Linear Quaternion Differential Equations: Basic Theory and Fundamental Results | |
Kou, Kit Ian; Xia, Yong-Hui | |
2018-07 | |
Source Publication | STUDIES IN APPLIED MATHEMATICS |
ISSN | 0022-2526 |
Volume | 141Issue:1Pages:3-45 |
Abstract | Quaternion-valued differential equations (QDEs) are a new kind of differential equations which have many applications in physics and life sciences. The largest difference between QDEs and ordinary differential equations (ODEs) is the algebraic structure. Due to the noncommutativity of the quaternion algebra, the set of all the solutions to the linear homogenous QDEs is completely different from ODEs. It is actually a right-free module, not a linear vector space. This paper establishes a systematic frame work for the theory of linear QDEs, which can be applied to quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modeling, attitude dynamics, Kalman filter design, spatial rigid body dynamics, etc. We prove that the set of all the solutions to the linear homogenous QDEs is actually a right-free module, not a linear vector space. On the noncommutativity of the quaternion algebra, many concepts and properties for the ODEs cannot be used. They should be redefined accordingly. A definition of Wronskian is introduced under the framework of quaternions which is different from standard one in the ODEs. Liouville formula for QDEs is given. Also, it is necessary to treat the eigenvalue problems with left and right sides, accordingly. Upon these, we studied the solutions to the linear QDEs. Furthermore, we present two algorithms to evaluate the fundamental matrix. Some concrete examples are given to show the feasibility of the obtained algorithms. Finally, a conclusion and discussion ends the paper. |
DOI | 10.1111/sapm.12211 |
URL | View the original |
Indexed By | SCI |
Language | 英语 |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied |
WOS ID | WOS:000436804900001 |
Publisher | WILEY |
The Source to Article | WOS |
Fulltext Access | |
Citation statistics | |
Document Type | Journal article |
Collection | University of Macau |
Recommended Citation GB/T 7714 | Kou, Kit Ian,Xia, Yong-Hui. Linear Quaternion Differential Equations: Basic Theory and Fundamental Results[J]. STUDIES IN APPLIED MATHEMATICS,2018,141(1):3-45. |
APA | Kou, Kit Ian,&Xia, Yong-Hui.(2018).Linear Quaternion Differential Equations: Basic Theory and Fundamental Results.STUDIES IN APPLIED MATHEMATICS,141(1),3-45. |
MLA | Kou, Kit Ian,et al."Linear Quaternion Differential Equations: Basic Theory and Fundamental Results".STUDIES IN APPLIED MATHEMATICS 141.1(2018):3-45. |
Files in This Item: | ||||||
There are no files associated with this item. |
Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.
Edit Comment