  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.
