Shannon wavelet regularization methods for a backward heat equation

We consider the backward heat equation u x x ( x , t ) = u t ( x , t ) , − ∞ < x < ∞ , 0 ≤ t < T . The solution u ( x , t ) on the final value t = T is an known function g T ( x ) . This is a typical ill-posed problem, since the solution–if it exists–does not depend continuously on the final data. In this paper, we shall give a Shannon wavelet regularization method and obtain some quite sharp error estimates between the exact solution and the approximate solution defined in the scaling space V j .