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)u(x,T)=\varphi(x), the temperature function u(x,t), x(0,π), t[0,T]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