Differential Equations and Mathematical Physics

Model problem of radial heating in a spherical layer with localized internal source

A. S. Zinchenko^a, A. M. Romanenkov<sup>ab

a</sup> Moscow Aviation Institute (National Research University), Moscow, 125993, Russian Federation
^b Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, 119333, Russian Federation

Abstract: This study presents a mathematical model for heat distribution in a spherical layer induced by a radially symmetric distributed heat source. The model is governed by an initial-boundary value problem for a linear parabolic equation in a spherically symmetric domain with three spatial variables, subject to thermal insulation boundary conditions.
By employing the method of separation of variables and exploiting radial symmetry, the three-dimensional problem is reduced to a one-dimensional formulation, yielding an exact analytical solution expressed as a convergent Fourier series. Explicit solutions for both homogeneous and inhomogeneous cases are derived by using the eigenfunctions of the associated Sturm–Liouville problem. Furthermore, the solution’s stability is rigorously established via a priori estimates.

Keywords: heat equation, spherical layer, Sturm–Liouville problem, separation of variables, Fourier series, a priori estimates, initial-boundary value problem

UDC: 517.956.4

MSC: 35K20, 35P10

Received: January 23, 2025
Revised: May 29, 2025
Accepted: June 23, 2025
First online: August 18, 2025

DOI: 10.14498/vsgtu2149

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