Electron. J. Diff. Eqns., Vol. 2008(2008), No. 33, pp. 1-14.

A nonhomogeneous backward heat problem: Regularization and error estimates

Dang Duc Trong, Nguyen Huy Tuan

Abstract:
We consider the problem of finding the initial temperature, from the final temperature, in the nonhomogeneous heat equation
$$\displaylines{ u_t-u_{xx}= f(x,t),\quad (x,t)\in (0,\pi)\times (0,T),\cr u(0,t)= u(\pi,t)= 0, \quad (x,t) \in (0,\pi)\times(0,T). }$$
This problem is known as the backward heat problem and is severely ill-posed. Our goal is to present a simple and convenient regularization method, and sharp error estimates for its approximate solutions. We illustrate our results with a numerical example.

Submitted November 9, 2007. Published March 6, 2008.
Math Subject Classifications: 35K05, 35K99, 47J06, 47H10.
Key Words: Backward heat problem; ill-posed problem; nonhomogeneous heat equation; contraction principle.

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Dang Duc Trong
Department of Mathematics and Computer Sciences
Hochiminh City National University
227 Nguyen Van Cu, Hochiminh City, Vietnam
email: ddtrong@mathdep.hcmuns.edu.vn
Nguyen Huy Tuan
Department of Information Technology and Applied Mathematics
Ton Duc Thang University
98 Ngo Tat To, Hochiminh City, Vietnam
email: tuanhuy_bs@yahoo.com

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