Abstract
This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.
Highlights
Let Ω ⊂ Rn (n ≥ 1) be a connected and bounded region, ∂Ω be the smooth boundary of Ω, TL x : H 2 (Ω) H01 (Ω) ⊂ L2 (Ω) → L2 (Ω) be a linear elliptic operator, which is densely defined, self-adjoint and positive-definite with regard to the variable x
Suppose that the eigenvalues of L x are λn (n ≥ 1), i.e., there is one nontrivial solution Xn ∈ L2 (Ω), and it satisfies the below boundary problem
This section establishes the conditional stability of problems (4), (5) by imposing the corresponding a-priori conditions for the exact solutions
Summary
Let Ω ⊂ Rn (n ≥ 1) be a connected and bounded region, ∂Ω be the smooth boundary of Ω, T. In [18], Hochstenbach-Reichel studied the ill-posed problems of discrete type by using a fractional Tikhonov regularization method. Inspired by the methods in [14,18], we develop a generalized-fractional Tikhonov-type regularization method to do with (4), (5), adopt the a-priori and a-posteriori rules to choose the regularized parameters, and give and prove some sharp estimates of convergence for our method. The proposed method is a generalization for the nonlocal boundary value problem method (or quasi-boundary value method), the boundary (or modified) Tikhonov method, and the generalized. Tikhonov method, it can be regarded as a modification on the fractional Tikhonov method The proofs of related Lemmas and Theorems are arranged in Appendix A
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
