Fourier truncation method for the non-homogeneous time fractional backward heat conduction problem

References

• Del Castillo Negrete D, Carreras BA, Lynch VE. Fractional diffusion in plasma turbulence. Phys Plasmas. 2004;11:3854–3864. doi: 10.1063/1.1767097
   Web of Science ®Google Scholar
• Del Castillo Negrete D, Carreras BA, Lynch VE. Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Phys Rev Lett. 2005;94:065–003. doi: 10.1103/PhysRevLett.94.065003
   Web of Science ®Google Scholar
• Gorenflo R, Mainardi F, Moretti D, et al. Fractional diffusion: probability distributions and random walk models. Phys A Stat Mech Appl. 2002;305:106–112. doi: 10.1016/S0378-4371(01)00647-1
   Google Scholar
• Metzler R, Klafter J. The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339:1–77. doi: 10.1016/S0370-1573(00)00070-3
   Web of Science ®Google Scholar
• Kneller G. Anomalous diffusion in biomolecular systems from the perspective of non-equilibrium statistical physics. Acta Phys Pol B. 2015;46:1167–1199. doi: 10.5506/APhysPolB.46.1167
   Web of Science ®Google Scholar
• Nigmatullin RR. The realization of the generalized transfer equation in a medium with fractal geometry. Phys Status Solidi B. 1986;133:425–430. doi: 10.1002/pssb.2221330150
   Google Scholar
• Gorenflo R, Luchko Y, Mainardi F. Wright functions as scale-invariant solutions of the diffusion-wave equation. J Comput Appl Math. 2000;118:175–191. doi: 10.1016/S0377-0427(00)00288-0
   Web of Science ®Google Scholar
• Gorenflo R, Luchko Y, Yamamoto M. Time-fractional diffusion equation in the fractional Sobolev spaces. Frac Calc Appl Anal. 2015;18:799–820.
   Web of Science ®Google Scholar
• Kubica A, Yamamoto M. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients. Frac Calc Appl Anal. 2018;21:276–311. doi: 10.1515/fca-2018-0018
   Web of Science ®Google Scholar
• Li Z, Liu Y, Yamamoto M. Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl Math Comput. 2015;257:381–397.
   Web of Science ®Google Scholar
• Liu Y, Rundell W, Yamamoto M. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Frac Calc Appl Anal. 2016;19:888–906.
   Web of Science ®Google Scholar
• Luchko Y. Initial boundary value problems for the one dimensional time fractional diffusion equation. Frac Calc Appl Anal. 2012;15:141–160.
   Web of Science ®Google Scholar
• Sakamoto K, Yamamoto M. Initial value boundary value problems for fractional diffusion wave equation and applications to some inverse problems. J Math Anal Appl. 2011;382:426–447. doi: 10.1016/j.jmaa.2011.04.058
   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. doi: 10.1088/0266-5611/25/11/115002
   Web of Science ®Google Scholar
• Li Z, Yamamoto M. Uniqueness for inverse problems of determining orders of multi-term time fractional derivatives of diffusion equation. Appl Anal. 2015;94:570–579. doi: 10.1080/00036811.2014.926335
   Web of Science ®Google Scholar
• Isakov V. Inverse problems for partial differential equations. Berlin: Springer-Verlag; 2006.
   Google Scholar
• Nair MT. Linear operator equation: approximation and regularization. Singapore: World Scientific; 2009.
   Google Scholar
• Liu JJ, Yamamoto M. A backward problem for the time fractional diffusion equation. Appl Anal. 2010;89:1769–1788. doi: 10.1080/00036810903479731
   Web of Science ®Google Scholar
• Wang L, Liu J. Data regularization for a backward time fractional diffusion problem. Comput Math Appl. 2012;64:3613–3626. doi: 10.1016/j.camwa.2012.10.001
   Web of Science ®Google Scholar
• Yang M, Liu J. Solving a final value fractional diffusion problem by boundary condition regularization. Appl Numer Math. 2013;66:45–58. doi: 10.1016/j.apnum.2012.11.009
   Web of Science ®Google Scholar
• Wang JG, Wei T, Zhou YB. Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Appl Math Model. 2013;37:8518–8532. doi: 10.1016/j.apm.2013.03.071
   Web of Science ®Google Scholar
• Wei T, Wang J. A modified quasi boundary value method for the backward time fractional diffusion problem. ESAIM Math Model Num. 2014;48:603–621. doi: 10.1051/m2an/2013107
   Web of Science ®Google Scholar
• Zhang H, Zhang X. Generalized Tikhonov method for the final value problem of time-fractional diffusion equation. Int J Comput Math. 2017;94:66–78. doi: 10.1080/00207160.2015.1089354
   Web of Science ®Google Scholar
• Tuan NH, Long LD, Nguyen VT, et al. On a final value problem for the time fractional diffusion equation with inhomogeneous source. Inverse Probl Sci En. 2016. doi:10.1080/17415977.2016.1259316.
   Web of Science ®Google Scholar
• Tuan NH, Long LD, Tatar S. Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation. Appl Anal. 2018;97:842–863. doi:10.1080/00036811.2017.1293815.
   Web of Science ®Google Scholar
• Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier; 2006.
   Google Scholar
• Podlubny I. Fractional differential equations. London: Academic Press; 1999.
   Google Scholar
• Wei T, Zhang ZQ. Robin coefficient identification for a time-fractional diffusion equation. Inverse Probl Sci Eng. 2014;24:647–666. doi: 10.1080/17415977.2015.1055261
   Web of Science ®Google Scholar
• Rudin W. Principles of mathematical analysis. Singapore: McGraw-Hill; 1976.
   Google Scholar
• Kress R. Linear integral equations. 3rd ed New York: Springer; 2014.
   Google Scholar
• YX Zhang, Fu CL, Deng ZL. An a posteriori truncation method for some Cauchy problems associated with Helmholtz-type equations. Inverse Probl Sci Eng. 2013;21:1151–1168. doi: 10.1080/17415977.2012.743538
   Web of Science ®Google Scholar