Overcoming Non-Commutativity: New Methods for Linear Quaternion Differential Equations

The non-commutativity of quaternion multiplication presents a fundamental obstacle in analyzing linear quaternion-valued differential equations (QDEs). While the exponential solution for homogeneous linear QDEs by Campos and Mawhin is a cornerstone of the field, its reliance on a restrictive commutativity condition limits it to a narrow, complex-like subclass of functions. This work overcomes this limitation by introducing a novel algorithmic framework that solves the homogeneous initial value problem without any commutativity assumptions. We first demonstrate that the commutativity condition is equivalent to confining the dynamics to a complex-valued subspace. Our primary contribution is a method that systematically reduces the QDE to a solvable real nonlinear differential equation. We further derive closed-form solutions for key non-commutative cases. These results dramatically expand the solvable landscape of linear QDEs, with direct applications in control theory, quantum mechanics, and hypercomplex signal processing, where non-commutative dynamics are intrinsic. We demonstrate the power of our approach by applying it to Robinson’s Quaternion Kinematical Differential Equations and a problem in medical image communication security.