A non-homogeneous generalization of Burgers equations

TL;DR

This paper extends the Burgers equation to inhomogeneous, multi-dimensional, and operator-based forms, establishing links with stochastic processes and geometric structures, and deriving exact solutions using advanced mathematical techniques.

Contribution

It introduces a non-homogeneous, multi-dimensional generalization of Burgers equations using operator methods and geometric analysis, providing new exact solutions and connections to stochastic processes.

Findings

Established relationships between solutions of hyperbolic Brownian motion and Burgers equations.

Derived exact solutions involving Hermite polynomials and special functions.

Extended solutions to Riemannian and pseudo-Riemannian manifolds, including Schwarzschild space.

Abstract

In this article we study generalizations of the inhomogeneous Burgers equation. First at the operator level, in the sense that we replace classical differential derivations by operators with certain properties, and then we increase the spatial dimensions of the Burgers equation, which is usually studied in one spatial dimension. This allows us, in one dimension, to find mathematical relationships between solutions of hyperbolic Brownian motion and the Burgers equations, which usually study the behaviour of mechanical fluids, and also, through appropriate transformations, to obtain in some cases exact solutions that depend on Hermite polynomials composed of appropriate functions. In the multi-dimensional case, this generalization allows us, by means of the method of invariant spaces, to find exact solutions on Riemannian and pseudo-Riemannian varieties, such as Schwarzschild and Ricci Solitons space, with time dictated by fractional derivatives, such as a Caputo-type operator of fractional evolution.