Constructive solutions of the heat equation with Stieltjes derivatives

TL;DR

This paper develops a constructive approach to solving the one-dimensional heat equation using Stieltjes calculus, extending to higher dimensions with multivariable derivators.

Contribution

It introduces a novel framework for solving the heat equation with Stieltjes derivatives and constructs explicit solutions for various boundary conditions and higher-dimensional cases.

Findings

Established existence of solutions using a constructive method.

Derived explicit solutions for specific classes of derivators.

Extended the framework to multivariable derivators for higher dimensions.

Abstract

In this work, we investigate the one-dimensional heat equation within the framework of Stieltjes calculus. We first consider the equation associated with two fixed derivators and develop a constructive approach to establish the existence of solutions. We then study the corresponding initial value problem and incorporate several types of boundary conditions. Finally, we introduce a notion of multivariable derivator, suitable for higher-dimensional settings, and obtain explicit solutions of the heat equation for relevant classes of such derivators.