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A nonhomogeneous backward heat problem: regularization and error estimates. (English) Zbl 1171.35488
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.

35R30Inverse problems for PDE
35K05Heat equation
65M30Improperly posed problems (IVP of PDE, numerical methods)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
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