Multilayer heat equations and their solutions via oscillating integral transforms

TL;DR

This paper introduces new oscillating integral transforms based on eigenfunction expansions to solve multilayer heat equations with moving boundaries, providing semi-analytical solutions for complex layered systems.

Contribution

The authors develop novel integral transforms for multilayer heat equations, offering alternative semi-analytical solutions that extend previous methods with different properties.

Findings

Constructed new oscillating integral transforms from eigenfunction expansions.

Applied transforms to solve multilayer heat equations with moving boundaries.

Provided semi-analytical solutions applicable to finance, physics, and mathematics.

Abstract

By expanding the Dirac delta function in terms of the eigenfunctions of the corresponding Sturm-Liouville problem, we construct some new (oscillating) integral transforms. These transforms are then used to solve various finance, physics, and mathematics problems, which could be characterized by the existence of a multilayer spatial structure and moving (time-dependent) boundaries (internal interfaces) between the layers. Thus, constructed solutions are semi-analytical and extend the authors' previous work (Itkin, Lipton, Muravey, Multilayer heat equations: application to finance, FMF, 1, 2021). However, our new method doesn't duplicate the previous one but provides alternative representations of the solution which have different properties and serve other purposes.