  In this paper a solution of the direct Cauchy problems for heat equation is
founded in the Hermite polynomial series form. A well-known classical solution
of direct problem is represented in the Poisson integral form. The author shows
the formulas for the solution of the inverse Cauchy problems have a symmetry
with respect to the formulas for the corresponding direct problems. The
obtained solution formulas for the inverse problems can serve as a basis for
regularizing computational algorithms while well-known classical formula for
the solution of inverse problem did not possess such properties and can't be a
basis for regularizing computational algorithms.

Classical Analysis and ODEs