A UNIFIED ANALYSIS OF LINEAR QUATERNION DYNAMIC EQUATIONS ON TIME SCALES

Dong Cheng; Kit Ian Kou; Yong Hui Xia

doi:10.11948/2018.172

2018 Volume 8 Issue 1

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Dong Cheng, Kit Ian Kou, Yong Hui Xia. A UNIFIED ANALYSIS OF LINEAR QUATERNION DYNAMIC EQUATIONS ON TIME SCALES[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 172-201. doi: 10.11948/2018.172

Citation:

Dong Cheng, Kit Ian Kou, Yong Hui Xia. A UNIFIED ANALYSIS OF LINEAR QUATERNION DYNAMIC EQUATIONS ON TIME SCALES[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 172-201. doi: 10.11948/2018.172

A UNIFIED ANALYSIS OF LINEAR QUATERNION DYNAMIC EQUATIONS ON TIME SCALES

1 Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, China;
2 School of Mathematical Sciences, Huaqiao University, 362021, Quanzhou, Fujian, China

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Abstract

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 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 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.
- Dynamic systems on time scales /
- difference equations /
- fundamental solution matrix /
- quaternions
MSC: 34N05;34A30;39A06;20G20

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