On a new formulas for a direct and inverse Cauchy problems of heat equation

Abstract

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.