# 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 $u(x,T)=φ(x)$, the temperature function $u(x,t),x∈(0,π),t∈[0,T]$ satisfies the following nonlinear system

$u_{t}−u_{xx}u(0,t) =f(x,t,u(x,t)),=u(π,t)=0, (x,t)∈(0,π)×(0,T)t∈(0,T). $

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