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A Unified Analysis of Linear Quaternion Dynamic Equations on Time Scales
Keywords:Dynamic systems on Time scales, linear equations, fundamental solution matrix, Quaternions
      Over the last years, considerable attentions have 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 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 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 formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer's algorithms for evaluating the fundamental solution matrix of homogeneous QDETS are investigated. Furthermore, the variation of constants formula for solving the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.