Reconstruct an unknown source term on the right hand side of the time-fractional diffusion equation with Caputo-Hadamard derivative
Keywords:
Caputo-Hadamard derivative, Inverse source problem, parabolic equation, memory term, regularization method.Abstract
This study considers an inverse source problem of a time-space fractional diffusion equation. In general this inverse problem is ill-posed in the sense of Hadamard. We provide a new non-stationary iterated quasi-boundary value regularization technique for reconstructing the source function. By choosing the regularization parameters (a priori; posterior decision rules), we determine the convergence rates.
References
B. D. Nghia, N. H. Luc, Xiaolan Qin, Yan Wang, On maximal solution to a degenerate parabolic equation involving in time fractional derivative. Electron. J. Appl. Math., 1 (1), 2023.
D. T. T. Xuan, V. T. T. Ha, Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data. Electron. J. Appl. Math., 1 (2), 2023.
X. Feng, L. Eld´en and C. Fu, A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous
Neumann data, J. Inverse Ill-Posed Probl., 18 (6): 617-645, 2010.
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(4): 045008, 30, 2020.
X. Feng, M. Zhao and Z. Qian, A Tikhonov regularization method for solving a backward time-space fractional diffusion problem. J. Comput. Appl. Math., 411: 114236, 20, 2022.
Y. Deng and Z. Liu, Iteration methods on sideways parabolic equations. Inverse Problems, 25(9), 095004, 14, 2005.
K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations. Math. Control Relat. Fields, 6(2): 251-269, 2016.
F. Yang and C. Fu, The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation. Appl. Math. Model., 39(5-6): 1500-1512, 2015.
K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. Journal of Mathematical Analysis and Applications, 382(1): 426-447, 2011.
K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations. Math. Control Relat. Fields, 6(2): 251-269, 2016.
T.Wei and J.Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem. ESAIM Math. Model. Numer. Anal., 48(2): 603-621, 2014.
T. Wei and J. Wang, A modified quasi-boundary value method for an inverse source problem of the time fractional diffusion equation. Appl. Numer. Math., 78: 95-111, 2014.
Yang, F., Cao, Y., & Li, X. X. (2023). Two regularization methods for identifying the source term of Caputo-Hadamard time-fractional diffusion equation. Mathematical Methods in the Applied Sciences, 46(15), 16170-16202.
F. Yang, C.L. Fu, A simplified Tikhonov regularization method for determining the heat source, Appl. Math. Model. 34 (2010), no. 11, 3286–3299.
F.F. Dou, C.L. Fu, F. Yang, Identifying an unknown source term in a heat equation Inverse Probl. Sci. Eng. 17 (2009), no. 7, 901–913.
Vo Thi Thanh Ha, Adv. Theory Nonlinear Anal. Appl. 7 (2023), 28–43. 42
Zhang, Y., & Feng, X. (2024). A nonstationary iterated quasi-boundary value method for reconstructing the source term in a time-space fractional diffusion equation. Journal of Computational and Applied Mathematics, 440, 115612.
M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math., 72 (1): 21-37, 1995.
J.Wang and T.Wei, An iterative method for backward time-fractional diffusion problem. Numer. Numer. Methods Partial Differ. Equ., 30(6): 2029-2041, 2014.
