Equivalence and Symmetries for Linear Parabolic Equations and Applications Revisited

TL;DR

This paper provides a comprehensive analysis of symmetries and transformations of second order linear parabolic PDEs, introducing new criteria for reduction to the heat equation and exploring applications in heat kernels and integral transforms.

Contribution

It introduces a unified approach to symmetries of parabolic equations, establishes new criteria for reduction to the heat equation, and re-examines applications with a broader perspective.

Findings

Derived Appell type transformations and Mehler's kernel in any dimension.

Established necessary and sufficient criteria for reduction to the heat equation.

Re-examined applications including heat kernels and Lie symmetries in recent literature.

Abstract

A systematic and unified approach to transformations and symmetries of general second order linear parabolic partial differential equations is presented. Equivalence group is used to derive the Appell type transformations, specifically Mehler's kernel in any dimension. The complete symmetry group classification is re-performed. A new criterion which is necessary and sufficient for reduction to the standard heat equation by point transformations is established. A similar criterion is also valid for the equations to have a four- or six-dimensional symmetry group (nontrivial symmetry groups). In this situation, the basis elements are listed in terms of coefficients. A number of illustrative examples are given. In particular, some applications from the recent literature are re-examined in our new approach. Applications include a comparative discussion of heat kernels based on group-invariant solutions and the idea of connecting Lie symmetries and classical integral transforms introduced by Craddock and his coworkers. Multidimensional parabolic PDEs of heat and Schr\"odinger type are also considered.