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

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