Open Access

ARTICLE

Cauchy Problem for the Heat Equation in a Bounded Domain Without Initial Value

Ji-Chuan Liu¹, Jun-Gang Wang²
1 Corresponding author. Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, PR China. Email: liujichuan2003@126.com
2 Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, PR China.

Computer Modeling in Engineering & Sciences 2014, 97(5), 437-462. https://doi.org/10.3970/cmes.2014.097.437

Abstract

We consider the determination of heat flux within a body from the Cauchy data. The aim of this paper is to seek an approach to solve the onedimensional heat equation in a bounded domain without initial value. This problem is severely ill-posed and there are few theoretic results. A quasi-reversibility regularization method is used to obtain a regularized solution and convergence estimates are given. For numerical implementation, we apply a method of lines to solve the regularized problem. From numerical results, we can see that the proposed method is reasonable and feasible.

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Keywords

Ill-posed problem, Quasi-reversibility method, Method of lines, Finite difference, Convergence estimate.

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