A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

  • 1 Department of Mathematics and Statistics, University of North Carolina Charlotte, NC, 28233, Charlotte, USA
  • 2 Department of Mathematics and Statistics, University of North Carolina Charlotte, NC, 28233, Charlotte, USA
Phuong Mai Nguyen
  • Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC, 28233, USA
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and Loc Hoang NguyenORCID iD: https://orcid.org/0000-0002-0172-8816

Abstract

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

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