Sun, Liangliang; Zhang, Zhaoqi Exponential Tikhonov regularization method for an inverse source problem in a sub-diffusion equation. (English) Zbl 07958102 Taiwanese J. Math. 28, No. 6, 1111-1136 (2024). Summary: In this paper, it is considered an inverse space-dependent source problem of time-space fractional diffusion equation from the noisy final data in a bounded domain. Such a problem is mildly ill-posed. A new regularization method called the exponential Tikhonov method with a parameter \(\gamma\) is utilized to solve the problem, and its convergence rates are analyzed under an a-priori and an a-posteriori regularization parameter choice rule. A novel result indicates that the optimal convergence rate can be obtained and it is independent of the regularity information of the unknown source term when \(\gamma\) is less than or equal to zero. However, when \(\gamma\) is greater than zero, the optimal convergence rate depends on the value of \(\gamma\) related to the regularity of the unknown source but it does not have convergence saturation limit and can theoretically approach 1, which is superior to Tikhonov’s regularization framework using the usual Sobolev space norm as a penalty term in a minimized functional. Numerical examples show that the proposed regularization method is effective and stable, and both parameter choice rules work well. MSC: 35R30 Inverse problems for PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35R11 Fractional partial differential equations 35R25 Ill-posed problems for PDEs 65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs Keywords:convergence rate; exponential regularization method; ill-posed problem; inverse source problem; time-space fractional diffusion equation × Cite Format Result Cite Review PDF Full Text: DOI References: [1] W. Audeh and F. Kittaneh, Singular value inequalities for compact operators, Linear Algebra Appl. 437 (2012), no. 10, 2516-2522. · Zbl 1263.47018 [2] C. Çelik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys. 231 (2012), no. 4, 1743-1750. · Zbl 1242.65157 [3] M. Chang, L. Sun and Y. 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