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Zbl 1171.35488
Dang Duc Trong; Nguyen Huy Tuan
A nonhomogeneous backward heat problem: regularization and error estimates.
(English)
[J] Electron. J. Differ. Equ. 2008, Paper No. 33, 14 p., electronic only (2008). ISSN 1072-6691/e

Summary: We consider the problem of finding the initial temperature, from the final temperature, in the nonhomogeneous heat equation $$\displaylines{ u_t-u_{xx}= f(x,t),\quad (x,t)\in (0,\pi)\times (0,T),\cr u(0,t)= u(\pi,t)= 0, \quad (x,t) \in (0,\pi)\times(0,T). }$$ This problem is known as the backward heat problem and is severely ill-posed. Our goal is to present a simple and convenient regularization method, and sharp error estimates for its approximate solutions. We illustrate our results with a numerical example.
MSC 2000:
*35R30 Inverse problems for PDE
35K05 Heat equation
65M30 Improperly posed problems (IVP of PDE, numerical methods)
65M60 Finite numerical methods (IVP of PDE)
65M15 Error bounds (IVP of PDE)

Keywords: backward heat problem; ill-posed problem; nonhomogeneous heat equation; contraction principle

Cited in: Zbl pre06150149 Zbl 1178.35374

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