Determining the random source and initial value simultaneously in stochastic fractional diffusion equations. (English) Zbl 07918264
Summary: In this paper, we investigate the inverse problem of recovering unknown random source and initial value simultaneously from statistical measurement data in a time-fractional stochastic diffusion equation. Based on the eigenfunction expansions, we first establish the statistical moments estimate for the solution of direct problem. Then the conditional stability for inverse problem is also proved. Furthermore, to address the issue of ill-posedness of inverse problem, the Tikhonov regularization method is adopted, and an a priori and a posteriori convergence rate estimates are obtained. Finally, several numerical results are presented to illustrate the effectiveness of the proposed method.
MSC:
| 58F15 | Hyperbolic structures (expanding maps, Anosov systems, etc.) (MSC1991) |
| 58F17 | Geodesic and horocycle flows (MSC1991) |
| 53C35 | Differential geometry of symmetric spaces |
Keywords:
time-fractional stochastic diffusion equation; random source; random initial value; Tikhonov regularizatio; convergence rate estimateReferences:
| [1] | A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280, 424-438 (2015) · Zbl 1349.65261 · doi:10.1016/j.jcp.2014.09.031 |
| [2] | G. C. P. Bao Chen Li, Inverse random source scattering problems in several dimensions, SIAM-ASA J. Uncertain. Quantif., 4, 1263-1287 (2016) · Zbl 1355.78025 · doi:10.1137/16M1067470 |
| [3] | J. M. Blackledge, Application of the fractional diffusion equation for predicting market behavior, Int. J. Appl. Math., 40, 130-158 (2010) · Zbl 1229.91371 |
| [4] | X. H. Cao Liu, Determining a fractional Helmholtz equation with unknown source and scattering potential, Commun. Math. Sci., 17, 1861-1876 (2019) · Zbl 1433.35015 · doi:10.4310/CMS.2019.v17.n7.a5 |
| [5] | X. Y.-H. H. Cao Lin Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schr \(\ddot{o}\) dinger operators, Inverse Probl. Imag., 13, 197-210 (2019) · Zbl 1407.35225 · doi:10.3934/ipi.2019011 |
| [6] | L. Chen, Nonlinear stochastic time-fractional diffusion equations on \(\mathbb{R} \): Moments, \(H \ddot{o}\) lder regularity and intermittency, Trans. Amer. Math. Soc., 369, 8497-8535 (2017) · Zbl 1406.60093 · doi:10.1090/tran/6951 |
| [7] | Z.-Q. K.-H. P. Chen Kim Kim, Fractional time stochastic partial differential equations, Stochastic Process. Appl., 125, 1470-1499 (2015) · Zbl 1322.60106 · doi:10.1016/j.spa.2014.11.005 |
| [8] | J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl., 25 (2009), 115002, 16 pp. · Zbl 1181.35322 |
| [9] | L. C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013. · Zbl 1416.60002 |
| [10] | X. Feng, P. Li and X. Wang, An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion, Inverse Probl., 36 (2020), 045008, 30 pp. · Zbl 1469.35251 |
| [11] | S. Fu and Z. Zhang, Application of the generalized multiscale finite element method in an inverse random source problem, J. Comput. Phys., 429 (2021), 110032, 17 pp. · Zbl 07500762 |
| [12] | M. S. H. Ginoa Cerbelli Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Phys. A: Stat. Mech. Appl., 191, 449-453 (1992) |
| [13] | Y. Gong, P. Li, X. Wang and X. Xu, Numerical solution of an inverse random source problem for the time fractional diffusion equation via phaselift, Inverse Probl., 37 (2021), 045001, 23 pp. · Zbl 1475.35432 |
| [14] | M. B. J. Gunzburger Li Wang, Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise, Math. Comp., 88, 1715-1741 (2019) · Zbl 1418.60077 · doi:10.1090/mcom/3397 |
| [15] | Y. N. Hatano Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resour. Res., 34, 1027-1034 (1998) |
| [16] | B. Jin and Z. Zhou, Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source, Inverse Probl., 37 (2021), 105009, 28 pp. · Zbl 1472.35452 |
| [17] | A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. · Zbl 1092.45003 |
| [18] | M. S. A. Kirane Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl. Math. Comput., 218, 163-170 (2011) · Zbl 1231.35289 · doi:10.1016/j.amc.2011.05.084 |
| [19] | G. Li, D. Zhang, X. Jia and M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Probl., 29 (2013), 065014, 36 pp. · Zbl 1281.65125 |
| [20] | K. Liao and T. Wei, Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously, Inverse Probl., 35 (2019), 115002, 23 pp. · Zbl 1442.65230 |
| [21] | Y. C. Lin Xu, Finite difference/spectral approximation for the time-fractional diffusion equation, J. Comp. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [22] | C. J. Z. Liu Wen Zhang, Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation, Inverse Probl. Imag., 14, 1001-1024 (2020) · Zbl 1464.35399 · doi:10.3934/ipi.2020053 |
| [23] | Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351, 218-223 (2009) · Zbl 1172.35341 · doi:10.1016/j.jmaa.2008.10.018 |
| [24] | Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59, 1766-1772 (2010) · Zbl 1189.35360 · doi:10.1016/j.camwa.2009.08.015 |
| [25] | Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations, Fract. Calc. Appl. Anal., 14, 110-124 (2011) · Zbl 1273.35297 · doi:10.2478/s13540-011-0008-6 |
| [26] | Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15, 141-160 (2012) · Zbl 1276.26018 · doi:10.2478/s13540-012-0010-7 |
| [27] | D. W. Nie Deng, A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussian noise with Hurst index \(H\in(0, 1)\), SIAM J. Numer. Anal., 60, 1548-1573 (2022) · Zbl 07556062 · doi:10.1137/21M1422616 |
| [28] | D. Nie and W. Deng, An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion, J. Inverse Ill-Posed Probl., (2023). · Zbl 1526.35322 |
| [29] | D. Nie and M. Yang, Inverse Random Source Problem for Parabolic Equation, Ph.D thesis (Chinese), Southeast University, 2019. |
| [30] | R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. (b), 133, 425-430 (1986) · doi:10.1002/pssb.2221330150 |
| [31] | P. Niu, T. Helin and Z, Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Probl., 36 (42020), 045002, 23 pp. · Zbl 1469.35259 |
| [32] | I. Pdlubny, Fractional Differential Equations (1999) · Zbl 0924.34008 |
| [33] | H. Pollard, The completely monotonic character of the Mittag-Leffler function \(E_{\alpha}(-x)\), Bull. Amer. Math. Soc., 54, 1115-1116 (1948) · Zbl 0033.35902 · doi:10.1090/S0002-9904-1948-09132-7 |
| [34] | Z. J. X. Ruan Yang Lu, 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 |
| [35] | Z. Ruan and S. Zhang, Simultaneous inversion of time-dependent source term and fractional order for a time-fractional diffusion equation, J. Comput. Appl. Math., 368 (2020), 112566, 15 pp. · Zbl 1454.65108 |
| [36] | Z. W. Z. Ruan Zhang Wang, Simultaneous inversion of the fractional order and the space-dependent source term for the time-fractional diffusion equation, Appl. Math. Comput., 328, 365-379 (2018) · Zbl 1427.65228 · doi:10.1016/j.amc.2018.01.025 |
| [37] | S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, 1993. · Zbl 0818.26003 |
| [38] | L. Sun, Y. Li and Y. Zhang, Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation, Inverse Probl., 37 (2021), 055007, 26 pp. · Zbl 1462.35469 |
| [39] | J. Z.-X. C.-W. S.-J. Wen Liu Yue Wang, Landweber iteration method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation, J. Appl. Math. Comput., 68, 3219-3250 (2022) · Zbl 07597397 · doi:10.1007/s12190-021-01656-0 |
| [40] | M. Yamamoto, Uniqueness in determining fractional orders of derivatives and initial values, Inverse Probl., 37 (2021), 095006, 34 pp. · Zbl 1478.35227 |
| [41] | X. Yan, Z. Zhang and T. Wei, Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation, Chaos Solitons Fractals, 157 (2022), 111901, 17 pp. · Zbl 1498.35600 |
| [42] | Y. T. Y.-X. Zhang Wei Zhang, Simultaneous inversion of two initial values for a time-fractional diffusion-wave equation, Numer. Methods Partial Differential Eq., 37, 24-43 (2021) · Zbl 1535.65175 · doi:10.1002/num.22517 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
