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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 26, Issue 2, 2007, pp. 231–245**

**DOI: 10.4171/ZAA/1321**

Published online: 2007-06-30

A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate

Dang Duc Trong^{[1]}, Pham Hoang Quan

^{[2]}, Tran Vu Khanh

^{[3]}and Nguyen Huy Tuan

^{[4]}(1) National University, Hochiminh City, Vietnam

(2) National University, Hochiminh City, Vietnam

(3) National University, Hochiminh City, Vietnam

(4) National University, Hochiminh City, Vietnam

We consider the problem of finding, from the final data $u(x,T)=\varphi(x)$, the temperature function $u(x,t),\ x\in (0,\pi),\ t\in [0,T]$ satisfies the following nonlinear system \vspace{-0.1cm} \begin{alignat*}{2} u_t-u_{xx}&= f(x,t,u(x,t)), &\quad &(x,t)\in (0,\pi)\times (0,T) \\ u(0,t)&= u(\pi,t)=0, &\quad &t\in (0,T). \end{alignat*} The nonlinear problem is severely ill-posed. We shall improve the quasi-boundary value method to regularize the problem and to get some error estimates. The approximation solution is calculated by the contraction principle. A numerical experiment is given.

*Keywords: *Backward heat problem, nonlinearly Ill-posed problem, quasi-boundary value methods, quasi-reversibility methods, contraction principle

Trong Dang Duc, Quan Pham Hoang, Khanh Tran Vu, Tuan Nguyen Huy: A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate. *Z. Anal. Anwend.* 26 (2007), 231-245. doi: 10.4171/ZAA/1321