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.