English abstract
The existing literature discusses different strategies to solve a scalar ordinary
differential equation using Lie point symmetries. This thesis focuses on three
of these strategies and discusses them in order to frame methods for finding solutions of non-linear systems of ordinary differential equations (ODEs). These
include Lie's integration theorem, the method of successive reduction of order
and the method of invariants of the admitted symmetry generators. The first
strategy based on Lie's integration theorem is applicable for a system of ODEs
admitting a solvable transitive Lie group. The second integration technique uses
the canonical forms of the generators in the space of the original variables. This
successively reduces the order of the system of ODEs provided that the admitted symmetry generators satisfy certain structure constants, allowing to have a
chain of derived sub-algebras of the admitted Lie algebra of the system. This
approach, which was earlier abandoned considering difficult to be applicable [41],
is successfully applied to finding solutions of the non-linear systems of ODEs. In
the third method, we investigate the use of invariant curves of the admitted Lie
groups of transformations in finding solutions of the systems of ODEs. Bluman's
theorem (1990) of invariant solutions for scalar ODEs is extended to design a
theorem which formulates conditions for systems of ODEs to admit invariant solutions. This method provides invariant solutions, if they exist, for system of
ODEs, without integrating the given system. Moreover, a system of ODEs can
be expressed in terms of differential invariants of the admitted symmetry generators from which the system may be reduced to an integrable form. We used
this approach successfully for finding solution of systems of second order ODEs.
Illustrative examples and those taken from mechanics are presented to highlight
the applications of these methods.
