The main subject of this paper is the regularization of the following homogeneous one-dimensional backwards heat equation with time-dependent coefficient:
It is assumed that the solution is given at final time with some noise, i. e., with is available, where denotes the norm on .
For the regularization of this problem, approximations of the form
are considered, where is arbitrary, but fixed, and , and denotes the Fourier transform of , i. e., . It is shown that
for small enough and a finite constant , provided that and for each , and is also required here.
A similar approach is presented for the regularization of a final value problem for the inhomogeneous backwards heat equation . The paper concludes with the presentation of some numerical results.