Expansion of the initial-boundary value problem solution for the heat equation into Hermite polynomial series

A. I. Nizhnikov^a, O. E. Yaremko^b, N. N. Yaremko<sup>c

a</sup> Moscow State Pedagogical University (Moscow)
^b Moscow State Technical University “Stankin” (Moscow)
^c National Research Technological University “MISiS” (Moscow)

Abstract: The initial boundary value problems solution for the heat equation is represented as a decomposition into a series of Hermite polynomials. For the Cauchy problem and the Cauchy retrospective problem, the coefficients of the solution expansion into series by Hermite polynomials are found. The relationship between the Laplace transform and the Hermite polynomial series is investigated. A new formula for inverting the integral Laplace transform with respect to the values of the image on the real half- axis is found. The original function is constructed as the sum of a quasi-series. A formula is obtained for the Laplace representation as the sum of a quasi-minor series.The problem of reconstructing the temperature field of an unlimited rod based on its moments has been solved.

Keywords: Hermite polynomial, Fourier transform, Laplace transform, Cauchy problem, retrospective problem, initial-boundary value problem.

UDC: 517.44

Received: 02.02.2025
Revised: 27.08.2025

DOI: 10.22405/2226-8383-2025-26-3-174-184