The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation

References

• Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339(1):1–77. doi:10.1016/S0370-1573(00)00070-3.
   Web of Science ®Google Scholar
• Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. San Diego (CA): Academic Press Inc.; 1999 (Mathematics in science and engineering; vol. 198).
   Google Scholar
• Scalas E, Gorenflo R, Mainardi F. Fractional calculus and continuous-time finance. Phys A. 2000;284(1–4):376–384. doi:10.1016/S0378-4371(00)00255-7.
   Web of Science ®Google Scholar
• Chi G, Li G, Jia X. Numerical inversions of a source term in the FADE with a Dirichlet boundary condition using final observations. Comput Math Appl. 2011;62(4):1619–1626. doi:10.1016/j.camwa.2011.02.029.
   Web of Science ®Google Scholar
• Zhang D, Li G, Chi G, et al. Numerical identification of multiparameters in the space fractional advection dispersion equation by final observations. J Appl Math. 2012;2012. 14 p. Art. ID 0740385.
   Google Scholar
• Benson DA, Wheatcraft SW, Meerschaert MM. The fractional-order governing equation of Lévy motion. Water Resour Res. 2000;36:1413–1423.
   Web of Science ®Google Scholar
• Mera NS. The method of fundamental solutions for the backward heat conduction problem. Inverse Probl Sci Eng. 2005;13(1):65–78.
   Web of Science ®Google Scholar
• Marin L, Lesnic D. The method of fundamental solutions for the cauchy problem in two-dimensional linear elasticity. Int J Solids Struct. 2004;41:3425–3438.
   Web of Science ®Google Scholar
• Marin L, Lesnic D. The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput Struct. 2005;83(4–5):267–278.
   Web of Science ®Google Scholar
• Jin B, Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation. Comput Methods Appl Mech Eng. 2006;195(19–22):2270–2288.
   Web of Science ®Google Scholar
• Hon YC, Wei T. A fundamental solution method for inverse heat conduction problem. Eng Anal Boundary Elem. 2004;28:489–495.
   Web of Science ®Google Scholar
• Hon YC, Wei T. The method of fundamental solution for solving multidimensional inverse heat conduction problems. CMES Comput Model Eng Sci. 2005;7(2):119–132.
   Web of Science ®Google Scholar
• Wei T, Yamamoto M. Reconstruction of a moving boundary from cauchy data in one-dimensional heat equation. Inverse Probl Sci Eng. 2009;17(4):551–567.
   Web of Science ®Google Scholar
• Karageorghis A, Lesnic D, Marin L. A survey of applications of the MFS to inverse problems. Inverse Probl Sci Eng. 2011;19(3):309–336. doi:10.1080/17415977.2011.551830.
   Web of Science ®Google Scholar
• Hon YC, Li M. A discrepancy principle for the source points location in using the MFS for solving the BHCP. Int J Comput Methods. 2009;6(2):181–197. doi:10.1142/S0219876209001759.
   Web of Science ®Google Scholar
• Liu CS. The method of fundamental solutions for solving the backward heat conduction problem with conditioning by a new post-conditioner. Numer Heat Transfer Part B: Fundam. 2011;60(1):57–72.
   Web of Science ®Google Scholar
• Wei T, Wang JC. Simultaneous determination for a space-dependent heat source and the initial data by the MFS. Eng Anal Boundary Elem. 2012;36(12):1848–1855. doi:10.1016/j.enganabound.2012.07.006.
   Web of Science ®Google Scholar
• Chen W, Wang F. Singular boundary method using time-dependent fundamental solution for transient diffusion problems. Eng Anal Bound Elem. 2016;68:115–123. doi:10.1016/j.enganabound.2016.04.004.
   Web of Science ®Google Scholar
• Kosec G, Šarler B. Solution of phase change problems by collocation with local pressure correction. CMES Comput Model Eng Sci. 2009;47(2):191–216.
   Web of Science ®Google Scholar
• Šarler B, Vertnik R. Meshfree explicit local radial basis function collocation method for diffusion problems. Comput Math Appl. 2006;51(8):1269–1282. doi:10.1016/j.camwa.2006.04.013.
   Web of Science ®Google Scholar
• Tsai CH, Young DL, Kolibal J. An analysis of backward heat conduction problems using the time evolution method of fundamental solutions. CMES Comput Model Eng Sci. 2010;66(1):53–72.
   Web of Science ®Google Scholar
• Tsai CH, Young DL, Kolibal J. Numerical solution of three-dimensional backward heat conduction problems by the time evolution method of fundamental solutions. Int J Heat Mass Transfer. 2011;54(11):2446–2458.
   Web of Science ®Google Scholar
• Dou FF, Hon YC. Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation. Eng Anal Bound Elem. 2012;36(9):1344–1352. doi:10.1016/j.enganabound.2012.03.003.
   Web of Science ®Google Scholar
• Dou FF, Hon YC. Numerical computation for backward time-fractional diffusion equation. Eng Anal Bound Elem. 2014;40:138–146. doi:10.1016/j.enganabound.2013.12.001.
   Web of Science ®Google Scholar
• Dou FF, Hon YC. Fundamental kernel-based method for backward space-time fractional diffusion problem. Comput Math Appl. 2016;71(1):356–367. doi:10.1016/j.camwa.2015.11.023.
   Web of Science ®Google Scholar
• Mainardi F, Luchko Y, Pagnini G. The fundamental solution of the space-time fractional diffusion equation. Fract Calc Appl Anal. 2001;4(2):153–192.
   Google Scholar
• Gorenflo R, Mainardi F. Some recent advances in theory and simulation of fractional diffusion processes. J Comput Appl Math. 2009;229(2):400–415. doi:10.1016/j.cam.2008.04.005.
   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(1):106–112.
   Google Scholar
• Cannon JR. Determination of an unknown heat source from overspecified boundary data. SIAM J Numer Anal. 1968;5:275–286. doi:10.1137/0705024.
   Web of Science ®Google Scholar
• Cannon JR, DuChateau P. Structural identification of an unknown source term in a heat equation. Inverse Probl. 1998;14(3):535–551. doi:10.1088/0266-5611/14/3/010.
   Web of Science ®Google Scholar
• Savateev EG. On problems of determining the source function in a parabolic equation. J Inverse Ill-Posed Probl. 1995;3(1):83–102. doi:10.1515/jiip.1995.3.1.83.
   Google Scholar
• Farcas A, Lesnic D. The boundary-element method for the determination of a heat source dependent on one variable. J Eng Math. 2006;54(4):375–388. doi:10.1007/s10665-005-9023-0.
   Web of Science ®Google Scholar
• Yan L, Fu CL, Yang FL. A fundamental solution method for inverse heat conduction problem. Eng Anal Boundary Elem. 2008;28:489–495.
   Google Scholar
• Wen J. A meshless method for reconstructing the heat source and partial initial temperature in heat conduction. Inverse Probl Sci Eng. 2011;19(7):1007–1022. doi:10.1080/17415977.2011.569711.
   Web of Science ®Google Scholar
• Wen J, Yamamoto M, Wei T. Simultaneous determination of a time-dependent heat source and the initial temperature in an inverse heat conduction problem. Inverse Probl Sci Eng. 2013;21(3):485–499. doi:10.1080/17415977.2012.701626.
   Web of Science ®Google Scholar
• Saichev AI, Zaslavsky GM. Fractional kinetic equations: solutions and applications. Chaos. 1997;7(4):753–764. doi:10.1063/1.166272.
   PubMed Web of Science ®Google Scholar
• Hansen PC. Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 1992;34(4):561–580.
   Web of Science ®Google Scholar
• Hansen PC. Numerical tools for analysis and solution of fredholm integral equations of the first kind. Inverse Probl. 1992;8(6):849–872.
   Web of Science ®Google Scholar
• Tikhonov AN, Arsenin VY. Solutions of ill-posed problems. Washington (DC), New York (NY): V. H. Winston & Sons, Wiley; 1977. Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics.
   Google Scholar
• Hansen PC. Rank-deficient and discrete ill-posed problems. In: Flaherty JE, editor. Numerical aspects of linear inversion. SIAM monographs on mathematical modeling and computation. Philadelphia (PA): Society for Industrial and Applied Mathematics; 1998.
   Google Scholar
• Hansen PC. Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer Algorithms. 1994;6(1–2):1–35.
   Google Scholar