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A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation

  • Zewen Wang ORCID logo EMAIL logo , Shuli Chen , Shufang Qiu and Bin Wu ORCID logo
Published/Copyright: May 12, 2020

Abstract

This paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise.

MSC 2010: 35R30; 65M32

Award Identifier / Grant number: 11961002

Award Identifier / Grant number: 11761007

Award Identifier / Grant number: 11661004

Funding statement: This work is supported by National Natural Science Foundation of China (11961002, 11761007, 11661004), the Ground Project of Science and Technology of Jiangxi Universities (KJLD14051), the Foundation of Academic and Technical Leaders Program for Major Subjects in Jiangxi Province (20172BCB22019), Graduate Innovation Project of East China University of Technology (DHYC-201830).

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Received: 2018-10-17
Revised: 2019-07-31
Accepted: 2020-04-09
Published Online: 2020-05-12
Published in Print: 2020-08-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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