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

TL;DR

This paper introduces new formulas for solving direct and inverse Cauchy problems of the heat equation, emphasizing symmetry and potential for regularization in computational algorithms.

Contribution

It presents novel solution formulas for inverse problems that are symmetric to direct problem formulas and suitable for regularization, unlike classical methods.

Findings

Solution in Hermite polynomial series form

Symmetry between direct and inverse problem formulas

Formulas suitable for regularization algorithms

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.