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

### Dang Duc Trong

National University, Hochiminh City, Vietnam### Pham Hoang Quan

National University, Hochiminh City, Vietnam### Tran Vu Khanh

National University, Hochiminh City, Vietnam### Nguyen Huy Tuan

National University, Hochiminh City, Vietnam

## Abstract

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.

## Cite this article

Dang Duc Trong, Pham Hoang Quan, Tran Vu Khanh, Nguyen Huy Tuan, A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate. Z. Anal. Anwend. 26 (2007), no. 2, pp. 231–245

DOI 10.4171/ZAA/1321