introduction to Inverse Scattering and Inverse Spectral Problems , 1987 .

[30]  Mark M. Meerschaert,et al.  A second-order accurate numerical method for the two-dimensional fractional diffusion equation , 2007, J. Comput. Phys..

[31]  M. Dentz,et al.  Modeling non‐Fickian transport in geological formations as a continuous time random walk , 2006 .

[32]  Masahiro Yamamoto,et al.  Inverse source problem with a finaloverdetermination for a fractional diffusionequation , 2011 .

[33]  Masahiro Yamamoto,et al.  A backward problem for the time-fractional diffusion equation , 2010 .

[34]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[35]  William Rundell,et al.  The determination of an unknown boundary condition in a fractional diffusion equation , 2013 .