Local stability for an inverse coefficient problem of a fractional diffusion equation

Abstract

Time-fractional diffusion equations are of great interest and importance on describing the power law decay for diffusion in porous media. In this paper, to identify the diffusion rate, i.e., the heterogeneity of medium, the authors consider an inverse coefficient problem utilizing finite measurements and obtain a local Hölder type conditional stability based upon two Carleman estimates for the corresponding differential equations of integer orders.

Keywords

Carleman estimate / Conditional stability / Inverse coefficient problem / Fractional diffusion equation

References

[1] Bisquert J. Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination. Phys. Rev. E, 2005, 72: 011109

[2] Brunner H, Ling L, Yamamoto M. Numerical simulations of 2D fractional subdiffusion problems. J. Comp. Phys., 2010, 229: 6613-6622

[3] Bukhgeim A, Klibanov M. Global uniqueness of a class of multidimensional inverse problems. Soviet Math. Doklady, 1981, 24: 244-247

[4] Cheng J, Nakagawa J, Yamamoto M, Yamazaki T. Uniqueness in an inverse problem for a onedimensional fractional diffusion equation. Inverse Problems, 2009, 25: 115002

[5] Hatano Y, Hatano N. Dispersive transport of ions in column experiments: an explanation of longtailed profiles. Water Resour. Res., 1998, 34: 1027-1033

[6] Hilfer R. Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B, 2000, 104: 3914-3917

[7] Lin Y, Xu C. Finite difference/spectral approximation for the time-fractional diffusion equation. J. Comp. Phys., 2007, 225: 1533-1552

[8] Liu J, Yamamoto M. A backward problem for the time-fractional diffusion equation. Appl. Anal., 2010, 89: 1769-1788

[9] Luchko Y. Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl., 2009, 351: 218-223

[10] Ren C, Xu X, Lu S. Regularization by projection for a backward problem of the time-fractional diffusion equation. Journal of Inverse and Ill-Posed Problems, 2013

[11] Rundell W, Xu X, Zuo L. The determination of an unknown boundary condition in a fractional diffusion equation. Appl. Anal., 2013

[12] Sakamoto K, Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl., 2011, 382: 426-447

[13] Sun Z, Wu X. A full discrete difference scheme for a diffusion-wave system. Appl. Numer. Math., 2006, 56: 193-209

[14] Xu X, Cheng J, Yamamoto M. Carleman estimate for fractional diffusion equation with half order and applicatioin. Appl. Anal., 2011, 90: 1355-1371

[15] Yamamoto M. Carleman estimates for parabolic equations and applications. Inverse Problems, 2009, 25: 123013

[16] Yamamoto M, Zhang Y. Conditional stability of coefficient in a fractional diffusion equation by Carleman estimate. Inverse Problems, 2012, 28: 105010

[17] Zhang Y, Xu X. Inverse source problem for a fractional diffusion equation. Inverse Problems, 2011, 27: 035010