[1] Metzler R, Klafter J, The random walk's guide to anomalous diffusion:a fractional dynamics approach. Phys Rep, 2000, 339:1-77 [2] Metzler R, Klafter J, Boundary value problems for frctional diffusion equations. Phys A, 2000, 278:107-125 [3] Agrawal O P, Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynam, 2002, 29:145-155 [4] Cao X, Liu H, Determining a fractional Helmholtz system with unknown source and medium parameter. Commun Math Sci, 2019, 17:1861-1876 [5] Cao X, Lin Y H, Liu H, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrodinger operators. Inverse Probl Imag, 2019, 13:197-210 [6] Yuste S B, Acedo L, Lindenberg K, Reaction front in an A +B → C reaction-subdiffusion process. Phys Rev E, 2004, 69:036126 [7] Santamaria F, Wils S, Schutter D E, et al, Anomalous diffusion in purkinje cell dendrites caused by spines. Neuron, 2006, 52:635-648 [8] Magin R, Feng X, Baleanu D, Solving the fractional order bloch equation. Concept Magn Reson A, 2009, 34A:16-23 [9] Henry B I, Langlands T A M, Wearne S L, Fractional cable models for spiny neuronal dendrites. Phys Rev Lett, 2008, 100:128103 [10] Hall M G, Barrick T R, From diffusion-weighted MRI to anomalous diffusion imaging. Magnet Reson Med, 2008, 59:447-455 [11] Raberto M, Scalas E, Mainardi F, Waiting-times and returns in high-frequency finan-cial data:an empirical study. Phys A, 2002, 314:749-755 [12] Scalas E, Gorenflo R, Mainardi F, Fractional calculus and continuous-time finance. Phys A, 2000, 284:376-384 [13] Wyss W, The fractional Black-Scholes equation. Fract Calc Appl Anal, 2000, 3:51-62 [14] Arran F, Dumitru B, Classes of operators in fractional calculus:A case study. Math Meth Appl Sci, 2021, 44:9143-9162 [15] Raoul N, Dumitru B, Arran F, Balance equations with generalised memory and the emerging fractional kernels. Nonlinear Dyn, 2021, 104:4149-4161 [16] Gu Q, Schiff E A, Grebner S, et al, Non-Gaussian Transport Measurements and the Einstein Relation in Amorphous Silicon. Phys Rev Lett, 1996, 76:3196-3199 [17] Klammler F, Kimmich R, Geometrical restrictions of incoherent transport of water by diffusion in protein of silica fineparticle systems and by flow in a sponge-a study of anomalous properties using an NMR field-gradient technique. Croat Chem Acta, 1992, 65:455-470 [18] Weber H W, Kimmich R, Anomalous segment diffusion in polymers and NMR relaxation spectroscopy. Macromolecules, 1993, 26:2597-2606 [19] Klemm A, Muller H P, Kimmich R, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects. Phys Rev E, 1997, 55:4413-4422 [20] Porto M, Bunde A, Havlin S, et al, Structural and dynamical properties of the percolation backbone in two and three dimensions. Phys Rev E, 1997, 56:1667-1675 [21] Weeks E R, Swinney H L, Anomalous diffusion resulting from strongly asymmetric random walks. Phys Rev E, 1998, 57:4915-4920 [22] Luedtke W D, Landman U, Slip diffusion and Levy fligths of an adsorbed gold nanocluster. Phys Rev Lett, 1999, 82:3835-3838 [23] Mclean W, Mustapha K, A second-order accurate numerical method for a fractional wave equation. Numer Math, 2006, 105:481-510 [24] Shlesinger M F, West B J, Klafter J, Levy dynamics of enhanced diffusion:Application to turbulence. Phys Rev Lett, 1987, 58:1100-1103 [25] Schaufler S, Schleich W P, Yakovlev V P, Scaling and asymptotic laws in subrecoil laser cooling. Europhys Lett, 2007, 39:383-388 [26] Zumofen G, Klafter J, Spectral random walk of a single molecule. Chem Phys Lett, 1994, 219:303-309 [27] Bychuk O V, Oshaughnessy B, Anomalous diffusion at liquid surfaces. Phys Rev Lett, 1995, 74:1795-1798 [28] Young D L, Tsai C C, Murugesan K, Fan C M, Chen C W, Time-dependent fundamental solutions for homogeneous diffusion problems. Eng Anal Bound Elem, 2004, 28:1463-1473 [29] Martin V, An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions. Appl Numer Math, 2005, 52:401-428 [30] Alcouffe R E, Brandt A, Dendy J E, Painter J W, The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J Sci Comput, 1981, 2:430-454 [31] Yang S, Liu Y, Liu H, Wang C. Numerical Methods for Semilinear Fractional Diffusion Equations with Time Delay. Adv Appl Math Mech, https://doi.org/https://doi.org/10.4208/aamm.OA-2020-0387 [32] Can N H, Luc N H, Baleanu D, et al, Inverse source problem for time fractional diffusion equation with Mittag-Leffler kernel. Adv Differ Equ, 2020, 2020:1-18 [33] Tuan N H, Zhou Y, Can N H, Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative. Comput Appl Math, 2020, 39:75 [34] Luchko Y, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract Calc Appl Anal, 2012, 15:141-160 [35] Kemppainen J, Existence and uniqueness of the solution for a time-fractional diffusion equation. Fract Calc Appl Anal, 2011, 14:411-417 [36] Wei T, Zhang Y, The backward problem for a time-fractional diffusion-wave equation in a bounded domain. Comput Math Appl, 2018, 75:3632-3648 [37] Tuan N H, Long L D, Tatar S, Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation. Appl Anal, 2018, 97:842-863 [38] Yang F, Zhang Y, Li X X, et al, The quasi-boundary value regularization method for identifying the initial value with discrete random value. Bound Value Probl, 2018, 2018:1-12 [39] Liu J J, Yamamoto M, A backward problem for the time-fractional diffusion equation. Appl Anal, 2010, 89:1769-1788 [40] Yang F, Ren Y P, Li X X, The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source. Math Method Appl Sci, 2018, 41:1774-1795 [41] Yang F, Zhang Y, Liu X, Li X X. The quasi-boundary value method for identifying the intital valuce of the space-time an fractional diffusion equation. Acta Math Sci, 2020, 40B(3):641-658 [42] Yang F, Fu C L, Li X X, A mollification regualrization method for unknown source in time-fractional diffusion equation. Int J Comput Math, 2014, 91:1516-1534 [43] Yang F, Fu C L, Li X X, The inverse source problem for time fractional diffusion equation:stability analysis and regularization. Inverse Probl Sci En, 2015, 23:969-996 [44] Wei T, Wang J G, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation. Appl Numer Math, 2014, 78:95-111 [45] Wei T, Zhang Z Q, Reconstruction of a time-dependent source term ina time-fractional diffusion equation. Eng Anal Bound Elem, 2013, 37:23-31 [46] Tuan N H, Nane E, Inverse source problem for time fractional diffusion with dicrete random noise. Stat Probabil Lett, 2017, 120:126-134 [47] Wang X, Guo Y, Zhang D, Liu H, Fourier method for recovering acoustic sources from multi-frequency far-field data. Inverse Probl, 2017, 33:035001 [48] Luc N H, Baleanu D, Agarwal R P, Identifying the source function for time fractional diffusion with nonlocal in time conditions. Comput Appl Math, 2021, 40:149 [49] Luc N H, Tatar S, Baleanu D, et al. An inverse source problem for pseudo-parabolic equation with Caputo derivative. J Appl Math Comput, 2021:1-27 [50] Karapinar E, Kumar D, Sakthivel R, et al, Identifying the space source term problem for time-space-fractional diffusion equation. Adv Differ Equ, 2020, 2020:1-23 [51] Ozbilge E, Demir A, Inverse problem for a time-fractional parabolic equation. J Inequal Appl, 2015, 2015:81 [52] Li G S, Zhang D L, Jia X Z, et al, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation. Inverse Probl, 2013, 29:065014(36pp) [53] Yang F, Liu X, Li X X, et al, Landweber iterative regularization method for identifying the unknown source of the time-fractional diffusion equation. Adv Differ Equ, 2017, 2017:388-402 [54] Yang F, Ren Y P, Li X X, et al, Landweber iterative method for identifying a space-dependent source for the time-fractional diffusion equation. Bound Value Probl, 2017, 2017:1-19 [55] Wang J G, Xiong X T, Gao X X, Fractional Tikhonov Regularization method for a Time-fractional Backward Heat Equation with a Fractional Laplacian. J Part Diff Eq, 2018, 31:333-342 [56] Xiong X T, Ma X J, A Backward Identifying Problem for an Axis-Symmetric Fractional Diffusion Equation. Math Model Anal, 2017, 22:311-320 [57] Xiong X T, Xue X X, Qian Z, A modified iterative regularization method for ill-posed problems. Appl Numer Math, 2017, 122:108-128 [58] Han Y Z, Xiong X T, Xue X M. A fractional Landweber method for solving backward time-fractional diffusion problem. Comput Math Appl, https://doi.org/https://doi.org/10.1016/j.camwa.2019.02.017 [59] Dimovski I, Operational calculus of a class of differential operators. C R Acad Bulg Sci, 1966, 19:1111-1114 [60] Garra R, Giusti A, Mainardi F and Pagnini G, Fractional relaxation with time-varying coefficient. Fract Calc Appl Anal, 2014, 17:424-439 [61] Garra R, Orsingher E and Polito F, Fractional diffusion with time-varying coefficients. J Math Phys, 2015, 56:1-19 [62] Tuan N H, Huynh L N, Baleanu D, Can N H, On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator. Math Method Appl Sci, 2020, 43:2858-2882 [63] Luc N H, Huynh L N, Baleanu D, et al. Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator. Adv Differ Equ. https://doi.org/10.1186/s13662-020-02712-y [64] 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 [65] Podlubny I. Fractional differential equations. San Diego:Academic Press, 1999. [66] Courant R, Hilbert D, Methods of Mathematical Physics:Partial Differential Equations. Nucl Sci Eng, 1963, 30:158 [67] Al-Musalhi F, Al-Salti N, Karimov E, Initial boundary value problems for a fractional differential equation with hyper-Bessel operator. Fract Calc Appl Anal, 2018, 21:200-219 [68] Wang J G, Wei T, Zhou Y B, Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. J Comput Appl Math, 2015, 279:277-292 [69] Liu S, Feng L. Optimal error bound and modified kernel method for a space-fractional backward diffusion problem. Adv Differ Equ, 2018, 268 [70] Hochstenbach M E, Reichel L, Fractional Tikhonov regularization for linear discrete ill-posed problems. Bit, 2011, 51:197-215 |