A non-homogeneous generalization of Burgers equations

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.