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Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

by Hongwu Zhang 1,*,† and Xiaoju Zhang 2,†
1
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2
Center for Faculty Development, North Minzu University, Yinchuan 750021, China
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(1), 48; https://doi.org/10.3390/math8010048
Received: 20 November 2019 / Revised: 22 December 2019 / Accepted: 23 December 2019 / Published: 1 January 2020
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems - Theory and Applications)
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. View Full-Text
Keywords: Cauchy problem; elliptic equation; regularization method; a-priori and a-posteriori convergence estimates; numerical simulation Cauchy problem; elliptic equation; regularization method; a-priori and a-posteriori convergence estimates; numerical simulation
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Zhang, H.; Zhang, X. Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation. Mathematics 2020, 8, 48.

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