Lie point symmetries of the biharmonic heat equation on surfaces of revolution

TL;DR

This paper applies Lie symmetry analysis to classify symmetries of the biharmonic heat equation on surfaces of revolution, revealing structural similarities with the harmonic heat equation and deriving exact solutions for various curvatures.

Contribution

It demonstrates that the biharmonic heat equation shares the same Lie symmetries as the harmonic heat equation on surfaces of revolution, providing new insights and methods for solving these equations.

Findings

Biharmonic heat equation admits the same Lie symmetries as the harmonic heat equation.

Derived similarity reductions lead to invariant solutions.

Constructed explicit solutions on surfaces with different Gaussian curvatures.

Abstract

This paper uses Lie symmetry analysis to investigate the biharmonic heat equation on a generalized surface of revolution. We classify the Lie point symmetries associated with this equation, allowing for the identification of surfaces and the corresponding infinitesimal generators. In a significant move, we demonstrate that the biharmonic heat equation on a surface of revolution admits the same Lie symmetries as the harmonic heat equation on the same surface, highlighting a profound structural relationship between the two equations. Utilizing these symmetry groups, we derive similarity reductions that yield invariant forms of the equation and facilitate the construction of exact solutions. Finally, we provide certain examples illustrating precise solutions on the related surfaces with positive, negative, and zero Gaussian curvatures, demonstrating the versatility of the approach. This work contributes to the understanding of biharmonic heat equations on symmetric surfaces.