Reliable numerical treatment of nonlinear singular Flierl–Petviashivili equations for unbounded domain using ANN, GAs, and SQP

Abstract

In this paper, a novel intelligent computational approach is developed for finding the solution of nonlinear singular system governed by boundary value problems of Flierl–Petviashivili equations using artificial neural networks optimized with genetic algorithms, sequential quadratic programming technique, and their combinations. The competency of artificial neural network for universal function approximation is exploited in formulation of mathematical modelling of the equation based on an unsupervised error with specialty of satisfying boundary conditions at infinity. The training of the weights of the networks is carried out with memetic computing based on genetic algorithm used as a tool for reliable global search method, hybridized with sequential quadratic programming technique used as a tool for rapid local convergence. The proposed scheme is evaluated on three variants of the two boundary problems by taking different values of nonlinearity operators and constant coefficients. The reliability and effectiveness of the design approaches are validated through the results of statistical analyses based on sufficient large number of independent runs in terms of accuracy, convergence, and computational complexity. Comparative studies of the proposed results are made with state of the art analytical solvers, which show a good agreement mostly and even better in few cases as well. The intrinsic worth of the schemes is simplicity in the concept, ease in implementation, to avoid singularity at origin, to deal with strong nonlinearity effectively, and their ability to handle exactly traditional initial conditions along with boundary condition at infinity.