Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 33, pp. 1-14.
Title: A nonhomogeneous backward heat problem:
Regularization and error estimates
Authors: Dang Duc Trong (Hochiminh City National Univ., Vietnam)
Nguyen Huy Tuan (Hochiminh City National Univ., Vietnam)
Abstract:
We consider the problem of finding the initial temperature,
from the final temperature, in the nonhomogeneous heat equation
$$\displaylines{
u_t-u_{xx}= f(x,t),\quad (x,t)\in (0,\pi)\times (0,T),\\
u(0,t)= u(\pi,t)= 0, \quad (x,t) \in (0,\pi)\times(0,T).
}$$
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 06, 2008.
Math Subject Classifications: 35K05, 35K99, 47J06, 47H10.
Key Words: Backward heat problem; ill-posed problem;
nonhomogeneous heat equation; contraction principle.