Academic Report by Xia Yonghui
Topic: Linear Quaternion Differential Equations: Basic Theory and Fundamental Results
Time: 9:00-11:00, May 13, 2019
Place: Conference Hall of Teaching Building 145 at the new campus of Central South University
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 report shall discuss 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 lecture.
Xia Yonghui, a distinguished professor and doctoral supervisor of Zhejiang Normal University, winner of 3 science and technology awards at the provincial and ministerial levels, once selected as Distinguished Professor of Minjiang Scholars in Fujian Province. In recent years, he has presided over 3 projects that were funded by National Natural Science Foundation of China (including 2 general projects) and has published over 60 academic papers in SCI journals of this discipline such asProc. Amer. Math. Soc.,J. Differential Equations,SIAM J. Appl. Math.,Studies. Appl. Math.,Proc. Edinburgh Math. Soc.,Phys. Rew. E.andScience in China.He has systematically established the basic framework of quaternion-valued differential equations (QDEs) and improved the major results of non-autonomous Hartman-Grobman linearization. He, along with his coworkers, has expanded the classic theoryon sufficient and necessary conditions for integrability of two-dimensional planar systems by Poincare and Lyapunov tospaces of all finite dimensions, and also improved a classic theory of generalized Lineard system by Professor Zhang Zhifen and expanded her results to the discontinuous system.