Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates

References

• Caputo M. Linear model of dissipation whose Q is almost frequency independent-II. Geophys J R Astr Soc. 1967;13:529–539.
   Web of Science ®Google Scholar
• Xu X, Cheng J, Yamamoto M. Carleman estimate for a fractional diffusion equation with half order and application. Appl Anal. 2011;90:1355–1371.
   Web of Science ®Google Scholar
• Machida M. The time-fractional radiative transport equation – Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion. J Math Phys. 2017;58:013301.
   Web of Science ®Google Scholar
• Cheng J, Nakagawa J, Yamamoto M, et al. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl. 2009;25:115002.
   Web of Science ®Google Scholar
• Yamamoto M, Zhang Y. Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate. Inverse Probl. 2012;28:105010.
   Web of Science ®Google Scholar
• Jin B, Rundell W. An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Probl. 2012;28:075010.
   Web of Science ®Google Scholar
• Hatano Y, Nakagawa J, Wang S, et al. Determination of order in fractional diffusion equation. J Math-for-Industry. 2013;5:51–57.
   Google Scholar
• Liu JJ, Yamamoto M, Yan L. On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process. Appl Num Math. 2015;87:1–19.
   Web of Science ®Google Scholar
• Lin C-L, Nakamura G. Unique continuation property for anomalous slow diffusion equation. Comm Partial Diff Eqs. 2016;41:749–758.
   Web of Science ®Google Scholar
• Carleman T. Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark Mat Astr Fys. 1939;26B:1–9.
   Google Scholar
• Bukhgeim AL, Klibanov MV. Global uniqueness of a class of multidimensional inverse problems. Soviet Math Dokl. 1981;24:244–7.
   Google Scholar
• Klibanov MV. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J Inverse Ill-Posed Probl. 2013;21:477–560.
   Web of Science ®Google Scholar
• Imanuvilov OY, Yamamoto M. Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Probl. 1998;14:1229–1245.
   Web of Science ®Google Scholar
• Yamamoto M. Carleman estimates for parabolic equations and applications. Inverse Probl. 2009;25:123013.
   Web of Science ®Google Scholar
• Klibanov MV, Pamyatnykh SE. Lipschitz stability of a non-standard problem for the non-stationary transport equation via a Carleman estimate. Inverse Probl. 2006;22:881–890.
   Web of Science ®Google Scholar
• Gaitan P, Ouzzane H. Inverse problem for a free transport equation using Carleman estimates. Appl Anal. 2013;93:1073–1086.
   Web of Science ®Google Scholar
• Machida M, Yamamoto M. Global Lipschitz stability in determining coefficients of the radiative transport equation. Inverse Probl. 2014;30:035010.
   Web of Science ®Google Scholar
• Acosta S. Recovery of the absorption coefficient in radiative transport from a single measurement. Inv Prob Imag. 2015;9:289–300.
   Web of Science ®Google Scholar
• Klibanov MV, Yamamoto M. Exact controllability for the time dependent transport equation. SIAM J Control Optim. 2007;46:2071–2195.
   Web of Science ®Google Scholar
• Prilepko AI, Ivankov AL. Inverse problems for the time-dependent transport equation. Sovviet Math Dokl. 1984;29:559–564.
   Google Scholar
• Bal G. Inverse transport theory and applications. Inverse Probl. 2009;25:053001.
   Web of Science ®Google Scholar
• Yuan G, Yamamoto M. Lipschitz stability in the determination of the principal part of a parabolic equation. ESAIM Control Optim Calc Var. 2009;15:525–554.
   Web of Science ®Google Scholar
• Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339:1–77.
   Web of Science ®Google Scholar
• Sokolov IM, Klafter J, Blumen A. Fractional kinetics. Phys Today. 2002;55:48–54.
   Web of Science ®Google Scholar
• Adams EE, Gelhar LW. Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis. Water Res Res. 1992;28:3293–3307.
   Web of Science ®Google Scholar
• Hatano Y, Hatano N. Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resour Res. 1998;34:1027–1033.
   Web of Science ®Google Scholar
• Yamakawa M. A study on the dependence of CTRW parameters on particle sizes in porus media (Japanese) [master thesis]. Tsukuba (Japan): Department of Risk Engineering, University of Tsukuba; 2012.
   Google Scholar
• Williams MMR. Stochastic problems in the transport of radioactive nuclides in fractured rock. Nucl Sci Eng. 1992;112:215–230.
   Web of Science ®Google Scholar
• Williams MMR. Radionuclide transport in fractured rock a new model: application and discussion. Ann Nucl Energy. 1993;20:279–297.
   Web of Science ®Google Scholar
• Arridge SR, Schotland JC. Optical tomography: forward and inverse problems. Inverse Probl. 2009;25:123010.
   Web of Science ®Google Scholar
• Èmanuilov OY. Controllability of parabolic equations. Sbornik Math. 1995;186:879–900.
   Web of Science ®Google Scholar
• Fursikov AV, Imanuvilov OY. Controllability of Evolution Equations. (Lecture Notes Series vol. 34). Seoul (Korea): Seoul National University; 1996.
   Google Scholar