Solutions of quaternion-valued differential equations with or without commutativity

TL;DR

This paper investigates quaternion-valued differential equations, identifying conditions for commutativity and proposing methods to solve equations without this property, thus broadening the understanding of quaternion differential equations.

Contribution

It characterizes quaternion functions that satisfy commutativity and develops a reduction method for non-commutative cases to real nonlinear differential equations.

Findings

Commutativity in quaternion functions implies they are complex-like.

Without commutativity, the homogeneous equation reduces to a real nonlinear differential equation.

Provides new approaches for solving quaternion differential equations without commutativity.

Abstract

Most results on quaternion-valued differential equation (QDE) are based on J. Campos and J. Mawhin's fundamental solution of exponential form for the homogeneous linear equation, but their result requires a commutativity property. In this paper we discuss with two problems: What quaternion function satisfies the commutativity property? Without the commutativity property, what can we do for the homogeneous equation? We prove that the commutativity property actually requires quaternionic functions to be complex-like functions. Without the commutativity property, we reduce the initial value problem of the homogeneous equation to a real nonautonomous nonlinear differential equation.