THESIS
2018
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...[
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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.
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