Abstract

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.