Inverse Problems and Imaging (IPI)

Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation
Pages: 125 - 149, Issue 1, February 2017

doi:10.3934/ipi.2017007      Abstract        References        Full text (533.1K)           Related Articles

Jaan Janno - Tallinn University of Technology, Ehitajate tee 5, Tallinn 19086, Estonia (email)
Kairi Kasemets - Tallinn University of Technology, Ehitajate tee 5, Tallinn 19086, Estonia (email)

1 M. Al-Refai and Y. Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives, Appl. Math. Comput., 257 (2015), 40-51.       
2 E. Beretta and C. Cavaterra, Identifying a space-dependent coefficient in a reaction-diffusion equation, Inverse Problems and Imaging, 5 (2011), 285-296.       
3 H. Brunner , H. Han and D. Yin, The maximum principle for time-fractional diffusion equations and its application, Numer. Funct. Anal Optim., 36 (2015), 1307-1321.       
4 J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002, 16pp.       
5 K. M. Furati, O. S. Iyiola and M. Kirane, An inverse problem for a generalized fractional diffusion, Appl. Math. Comput., 249 (2014), 24-31.       
6 V. Gafiychuk, B. Datsko and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math., 220 (2008), 215-225.       
7 R. Gorenflo and F. Mainardi, Some recent advances in theory and simulation of fractional diffusion processes, J. Comput. Appl. Math., 229 (2009), 400-415.       
8 G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, Math. Zeitschrift, 27 (1928), 565-606.       
9 V. Isakov, Inverse parabolic problems with final overdetermination, Commun. Pure Appl. Math., 44 (1991), 185-209.       
10 V. Isakov, Inverse Problems for Partial Differential Equations, $2^{nd}$ edition, Springer, New York, 2006.       
11 J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electron. J. Diff. Eqns., 2016 (2016), 28pp.       
12 J. Janno and K. Kasemets, A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination, Inverse Problems and Imaging, 3 (2009), 17-41.       
13 B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012), 075010, 19pp.       
14 M. Kirane, A. S. Malik and M. A. Al-Gwaizb, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Meth. Appl. Sci., 36 (2013), 1056-1069.       
15 M. Krasnoschok and N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces, Nonlin. Stud., 20 (2013), 591-621.       
16 O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, Rhode Island, 1968.
17 Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548.       
18 Y. Luchko, W. Rundell, M. Yamamoto and L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Problems, 29 (2013), 065019, 16pp.       
19 A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.       
20 R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Computers Math. Appl., 59 (2010), 1586-1593.       
21 C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York, 1970.       
22 J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Berlin, 1993.       
23 S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, On the solutions of time-fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 3847-3854.       
24 K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields, 1 (2011), 509-518.       
25 K. Seki, M. Wojcik and M. Tachiya, Fractional reaction-diffusion equation, J. Chem. Phys., 119 (2003), 2165-2170.
26 H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc., 259 (1980), 299-310.       
27 V. E. Tarasov, Fractional Dynamics. Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, 2010.       
28 V. Turut and N. Güzel, Comparing numerical methods for solving time-fractional reaction-diffusion equations, Intern. Scholar. Res. Notices, 2012 (2012), Art. ID 737206, 28 pp.       
29 T. Wei and J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math., 78 (2014), 95-111.       
30 R. Zacher, Quasilinear Parabolic Problems with Nonlinear Boundary Conditions, Ph.D thesis, Martin-Luther-Universität Halle-Wittenberg, 2003. Available from: https://www.yumpu.com/en/document/view/4926858/quasilinear-parabolic-problems-with-nonlinear-boundary-conditions
31 R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.       
32 R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149.       
33 G. M. Zaslavsky, Fractional kinetics and anomalous transport, Physics Reports, 371 (2002), 461-580.       
34 Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2011), 035010, 12pp.       

Go to top