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.