Equivalence and symmetries for variable coefficient linear heat type equations. II. Fundamental solutions

Abstract

We present a comparative study of fundamental solutions (heat kernels) of variable coefficient heat type partial differential equations based on Lie symmetry group methods and equivalence transformations discussed in the work of Güngör [J. Math. Phys. 59, 051507 (2018)]. Applications will include both one- and two-dimensional equations.

Notes on Lie symmetry group methods for differential equations

“…Definition 2.1. A local Lie group of point transformations G is called a symmetry group of the system of partial differential equations (11) if f = g.f is a solution whenever f is.…”

Calculation of the propagator of Schrödinger's equation on $(0,\infty)$ with the potential $kx^{-2} + ω^2x^2$ by Lie symmetry group method

“…For more details of this and another related method the reader is directed to Ref. [2]. We know from the results of [3] that this equation admits a four-dimensional Lie point symmetry algebra g, excluding the obvious infinite-dimensional one, because k = 0 (otherwise 6-dimensional).…”

“…Green function for the special potential when ω = 0 has already be obtained in Ref. [2] (page 6) using methods within the symmetry context (with k = −µ in notation of [2]). We reproduce it here for the purpose of reference (the replacement t → t/2 is done)…”

“…In another recent paper [2] with a reliance on the results of [3], we studied calculation of fundamental solutions of variable coefficient linear parabolic equations allowing sufficiently enough symmetry groups.…”

“…The purpose of this paper is to make use of methods from [2,3] as an alternative natural approach to the one pursued in [1] to derive fundamental solutions for the parabolic operator (1.1) and its Schrödinger variant.…”