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

Abstract

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 $\phi$ with respect to which the components of all the functions $\mathcal{L}\circ \phi$ 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 $\phi$ we get families of solutions for the Laplace equation with three independent variables.