Elifalet López-González

arXiv2023

Potentials for solenoidal fields using the three-dimensional $φ$-harmonic cyclic algebra

Given a PDE in [10] it is proposed a method for constructing solutions by considering an associative real algebra A, and a suitable affine vector field $φ$ with respect to which the components of all the functions $\mathcal{L}\circφ$ are solutions, where $\mathcal L$ is differentiable in the sense of Lorch with respect to A. When we consider the 3D cyclic algebra and a suitable 3D affine map $φ$ we get families of solutions for the Laplace equation with three independent variables.

math.AP↗

arXiv2021

Pre-twisted algebrizable differential equations

We introduce the \emph{$φ\mathbb{A}$-differentiability} for functions $f:U\subset \mathbb R^{k}\to\mathbb A$ where $\mathbb A$ is the linear space $\mathbb R^{n}$ endowed with an algebra product which is unital, associative, commutative, $U$ is an open set, and $φ:U\subset \mathbb R^{k}\to\mathbb A$ is a differentiable function in the usual sense. We also introduce the corresponding generalized Cauchy-Riemann equations ($φ\mathbb{A}$-CREs), the Cauchy-integral theorem, and the \emph{$φ\mathbb A$-differential equations}. The four-dimensional vector fields associated with triangular billiards are $φ\mathbb{A}$-differentiable. The \emph{$φ\mathbb A$-differential equations} can be used for constructing exact solutions of partial differential equations like the three-dimensional heat equation.

math.CA↗