Abstract
Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and it has wide applications in physics and life science. This paper establishes a systematic frame work for the theory of linear quaternion dynamic equations on time scales (QDETS), which can be applied to wave phenomena modeling, fluid dynamics and filter design. The algebraic structure of the solutions to the QDETS is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties of the classical dynamic equations on time scales (DETS) can not be applied. They should be redefined accordingly. Using $q$-determinant, a novel definition of Wronskian is introduced under the framework of quaternions which is different from the standard one in DETS. Liouville's formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer's algorithms to evaluate the fundamental solution matrix for homogeneous QDETS are presented. Furthermore, the variation of constants formula to solve the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.
Highlights
The theory of dynamic equations on time scales (DETS) has enormous applications [7, 29]
The main purpose of this paper is to study the basic theory of linear quaternion dynamic equations on time scales (QDETS)
Employing the newly established Wronskian and Liouville’s formula for QDETS, we obtain the algebraic structure of general solutions of n dimensional linear QDETS
Summary
The theory of dynamic equations on time scales (DETS) has enormous applications [7, 29]. We will show that the quaternion exponential function, in general, is not the solution of one-dimensional homogeneous linear QDETS: y∆ = p(t)y, unless p(t) is either a real-valued function or a quaternion constant function This is a striking difference between DETS and QDETS. Employing the newly established Wronskian and Liouville’s formula for QDETS, we obtain the algebraic structure of general solutions of n dimensional linear QDETS Explicit formulations of the fundamental solution matrices (in particular, eAt) for the linear QDEs with quaternion constant coefficient matrix were derived in [27]. Some conclusions are drawn at the end of the paper
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