Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 77, pp. 1-13.

Title: Remarks on a 2-D nonlinear backward heat problem
       using a truncated Fourier series method

Authors: Dang Duc Trong  (Hochiminh City National Univ., Vietnam)
         Nguyen Huy Tuan (Ton Duc Thang Univ., Hochiminh, Vietnam)

Abstract:
 The inverse conduction problem arises when experimental measurements
 are taken in the interior of a body, and it is desired to calculate
 temperature  on the surface. We consider the problem of finding,
 from the final data $u(x,y,T)=\varphi(x,y)$, the initial data
 $u(x,y,0)$ of the temperature function
 $u(x,y,t)$, $(x,y) \in U\equiv (0,\pi)\times (0,\pi)$,
 $t\in [0,T]$ satisfying the nonlinear system
 $$\displaylines{
 u_t-\Delta u= f(x,y,t,u(x,y, t)),\quad (x,y,t)\in U\times (0,T),\cr
 u(0,y,t)= u(\pi,y,t)= u(x,0,t) = u(x,\pi,t) = 0,\quad
 (x,y,t) \in U\times(0,T).
 }$$
 This problem is known to be ill-posed, as the solution exhibits
 unstable dependence on the given data functions.
 Using the Fourier series method, we regularize the problem and
 to get some new error estimates.
 A numerical experiment is given.
 
Submitted December 11, 2008. Published June 16, 2009.
Math Subject Classifications: 35K05, 35K99, 47J06, 47H10.
Key Words: Backward heat problem; nonlinearly ill-posed problem;
           Fourier series; contraction principle.