Solution of Partial Differential Equations by Method of Hyperholomorphic functions

TL;DR

This paper generalizes the relationship between holomorphic functions and harmonic functions to hyperholomorphic functions in finite-dimensional commutative algebras, linking algebraic properties to PDE solutions.

Contribution

It introduces a framework connecting hyperholomorphic functions in commutative algebras with solutions to partial differential equations, extending classical complex analysis.

Findings

Components of hyperholomorphic functions satisfy specific PDEs

Polynomial equations in algebra basis imply PDE solutions

Generalization of holomorphic-harmonic function relationship

Abstract

It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a subspace of a commutative algebra satisfies a polynomial equation then components of a hyperholomorphic function on the subspace are solutions of the respective partial differential equation.