Visualization of The Motion of Singularities for The Solutions of Nonlinear Partial Differential Equation via Computing Their Locations in Complex Plane: Algorithmic Approach

Abstract

Singularities often occur in solutions of partial differential equations, In this paper, two new algorithms have been presented here, by which one can visualize the motion of poles of solution of partial differential equation, through detecting the position of the poles as time varies. We start by solving the given partial differential equation by spectral method, then continue this solution into complex plane through Padѐ approximation, and then compute the singularity of the resulting solution. Subsequently, we apply both algorithms to some Cauchy problems of Constantin-Lax-Majda and Burgers and Sharma – Tasso – Olver equations.