Multilayer heat equations and their solutions via oscillating integral transforms

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 does not duplicate the previous one but provides alternative representations of the solution which have different properties and serve other purposes.