Eur. Phys. J. Plus (2018) 133: 254
https://doi.org/10.1140/epjp/i2018-12080-4

Regular Article

A new stochastic computing paradigm for nonlinear Painlevé II systems in applications of random matrix theory

¹ Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock Campus, 43600, Attock, Pakistan
² Department of Mathematics, Preston University, Islamabad Campus, Kohat, Pakistan
³ Hamdard Institute of Engineering and Technology, Hamdard University, Islamabad, Pakistan
⁴ Department of Mathematics, University of Gujrat, 50700, Guirat, Pakistan
⁵ Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, 43600, Attock, Pakistan
⁶ Department of Mathematics, Cankaya University, Ankara, Turkey
⁷ Institute of Space Sciences, Magurele-Bucharest, Romania

^* e-mail: muhammad.asif@ciit-attock.edu.pk

Received: 17 April 2018
Accepted: 29 May 2018
Published online: 10 July 2018

Abstract

The aim of the present work is to investigate the stochastic numerical solutions of nonlinear Painlevé II systems arising from studies of two-dimensional Yang-Mills theory, growth processes through fluctuation formulas in statistical physics, soft-edge random matrix distributions using the strength of bio-inspired heuristics through artificial neural networks (ANNs), genetic algorithm (GA)-based evolutionary computations and interior-point techniques (IPTs). A new mathematical modelling of the system is formulated through ANNs by defining an error function that exactly satisfies the initial conditions. The weights of ANN models optimized through a memetic computing approach that is based on a global search with GAs, and IPTs are used for an efficient local search. The designed scheme is substantiated through comparative analysis with a fully explicit Range-Kutta numerical procedure on nonlinear Painlevé II systems by taking different magnitudes of forcing factors. The accuracy and convergence of the proposed scheme are validated through statistics performed on large numbers of simulations.