Liouvillian solutions of certain differential equations

Abstract

In this thesis, we apply the Kovacic’s algorithm, a tool that is developed from differential Galois theory, to determine whether the Whittaker-Ince equation, ellipsoidal wave equation and the Picard-Fuchs equation of a K3 surface have Liouvillian solutions or not. We have determined the necessary and sufficient conditions of having Liouvillian solutions for the Whittaker-Ince equation when one parameter is equal to zero. Also, we will give a sufficient condition of having a Liouvillian solution for the Whittaker-Ince equation when this parameter is non-zero. On the other hand, we have discovered that the ellipsoidal wave equation has no Liouvillian solution. We generalize a Picard-Fuchs equation for certain K3 surface and show that a particular case of the Picard-Fuchs equation cannot have any Liouvillian solutions.

Date of Award	2018
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology