References
- Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198, Mathematics in science and engineering. San Diego (CA): Academic Press; 1999.
- Kenichi S, Masahiro Y. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 2011;382:426–447.
- Jin BT, Lazarov R, Zhou Z. error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 2013;55:445–466.
- Eidelman SD, Kochubei AN. Cauchy problem for fractional diffusion equations. J. Differ. Equ. 2004;199:211–255.
- Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 2007;225:1533–1552.
- Jiang YJ, Ma JT. High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 2011;235:3285–3290.
- Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 1996;9:23–28.
- Li XJ, Xu CJ. a space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 2009;47:2108–2131.
- Agarwal OP. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 2002;29:145–155.
- Liu JJ, Yamamoto M. A backward problem for the time-fractional diffusion equation. Applicable Anal. 2010;89:1769–1788.
- Ren C, Xu X, Lu S. Regularization by projection for a backward problem of the time-fractional diffusion equation. J. Inverse Ill-posed Prob. 2014;22:121–139.
- Xiong XT, Wang JX, Li M. An optimal method for fractional heat conduction problem backward in time. Applicable Anal. 2012;91:823–840.
- Wang LY, Liu JJ. Data regularization for a backward time-fractional diffusion problem. Comput. Math. Appl. 2012;64:3613–3626.
- Zhang Y, Xu X. Inverse source problem for a fractional diffusion equation. Inverse Prob. 2011;27:1–12.
- Wang W, Yamamoto M, Han B. Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation. Inverse Prob. 2013;29:095009.
- Wang JG, Zhou YB, Wei T. Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation. Appl. Numer. Math. 2013;68:39–57.
- Jin BT, William R. An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Prob. 2012;28:1–19.
- Liu JJ, Yamamoto M, Yan LL. On the reconstruction of unknown time-dependent boundary sources for time fractional diffusion process by distributing measurement. Inverse Prob. 2015;32:015009.
- Hasanov A, Pektaş B. Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method. Comput. Math. Appl. 2013;65:42–57.
- Yang F, Fu CL, Li XX. A mollification regularization method for identifying the time-dependent heat source problem. J. Eng. Math. Forthcoming. doi: 10.1007/s10665-015-9834-6.
- Wei T, Zhang ZQ. Reconstruction of a time-dependent source term in a time-fractional diffusion equation. Eng. Anal. Boundary Elem. 2013;37:23–31.
- Aleroev TS, Kirane M, Malik SA. Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition. Electron. J. Differ. Equ. 2013;270:1–16.
- Demir A, Kanca F, Ozbilge E. Numerical solution and distinguishability in time fractional parabolic equation. Boundary Value Prob. 2015;1:1–12.
- Liu Y, Rundell W, Yamamoto M. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fractional Calculus Appl. Anal. Forthcoming 2016;19.
- Gorenflo R, Luchko Y, Yamamoto M. Time-fractional diffusion equation in the fractional Sobolev spaces. Fractional Calculus Appl. Anal. 2015;18:799–820.
- Shimakura N. Partial differential operators of elliptic type. Providence (RI): American Mathematical Society; 1992.
- Murio DA, Mejía CE, Zhan S. Discrete mollification and automatic numerical differentiation. Comput. Math. Appl. 1998;35:1–16.