Electron. J. Diff. Eqns., Vol. 2009(2009), No. 77, pp. 1-13.

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

Dang Duc Trong, Nguyen Huy Tuan

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.

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Dang Duc Trong
Department of Mathematics and Informatics
Hochiminh City National University
227 Nguyen Van Cu, Q. 5, Hochiminh City, Vietnam
email: ddtrong@mathdep.hcmuns.edu.vn
Nguyen Huy Tuan
Department of Information Technology and Mathematics
Ton Duc Thang University
98 Ngo Tat To, Hochiminh City, Vietnam
email: tuanhuy_bs@yahoo.com

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