Potentials for Solenoidal Fields Using the Three-Dimensional φ-Harmonic Cyclic Algebra

Abstract

Given a PDE, in [E. López-González, E. A. Martínez-García, and R. Torres-Córdoba, Chaos Solitons Fractals, 73, Article 113757 (2023)], the authors proposed a method for the construction of solutions by considering an associative real algebra A and a suitable affine vector field φ with respect to which the components of all functions ℒ ○ φ are solutions, where ℒ is differentiable in Lorch’s sense with respect to 𝔸. If we consider the 3D cyclic algebra and a suitable 3D affine map φ, then we get families of solutions for the Laplace equation with three independent variables.