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

Dang Duc Trong; Pham Hoang Quan; Tran Vu Khanh; Nguyen Huy Tuan

doi:10.4171/zaa/1321

JournalszaaVol. 26, No. 2pp. 231–245

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

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Abstract

We consider the problem of finding, from the final data $u (x, T) = φ (x)$ , the temperature function $u (x, t), x \in (0, π), t \in [0, T]$ satisfies the following nonlinear system

u_{t} - u_{xx} u (0, t) = f (x, t, u (x, t)), = u (π, t) = 0, (x, t) \in (0, π) \times (0, T) t \in (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