Abstract:
The Painleve' equation and their solutions are used in describing various phenomina in plasma
physics, nonlinear waves, quantum field theory, general relativity, nonlinear optics, fibre optics, etc.
Paul Painlevé, French mathematician and politician built Poincaré's work in investigating
nonlinear second order ordinary differential equations with or without singularities to classify their
analytic properties. In the process, he discovered the Painlevé transcendents, written in terms of
entire functions, are solution to nonlinear second order ordinary differential equations with the
Painlevé property, the solutions are free from movable singularities. This property is a strong
indicator of the integrability. Here, we consider about solving the first and second Painlevé
equations by using a new computational approach based on Python programs. First, the solutions
of first and second Painlevé equations are expressed as Laurent series expansions. Then recurrence
relation is obtained for each equation. Finally, solutions are obtained with use of aforementioned
recurrence relations and Python codes.
