报告人：夏永辉教授

华侨大学

报告题目：Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

报告时间：2016年11月22日上午09:30

报告地点：海韵行政楼B313

Abstract:This paper establishes a systematic frame work for the theory of linear quaternion-valueddifferential equations (QDEs), which can be applied to quantum mechanics, Frenet frame in differential geometry, kinematic modelling, attitude dynamics, Kalman filter design, spatial rigid body dynamics and fluid mechanics, etc. On the non-commutativity of the quaternion algebra, the algebraic structure of the solutions to the QDEs is not a linear vector space. It is actually a right-free module.Moreover, many concepts and properties for the ordinary differential equations (ODEs) can not 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 ordinary differential equations. 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.

An algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented. The fundamental matrix can be constructed differently providing that the eigenvalues are simple and multiple eigenvalues. If the linear system has multiple eigenvalues, how to construct the fundamental matrix?In particular, if the number of independent eigenvectors might be less than the dimension of the system. That is, the numbers of the eigenvectors is not enough to construct a fundamental matrix. How to find the ``missing solutions"?

   Moreover, we presented an algorithm for finding a solution of the linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations.

报告人简介：夏永辉，男、博士、闽江学者特聘教授，现为华侨大学特聘教授，主要研究方向为微分方程和动力系统，研究兴趣包括微分方程的线性化理论、微分方程的周期解和稳定性、概周期微分方程、差分方程理论等方面。曾于2012年7月-2013年7月担任斯洛文尼亚Maribor大学研究员，于2015.7.1-2016.8.31担任澳门大学兼职研究人员。2009年度福建省科学技术奖三等奖1项（排名第一），2011年度浙江省科学技术奖一等奖1项（排名第三），2012年入选浙江省“新世纪151人才工程”，2013年获 “浙江省优秀科技工作者” 荣誉称号，2015年入选“中国高被引学者名单”。近年来主持国家自然科学基金3项（面上项目2项，青年项目各1项），主持浙江省自然科学基金2项，主持欧盟研究基金项目(MSCA-IF-2014-EF：Marie Curie Individual Fellowship) 1项。

联系人：张剑文教授

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