Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems. (English) Zbl 1518.35678
Summary: In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value problem, such as the existence, uniqueness and regularity estimates. Moreover, we apply the obtained theories to an inverse source problem. The uniqueness of this inverse problem under the boundary measurements is proved.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35R11 | Fractional partial differential equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
stochastic time-fractional diffusion-wave equations; regularity; inverse random source problem; uniquenessReferences:
| [1] | Adams, E. E.; Gelhar, L. W., Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis, Water Resour. Res., 28, 3293-307 (1992) · doi:10.1029/92WR01757 |
| [2] | Adams, R. A., Sobolev Spaces (Pure and Applied Mathematics, vol 65 (1975), New York: Academic [Harcourt Brace Jovanovich, Publishers], New York · Zbl 0314.46030 |
| [3] | Barkai, E., Fractional fokker-planck equation, solution and application, Phys. Rev. E, 63 (2001) · doi:10.1103/PhysRevE.63.046118 |
| [4] | Chen, H.; Stynes, M., Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem, J. Sci. Comput., 79, 624-47 (2019) · Zbl 1419.65010 · doi:10.1007/s10915-018-0863-y |
| [5] | Chen, L.; Hu, G., Hölder regularity for the nonlinear stochastic time-fractional slow fast diffusion equations on \(####\), Fract. Calc. Appl. Anal., 25, 608-29 (2022) · Zbl 1503.60080 · doi:10.1007/s13540-022-00033-3 |
| [6] | Cheng, J.; Nakagawa, J.; Yamamoto, M.; Yamazaki, T., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009) · Zbl 1181.35322 · doi:10.1088/0266-5611/25/11/115002 |
| [7] | Cheng, X.; Li, Z., Uniqueness and stability for inverse source problem for fractional diffusion-wave equations, J. Inverse Ill-Posed Problems (2023) · Zbl 1528.35231 · doi:10.1515/jiip-2021-0078 |
| [8] | Del Castillo Negrete, D.; Carreras, B. A.; Lynch, V. E., Fractional diffusion in plasma turbulence, Phys. Plasmas, 11, 3854-64 (2004) · doi:10.1063/1.1767097 |
| [9] | Del Castillo Negrete, D.; Carreras, B. A.; Lynch, V. E., Nondiffusive transport in plasma turbulence: a fractional diffusion approach, Phys. Rev. Lett., 94 (2005) · doi:10.1103/PhysRevLett.94.065003 |
| [10] | Evans, L. C., An Introduction to Stochastic Differential Equations (2013), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1416.60002 |
| [11] | Feng, X.; Li, P.; Wang, X., An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion, Inverse Problems, 36 (2020) · Zbl 1469.35251 · doi:10.1088/1361-6420/ab6503 |
| [12] | Fu, S.; Zhang, Z., Application of the generalized multiscale finite element method in an inverse random source problem, J. Comput. Phys., 429, 17 (2021) · Zbl 07500762 · doi:10.1016/j.jcp.2020.110032 |
| [13] | Giona, M.; Cerbelli, S.; Roman, H. E., Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191, 449-53 (1992) · doi:10.1016/0378-4371(92)90566-9 |
| [14] | Gong, Y.; Li, P.; Wang, X.; Xu, X., Numerical solution of an inverse random source problem for the time fractional diffusion equation via phaselift, Inverse Problems, 37 (2021) · Zbl 1475.35432 · doi:10.1088/1361-6420/abe6f0 |
| [15] | Gorenflo, R.; Kilbas, A. A.; Mainardi, F.; Rogosin, S. V., Mittag-Leffler Functions, Related Topics and Applications (Springer Monographs in Mathematics) (2014), Heidelberg: Springer, Heidelberg · Zbl 1309.33001 |
| [16] | Hatano, Y.; Hatano, N., Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resour. Res., 34, 1027-33 (1998) · doi:10.1029/98WR00214 |
| [17] | He, J. W.; Peng, L., Approximate controllability for a class of fractional stochastic wave equations, Comput. Math. Appl., 78, 1463-76 (2019) · Zbl 1442.93007 · doi:10.1016/j.camwa.2019.01.012 |
| [18] | Ishimaru, A., Wave Propagation and Scattering in Random Media (IEEE/OUP Series on Electromagnetic Wave Theory) (1997), New York: IEEE Press, New York · Zbl 0873.65115 |
| [19] | Jiang, D.; Li, Z.; Liu, Y.; Yamamoto, M., Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017) · Zbl 1372.35364 · doi:10.1088/1361-6420/aa58d1 |
| [20] | Jiang, D.; Li, Z.; Pauron, M.; Yamamoto, M., Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem, Math. Methods Appl. Sci., 46, 2275-87 (2023) · Zbl 1529.35549 · doi:10.1002/mma.8644 |
| [21] | Jin, B.; Lazarov, R.; Zhou, Z., An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36, 197-221 (2016) · Zbl 1336.65150 · doi:10.1093/imanum/dru063 |
| [22] | Kian, Y.; Soccorsi, E.; Triki, F., Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations (2022) |
| [23] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (North-Holland Mathematics Studies vol 204) (2006), Amsterdam: Elsevier Science B.V, Amsterdam · Zbl 1092.45003 |
| [24] | Kirane, M.; Malik, S. A., Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl. Math. Comput., 218, 163-70 (2011) · Zbl 1231.35289 · doi:10.1016/j.amc.2011.05.084 |
| [25] | Kneller, G., Anomalous diffusion in biomolecular systems from the perspective of nonequilibrium statistical physics, Acta Phys. Pol. B, 46, 1167-99 (2015) · Zbl 1371.82008 · doi:10.5506/APhysPolB.46.1167 |
| [26] | Kubica, A.; Yamamoto, M., Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal., 21, 276-311 (2018) · Zbl 1398.35271 · doi:10.1515/fca-2018-0018 |
| [27] | Levy, M.; Berkowitz, B., Measurement and analysis of non-fickian dispersion in heterogeneous porous media, J. Contaminant Hydrol., 64, 203-26 (2003) · doi:10.1016/S0169-7722(02)00204-8 |
| [28] | Li, Y.; Wang, Y.; Deng, W., Galerkin finite element approximations for stochastic space-time fractional wave equations, SIAM J. Numer. Anal., 55, 3173-202 (2017) · Zbl 1380.65017 · doi:10.1137/16M1096451 |
| [29] | Li, Z.; Liu, Y.; Yamamoto, M., Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput., 257, 381-97 (2015) · Zbl 1338.35471 · doi:10.1016/j.amc.2014.11.073 |
| [30] | Li, Z.; Liu, Y.; Yamamoto, M., Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions, Inverse Problems Imaging, 17, 1-22 (2023) · Zbl 1512.35626 · doi:10.3934/ipi.2022027 |
| [31] | Li, Z.; Yamamoto, M., Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Appl. Anal., 94, 570-9 (2015) · Zbl 1327.35408 · doi:10.1080/00036811.2014.926335 |
| [32] | Li, Z.; Zhang, Z., Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data, Inverse Problems, 36 (2020) · Zbl 1452.35259 · doi:10.1088/1361-6420/abbc5d |
| [33] | Lions, J-L; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications (Die Grundlehren der Mathematischen Wissenschaften, Band 181 vol I) (1972), New York: Springer, New York · Zbl 0223.35039 |
| [34] | Lions, J-L; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications (Die Grundlehren der Mathematischen Wissenschaften, Band 182 vol II) (1972), New York-Heidelberg: Springer, New York-Heidelberg · Zbl 0223.35039 |
| [35] | Lions, J-L; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications (Die Grundlehren der Mathematischen Wissenschaften, Band 183 vol III) (1973), New York: Springer, New York · Zbl 0251.35001 |
| [36] | Liu, C.; Wen, J.; Zhang, Z., Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation, Inverse Probl. Imaging, 14, 1001-24 (2020) · Zbl 1464.35399 · doi:10.3934/ipi.2020053 |
| [37] | Liu, Y.; Hu, G.; Yamamoto, M., Inverse moving source problem for time-fractional evolution equations: determination of profiles, Inverse Problems, 37 (2021) · Zbl 1469.35257 · doi:10.1088/1361-6420/ac0c20 |
| [38] | Liu, Y.; Rundell, W.; Yamamoto, M., Strong maximum principle for fractional diffusion equations and an application to an inverse source problem, Fract. Calc. Appl. Anal., 19, 888-906 (2016) · Zbl 1346.35216 · doi:10.1515/fca-2016-0048 |
| [39] | Luchko, Y.; Yamamoto, M., On the maximum principle for a time-fractional diffusion equation, Fract. Calc. Appl. Anal., 20, 1131-45 (2017) · Zbl 1374.35426 · doi:10.1515/fca-2017-0060 |
| [40] | Luchko, Y.; Yamamoto, M., Maximum principle for the time-fractional PDEs, Handbook of Fractional Calculus With Applications, vol 2, pp 299-325 (2019), Berlin: De Gruyter, Berlin · Zbl 1410.26004 |
| [41] | Magdziarz, M.; Weron, A.; Weron, K., Fractional fokker-planck dynamics: Stochastic representation and computer simulation, Phys. Rev. E, 75 (2007) · Zbl 1123.60045 · doi:10.1103/PhysRevE.75.016708 |
| [42] | Metzler, R.; Barkai, E.; Klafter, J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional fokker-planck equation approach, Phys. Rev. Lett., 82, 3563 (1999) · doi:10.1103/PhysRevLett.82.3563 |
| [43] | Metzler, R.; Klafter, J., The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys Rep., 239, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 |
| [44] | Metzler, R.; Klafter, J.; Sokolov, I. M., Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended, Phys. Rev. E, 58, 1621 (1998) · doi:10.1103/PhysRevE.58.1621 |
| [45] | Mijena, J. B.; Nane, E., Space-time fractional stochastic partial differential equations, Stochastic Process. Appl., 125, 3301-26 (2015) · Zbl 1329.60216 · doi:10.1016/j.spa.2015.04.008 |
| [46] | Nie, D.; Deng, W., An inverse random source problem for the time-space fractional diffusion equation driven by fractional brownian motion (2021) |
| [47] | Nigmatullin, R. R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys Status Solidi b, 133, 425-30 (1986) · doi:10.1002/pssb.2221330150 |
| [48] | Niu, P.; Helin, T.; Zhang, Z., An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020) · Zbl 1469.35259 · doi:10.1088/1361-6420/ab532c |
| [49] | Øksendal, B., Stochastic Differential Equations (An Introduction With Applications) (2003), Berlin: Universitext. Springer, Berlin · Zbl 1025.60026 |
| [50] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (Applied Mathematical Sciences vol 44) (1983), New York: Springer, New York · Zbl 0516.47023 |
| [51] | Podlubny, I., Fractional Differential Equations (Mathematics in Science and Engineering vol 198) (1999), San Diego, CA: Academic, San Diego, CA · Zbl 0918.34010 |
| [52] | Pollard, H., The completely monotonic character of the Mittag-Leffler function \(####\), Bull. Am. Math. Soc., 54, 1115-6 (1948) · Zbl 0033.35902 · doi:10.1090/S0002-9904-1948-09132-7 |
| [53] | Ruan, Z.; Yang, J. Z.; Lu, X., Tikhonov regularisation method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation, East Asian J. Appl. Math., 5, 273-300 (2015) · Zbl 1457.65086 · doi:10.4208/eajam.310315.030715a |
| [54] | Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426-47 (2011) · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058 |
| [55] | Sakamoto, K.; Yamamoto, M., Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields, 1, 509-18 (2011) · Zbl 1241.35220 · doi:10.3934/mcrf.2011.1.509 |
| [56] | Sokolov, I. M., Models of anomalous diffusion in crowded environments, Soft Matter, 8, 9043-52 (2012) · doi:10.1039/c2sm25701g |
| [57] | Sun, C.; Liu, J., An inverse source problem for distributed order time-fractional diffusion equation, Inverse Problems, 36 (2020) · Zbl 1469.35263 · doi:10.1088/1361-6420/ab762c |
| [58] | Uchaikin, V. V., Fractional Derivatives for Physicists and Engineers (Nonlinear Physical Science vol II) (2013), Beijing: Higher Education Press, Springer, Beijing · Zbl 1312.26002 |
| [59] | Vergara, V.; Zacher, R., Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47, 210-39 (2015) · Zbl 1317.45006 · doi:10.1137/130941900 |
| [60] | Wei, T.; Sun, L.; Li, Y., Uniqueness for an inverse space-dependent source term in a multi-dimensional time-fractional diffusion equation, Appl. Math. Lett., 61, 108-13 (2016) · Zbl 1386.35481 · doi:10.1016/j.aml.2016.05.004 |
| [61] | Wen, J.; Liu, Z-X; Wang, S-S, A non-stationary iterative Tikhonov regularization method for simultaneous inversion in a time-fractional diffusion equation, J. Comput. Appl. Math., 426, 18 (2023) · Zbl 1517.35267 · doi:10.1016/j.cam.2023.115094 |
| [62] | Zacher, R., Maximal regularity of type L_p for abstract parabolic Volterra equations, J. Evol. Equ., 5, 79-103 (2005) · Zbl 1104.45008 · doi:10.1007/s00028-004-0161-z |
| [63] | Zhang, Y.; Wei, T.; Zhang, Y-X, Simultaneous inversion of two initial values for a time-fractional diffusion-wave equation, Numer. Methods Partial Differ. Equ., 37, 24-43 (2021) · Zbl 1535.65175 · doi:10.1002/num.22517 |
| [64] | Zhang, Y.; Xu, X., Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2011) · Zbl 1211.35280 · doi:10.1088/0266-5611/27/3/035010 |
| [65] | Zhang, Z., An undetermined coefficient problem for a fractional diffusion equation, Inverse Problems, 32 (2016) · Zbl 1332.35403 · doi:10.1088/0266-5611/32/1/015011 |
| [66] | Zou, G-A, Galerkin finite element method for time-fractional stochastic diffusion equations, Comput. Appl. Math., 37, 4877-98 (2018) · Zbl 1432.65153 · doi:10.1007/s40314-018-0609-3 |
| [67] | Zou, G-A; Atangana, A.; Zhou, Y., Error estimates of a semidiscrete finite element method for fractional stochastic diffusion-wave equations, Numer. Methods Partial Differ. Equ., 34, 1834-48 (2018) · Zbl 1407.74098 · doi:10.1002/num.22252 |
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