Inverse problem for a space-time fractional diffusion equation: application of fractional Sturm-Liouville operator. (English) Zbl 1388.35205
Summary: An inverse problem of determining a time-dependent source term from the total energy measurement of the system (the over-specified condition) for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional-order time derivative and Sturm-Liouville operator by fractional-order Sturm-Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.
MSC:
| 35R11 | Fractional partial differential equations |
References:
| [1] | IonescuC, LopesA, CopotD, MachadoJAT, BatesJHT. The role of fractional calculus in modelling biological phenomena: a review. Commun Nonlinear Sci Numer Simul. 2017;51:141‐159. · Zbl 1467.92050 |
| [2] | HilferR. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000. · Zbl 0998.26002 |
| [3] | TarasovVE. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Beijing: Springer‐Verlag; 2010. · Zbl 1214.81004 |
| [4] | MachadoJAT, LopesAM. Relative fractional dynamics of stock markets. Nonlinear Dyn. 2016;86:1613‐1619. |
| [5] | BaleanuD, GuvencZB, MachadoJAT. New Trends in Nanotechnology and Fractional Calculus Applications. Netherlands: Springer; 2009. |
| [6] | BagleyRL, TorvikPJ. A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol. 1983;27:201‐210. · Zbl 0515.76012 |
| [7] | MainardiF. Fractional Calculus and Waves in Linear Viscoelaticity. London: Imperial College Press; 2010. |
| [8] | CaputoM, CarcioneJM, BotelhoMAB. Modeling extreme‐event precursors with the fractional diffusion equation. Fract Calc Appl Anal. 2015;18:208‐222. · Zbl 1515.35308 |
| [9] | SokolovIM, KlafterJ. From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos. 2005;15:1‐7. · Zbl 1080.82022 |
| [10] | HatanoY, HatanoN. Dispersive transport of ions in column experiments: an explanation of long tailed profiles. Water Resour Res. 1998;34:1027‐1033. |
| [11] | FedotovS, KorabelN. Subdiffusion in an external potential: anomalous effects hiding behind normal behavior. Phys Rev E. 2015;91:042112‐1-042112‐7. |
| [12] | MeerschaertMM, SikorskiiA. Stochastic Models for Fractional Calculus. De Gruyter Berlin; 2010. |
| [13] | KlagesR, RadonsG, SokolovIM. Anomalous Transport. WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim; 2008. |
| [14] | SierociukD, SkovranekT, MaciasM, et al. Diffusion process modeling by using fractional‐order models. Appl Math Comp. 2015;257:2‐11. |
| [15] | SalariL, RondoniL, GibertiC, KlagesR. A simple non‐chaotic map generating subdiffusive, diffusive and superdiffusive dynamics. Chaos. 2015;25:1‐11. · Zbl 1374.37063 |
| [16] | BirdRB, KlingenbergDJ. Multicompenent diffusion‐ A brief review. Adv Water Res. 2013;62:238‐242. |
| [17] | WeissGH. Aspects and Applications of the Random Walk. Amsterdam: North‐Holland; 1994. · Zbl 0925.60079 |
| [18] | HughesDB. Random Walks and Random Environments, Volume I: Random Walks. New York: Oxford University Press; 1995. · Zbl 0820.60053 |
| [19] | MuraA, PagniniG. Characterization and simulations of a class of stochastic processes to model anomalous diffusion. J Phys A: Math Theor. 2008;41:1‐22. · Zbl 1143.82028 |
| [20] | MetzlerR, JeonJH, CherstvyAG, BarkaiE. Anomalous diffusion models and their properties: non‐stationarity, non‐ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys. 2014;16:24128‐24164. |
| [21] | PovstenkoY. Linear Fractional Diffusion‐Wave Equation for Scientists and Engineers. Springer Switzeland; 2015. · Zbl 1331.35004 |
| [22] | ChechkinAV, GorenfloR, SokolovIM. Fractional diffusion in inhomogenous media. J Phys A: Math Gen. 2005;38:670‐684. · Zbl 1082.76097 |
| [23] | RomanHE, AlemanyPA. Continuous‐time random walks and the fractional diffusion equation. J Phys A: Math Gen. 1994;27:3407‐3410. · Zbl 0827.60057 |
| [24] | HilferR, AntonL. Fractional master equations and fractal time random walks. Phys Rev E. 1995;51:R848‐R851. |
| [25] | SchneiderWR, WyssW. Fractional diffusion in one dimension. J Math Phys. 1989;30:134‐144. · Zbl 0692.45004 |
| [26] | ChenC, RaghavanR. Transient flow in a linear reservior for space‐time fractional diffusion. J Pet Sci Eng. 2015;128:194‐202. |
| [27] | LaskinN. Some applications of the fractional Poisson probability distribution. J Math Phys. 2009;50:1‐12. · Zbl 1304.81100 |
| [28] | LaskinN. Time fractional quantum mechanics. Chaos, Solitons Fractals. 2017;102:16‐28. · Zbl 1374.81059 |
| [29] | SandevT, SchulzA, KantzH, IominA. Heterogeneous diffusion in comb and fractal grid structure. Chaos, Solitons Fractals. 2017. https://doi.org/10.1016/j.chaos.2017.04.041. · Zbl 1415.82011 · doi:10.1016/j.chaos.2017.04.041 |
| [30] | KlimekM, MalinowskaAB, OdzijewiczT. Variational methods for the fractional Sturm‐Liouville problem. J Math Anal Appl. 2014;416:402‐428. · Zbl 1297.65087 |
| [31] | KlimekM, MalinowskaAB, OdzijewiczT. Applications of the fractional Sturm‐Liouville Problem to the space‐time fractional diffusion in a finite domain. Fract Calc Appl Anal. 2016;19:516‐550. · Zbl 1381.34017 |
| [32] | IsmailovMI, ÇiçekM. Inverse source problem for a time‐fractional diffusion equation with nonlocal boundary conditions. Applied Mathematical Modelling. 2016;40:4891‐4899. · Zbl 1459.35395 |
| [33] | FuratiKM, IyiolaOS, KiraneM. An inverse problem for a generalized fractional diffusion. Appl Math Comp. 2014;249:24‐31. · Zbl 1338.35493 |
| [34] | TatarS, UlusoyS. An inverse source problem for a one dimensional space‐time fractional diffusion equation. Appl Anal. 2015;94:2233‐2244. · Zbl 1327.35416 |
| [35] | WeiT, SunL, LiY. Uniqueness for an inverse space‐dependent source term in a multi‐dimensional time‐fractional diffusion equation. Appl Math Lett. 2016;61:108‐113. · Zbl 1386.35481 |
| [36] | LiuCS, ChenW, FuZ. A multiple‐scale MQ‐RBF for solving the inverse Cauchy problems in arbitrary plane domain. Eng Anal Boundary Elem. 2016;68:11‐16. · Zbl 1403.65116 |
| [37] | KiraneM, MalikSA. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time. Appl Math Comp. 2011;218:163‐170. · Zbl 1231.35289 |
| [38] | KiraneM, MalikSA, Al‐GwaizMA. An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Math Meth Appl Sci. 2013;36:1056‐1069. · Zbl 1267.80013 |
| [39] | FuratiKM, IyiolaOS, MustaphaK. An inverse source problem for a two‐parameter anomalous diffusion with local time datum. Comput Math Appl. 2017;73:1008‐1015. · Zbl 1409.35214 |
| [40] | Al‐JamalMF. A backward problem for the time‐fractional diffusion equation. Math Meth Appl Sci. 2017;40:2466‐2474. · Zbl 1366.65086 |
| [41] | AliM, MalikSA. An inverse problem for a family of time fractional diffusion equations. Inverse Prob Sci Eng. 2017;25:1299‐1322. · Zbl 1398.65232 |
| [42] | AzizS, MalikSA. Identification of an unknown source term for a time fractional fourth‐order parabolic equation. Electron J Differ Equ. 2016;293:1‐20. · Zbl 1358.80004 |
| [43] | LukashchukSY. Estimation of parameters in fractional subdiffusion equations by the time integral characteristics method. Comput Math Appl. 2011;62:834‐844. · Zbl 1228.35265 |
| [44] | JannoJ. Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation. Electron J Differ Equ. 2016;199:1‐28. · Zbl 1342.35452 |
| [45] | ChenS, JiangXY. Parameter estimation for a new anomalous thermal diffusion model in layered media. Comput Math Appl. 2017;73:1172‐1181. · Zbl 1409.35235 |
| [46] | TatarS, TinaztepeR, UlusoyS. Simultaneous inversion for the exponents of the fractional time and space derivatives in the space‐time fractional diffusion equation. Appl Anal. 2016;95:1‐23. · Zbl 1334.35401 |
| [47] | MalikSA, AzizS. An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions. Comput Math Appl. 2017;73:2548‐2560. · Zbl 1386.35479 |
| [48] | JiaJ, PengJ, YangJ. Harnack’s inequality for a space‐time fractional diffusion equation and application to an inverse source problem. J Differ Equ. 2017;262:4415‐4450. · Zbl 1357.35284 |
| [49] | SamkoGS, KilbasAA, MarichevDI. Fractional Integrals and Derivatives: Theory and Applications. Amsterdam: Gordon and Breach Science Publishers; 1993. · Zbl 0818.26003 |
| [50] | KilbasAA, SrivastavaHM, TrujilloJJ. Theory and Applications of Fractional Differential Equations, Vol. 204. Amsterdam: Elsevier; 2006. · Zbl 1092.45003 |
| [51] | GorenfloR, KilbasAA, MainardiF, RogosinSV. Mittag‐Leffler Functions, Related Topics and Application. Berlin, Heidelberg: Springer; 2014. · Zbl 1309.33001 |
| [52] | PodlubnyI. Fractional Differential Equations, Vol. 198. San Diego, California: Mathematics in Science and Engineering, Acad. Press; 1999. · Zbl 0924.34008 |
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