An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion

References

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amsterdam) 140, Elsevier/Academic Press, Amsterdam, 2003. Search in Google Scholar

[2] A. Babaei and S. Banihashemi, Reconstructing unknown nonlinear boundary conditions in a time-fractional inverse reaction-diffusion-convection problem, Numer. Methods Partial Differential Equations 35 (2019), no. 3, 976–992. 10.1002/num.22334Search in Google Scholar

[3] I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis. Vol. II, North-Holland, Amsterdam (1991), 641–787. 10.1016/S1570-8659(05)80042-0Search in Google Scholar

[4] X. Bardina and M. Jolis, Multiple fractional integral with Hurst parameter less than 1 2 , Stochastic Process. Appl. 116 (2006), no. 3, 463–479. 10.1016/j.spa.2005.09.009Search in Google Scholar

[5] E. Barkai, Fractional Fokker–Planck equation, solution, and application, Phys. Rev. E 63 (2001), Article ID 046118. 10.1103/PhysRevE.63.046118Search in Google Scholar PubMed

[6] E. Barkai, R. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker–Planck equation, Phys. Rev. E 61 (2000), no. 1, 132–138. 10.1103/PhysRevE.61.132Search in Google Scholar

[7] E. Cancès, R. Chakir and Y. Maday, Numerical analysis of nonlinear eigenvalue problems, J. Sci. Comput. 45 (2010), no. 1–3, 90–117. 10.1007/s10915-010-9358-1Search in Google Scholar

[8] Y. Cao, J. Hong and Z. Liu, Approximating stochastic evolution equations with additive white and rough noises, SIAM J. Numer. Anal. 55 (2017), no. 4, 1958–1981. 10.1137/16M1056122Search in Google Scholar

[9] Y. Cao, J. Hong and Z. Liu, Finite element approximations for second-order stochastic differential equation driven by fractional Brownian motion, IMA J. Numer. Anal. 38 (2018), no. 1, 184–197. 10.1093/imanum/drx004Search in Google Scholar

[10] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations 22 (2006), no. 3, 558–576. 10.1002/num.20112Search in Google Scholar

[11] 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 Problems 36 (2020), no. 4, Article ID 045008. 10.1088/1361-6420/ab6503Search in Google Scholar

[12] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monogr. Math., Springer, Heidelberg, 2014. 10.1007/978-3-662-43930-2Search in Google Scholar

[13] M. Gunzburger, B. Li and J. Wang, Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise, Numer. Math. 141 (2019), no. 4, 1043–1077. 10.1007/s00211-019-01028-8Search in Google Scholar

[14] M. Gunzburger, B. Li and J. Wang, Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise, Math. Comp. 88 (2019), no. 318, 1715–1741. 10.1090/mcom/3397Search in Google Scholar

[15] X. Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Problems 35 (2019), no. 4, Article ID 045003. 10.1088/1361-6420/ab0138Search in Google Scholar

[16] B. Jin and Z. Zhou, An inverse potential problem for subdiffusion: Stability and reconstruction, Inverse Problems 37 (2021), no. 1, Article ID 015006. 10.1088/1361-6420/abb61eSearch in Google Scholar

[17] B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems 35 (2019), no. 6, Article ID 065004. 10.1088/1361-6420/ab109eSearch in Google Scholar

[18] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Appl. Math. (New York) 23, Springer, Berlin, 1992. 10.1007/978-3-662-12616-5Search in Google Scholar

[19] Z. Li, X. Cheng and G. Li, An inverse problem in time-fractional diffusion equations with nonlinear boundary condition, J. Math. Phys. 60 (2019), no. 9, Article ID 091502. 10.1063/1.5047074Search in Google Scholar

[20] C. Liu, J. Wen and Z. Zhang, Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation, Inverse Probl. Imaging 14 (2020), no. 6, 1001–1024. 10.3934/ipi.2020053Search in Google Scholar

[21] X. Liu and W. Deng, Higher order approximation for stochastic space fractional wave equation forced by an additive space-time Gaussian noise, J. Sci. Comput. 87 (2021), no. 1, Paper No. 11. 10.1007/s10915-021-01415-0Search in Google Scholar

[22] R. Metzler, J. Klafter and I. M. Sokolov, Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended, Phys. Rev. E 58 (1998), 1621–1633. 10.1103/PhysRevE.58.1621Search in Google Scholar

[23] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Math. 1929, Springer, Berlin, 2008. 10.1007/978-3-540-75873-0Search in Google Scholar

[24] D. Nie and W. Deng, A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussian noise with Hurst index H ∈ ( 0 , 1 ) , SIAM J. Numer. Anal. 60 (2022), no. 3, 1548–1573. 10.1137/21M1422616Search in Google Scholar

[25] P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems 36 (2020), no. 4, Article ID 045002. 10.1088/1361-6420/ab532cSearch in Google Scholar

[26] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[27] J. Prüss, Evolutionary Integral Equations and Applications, Springer, Basel, 2012. 10.1007/978-3-0348-0499-8Search in Google Scholar

[28] S. Qasemi, D. Rostamy and N. Abdollahi, The time-fractional diffusion inverse problem subject to an extra measurement by a local discontinuous Galerkin method, BIT 59 (2019), no. 1, 183–212. 10.1007/s10543-018-0731-zSearch in Google Scholar

[29] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. 10.1016/j.jmaa.2011.04.058Search in Google Scholar

[30] T. Tran Ngoc, T. Nguyen Huy, T. Pham Thi Minh, M. Mach Nguyet and C. Nguyen Huu, Identification of an inverse source problem for time-fractional diffusion equation with random noise, Math. Methods Appl. Sci. 42 (2019), no. 1, 204–218. 10.1002/mma.5334Search in Google Scholar

[31] N. H. Tuan, V. C. H. Luu and S. Tatar, An inverse problem for an inhomogeneous time-fractional diffusion equation: A regularization method and error estimate, Comput. Appl. Math. 38 (2019), no. 2, Paper No. 32. 10.1007/s40314-019-0776-xSearch in Google Scholar

[32] X. Wu, Y. Yan and Y. Yan, An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise, Appl. Numer. Math. 157 (2020), 69–87. 10.1016/j.apnum.2020.05.014Search in Google Scholar

[33] L. Yan and X. Yin, Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 2, 615–635. 10.3934/dcdsb.2018199Search in Google Scholar

[34] X. B. Yan and T. Wei, Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach, J. Inverse Ill-Posed Probl. 27 (2019), no. 1, 1–16. 10.1515/jiip-2017-0091Search in Google Scholar

[35] Z. Zhang and Z. Zhou, Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math. 82 (2017), no. 3, 579–600. 10.1093/imamat/hxx004Search in Google Scholar