Recovering a time-dependent potential function in a time fractional diffusion equation by using a nonlinear condition

References

• Slodicka M, Van Keer R. Determination of a robin coefficient in semilinear parabolic problems by means of boundary measurements. Inverse Probl. 2002;18:139–152. doi: 10.1088/0266-5611/18/1/310
   Google Scholar
• Kubica A, Yamamoto M. Initial-booundary value problems for fractional diffusion equations with time-dependent coefficients. Fract Calc Appl Anal. 2018;21(2):276–311. doi: 10.1515/fca-2018-0018
   Google Scholar
• Li Z, Yamamoto M. Unique continuation principle for the one dimensional time fractional diffusion equation. Fract Calcul Appl Anal. 2019;22(3):644–657. doi: 10.1515/fca-2019-0036
   Google Scholar
• McLean W, Mustapha K, Ali R, et al. Well-posedness of time-fractional, advection-diffusion-reaction equations. Fract Calc Appl Anal. 2019;22(4):918–994. doi: 10.1515/fca-2019-0050
   Google Scholar
• Gorenflo R, Luchko R, Yamamoto M. Time fractional diffusion equation in the fractional Sobolev spaces. Fract Calc Appl Anal. 2015;18:799–820. doi: 10.1515/fca-2015-0048
   Google Scholar
• Liu Y, Rundell W, Yamamoto M. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fract Calc Appl Anal. 2016;19:888–906.
   Google Scholar
• Luchko Y. Maximum principle for the generalized time fractional diffusion equation. J Math Anal Appl. 2009;351:218–223. doi: 10.1016/j.jmaa.2008.10.018
   Google Scholar
• 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. doi: 10.1016/j.jmaa.2011.04.058
   Google Scholar
• Zhang Z. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Probl Imaging. 2017;11(5):875–900. doi: 10.3934/ipi.2017041
   Google Scholar
• Zhang ZD. An undetermined coefficient problem for a fractional diffusion equation. Inverse Probl. 2016;32:015011.
   Google Scholar
• Fujishiro K, Kian Y. Determination of time dependent factors of coefficients in fractional diffusion equations. Math Control Relat Fields. 2016;6(2):251–269. doi: 10.3934/mcrf.2016003
   Google Scholar
• Sun LL, Zhang Y, Wei T. Recivering the time dependent potential function in a multi-term time fractional diffusion equation. Appl Numer Math. 2019;135:228–245. doi: 10.1016/j.apnum.2018.09.001
   Google Scholar
• Jin B, Rundell W. A tutorial on inverse problem for anomalous diffusion processes. Iverse Probl. 2015;31:035003.
   Google Scholar
• Jin BT, Rundell W. An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Probl. 2012;28:075010.
   Google Scholar
• Miller L, Yamamoto M. Coefficient inverse problem for a fractional diffusion equation. Inverse Probl. 2013;29(7):0750013. doi: 10.1088/0266-5611/29/7/075013
   Google Scholar
• Sun L, Wei T. Identification of zero order coefficient in a time fractional diffusion equation. Appl Numer Math. 2017;111:160–180. doi: 10.1016/j.apnum.2016.09.005
   Google Scholar
• Tuan VK. Inverse problem for fractional diffusion equation. Fract Calc Appl Anal. 2011;14(1):31–55. doi: 10.2478/s13540-011-0004-x
   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(10):105010. doi: 10.1088/0266-5611/28/10/105010
   Google Scholar
• Zhang Z, Zhou Z. Recovering the potential term in a fractional diffusion equation. IMA J Appl Math. 2017;82:579–600. doi: 10.1093/imamat/hxx004
   Google Scholar
• Li Z, Yamamoto M. Inverse problems of determining coefficients of the fractional partial differential equations. In: Kochubei A, Luchko Y, editors, Handbook of fractional calculus with applications. Vol. 2; 2019. p. 443–464.
   Google Scholar
• Li Z, Yamamoto M. Inverse problems of determining sources of the fractional partial differential equations. Kochubei A, Luchko, Y, editors, Handbook of fractional calculus with applications. Vol. 2; 2019. p. 411–430.
   Google Scholar
• Liu Y, Li Z, Yamamoto M. Inverse problems of determining parameters of the fractional partial differential equations. Kochubei A, Luchko Y, editors, Handbook of fractional calculus with applications. Vol. 2; 2019. p. 431–442.
   Google Scholar
• Jiang D, Li Z, Liu Y, et al. Weak unique continuation and a related inverse source problem for time fractional diffusion equations. Inverse Probl. 2017;33(5):055013. doi: 10.1088/1361-6420/aa58d1
   Google Scholar
• Li Z, Imanuvilov OY, Yamamoto M. Uniqueness in inverse boundary value problems for fractional diffusion equations. Inverse Probl. 2016;32(1):015004. doi: 10.1088/0266-5611/32/1/015004
   Google Scholar
• Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations. Vol. 204. Amsterdam: Elsevier; 2006. 2453–2461.
   Google Scholar
• Alikhanov AA. Boundary value problems for the diffusion equation of the variable order in differential and difference settings. Appl Math Comput. 2012;219:3938–3946.
   Google Scholar
• Podlubny I. Fractional differential equations. San Diego: Acad Press. 1999.
   Google Scholar
• Brezis H. Functional analysis, sobolev spaces and partial differential equations. Universitext. 2012;88(2):117–126.
   Google Scholar
• Henry D. Geometric theory of semilinear parabolic equations. Berlin: Springer-Verlag; 1981. (Lecture Notes in Mathematics; 840).
   Google Scholar
• Pazy A. Semigroups of linear operators and applications to partial differential equations. Berlin,: Springer-Verlag; 1983.
   Google Scholar
• Fujiwara D. Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc Jpn Acad. 1967;43(2):82–86. doi: 10.3792/pja/1195521686
   Google Scholar
• Gorenflo R, Yamamoto M. Operator-theoretic treatment of linear abel integral equation of first kind. Jpn J Ind Appl Math. 1999;16(1):137–161. doi: 10.1007/BF03167528
   Google Scholar
• Courant R, Hilbert D. Methods of mathematical physics. Vol. 1, New York: Interscience; 1953.
   Google Scholar
• Royden HL. Real analysis. Vol. 71. Princeton: Princeton University Press; 1963. 402.
   Google Scholar
• Morozov V, Nashed Z, Aries A. Methods for solving incorrectly posed problems. New York: Springer-Verlag; 1984.
   Google Scholar
• Hanke M, Hansen P. Regularization methods for large-scale problems. Surv Math Ind. 1993;3(4):253–315.
   Google Scholar
• Murio DA. Implicit finite difference approximation for time fractional diffusion equations. Comput Math Appl. 2008;56(4):1138–1145. doi: 10.1016/j.camwa.2008.02.015
   Google Scholar