ORIGINAL ARTICLE Numerical treatment of nonlinear singular Flierl–Petviashivili systems using neural networks models Muhammad Asif Zahoor Raja 1 • Junaid Ali Khan 2 • Aneela Zameer 3 • Najeeb Alam Khan 4 • Muhammad Anwaar Manzar 5 Received: 20 December 2016 / Accepted: 14 August 2017 Ó The Natural Computing Applications Forum 2017 Abstract In this study, new intelligent computing methodologies have been developed for highly nonlinear singular Flierl–Petviashivili (FP) problem having boundary condition at infinity by exploiting three different neural network models integrated with active-set algorithm (ASA). A modification in the modeling is introduced to cater the singularity, avoid divergence in results for unbounded inputs and capable of dealing with strong nonlinearity. Three models have been constructed in an unsupervised manner for solving the FP equation using log- sigmoid, radial basis and tan-sigmoid transfer functions in the hidden layers of the network. The training of adaptive adjustable variables of each model is carried out with a constrained optimization technique based on ASA. The proposed models have been evaluated on three variants of the two FP equations. The designed models have been examined with respect to precision, stability and com- plexity through statistics. Keywords Flierl–Petviashivili problem  Artificial neural networks  Radial basis function  Nonlinear ODEs  Nonlinear singular systems  Active-set algorithm 1 Introduction A well-known nonlinear singular ordinary differential equation named as Flierl–Petviashivili equation has been solved in this study using artificial intelligence method- ologies based on neural networks. The generic form of the equation with boundary conditions can be written as [1–3]: d 2 u dx 2 þ r x du dx  u n  u nþ1 ¼ 0; n  1; r  1 ð1Þ uð0Þ¼ a; duð0Þ dx ¼ uð1Þ ¼ 0; where n is nonlinear operator, r and a are real constants. The FP equation is special case of singular Emden–Fowler- type equation arises extensively in astrophysics and astronomy applications [4–7]. The research community exhibits special interest in these nonlinear singular systems in order to handle not only the singularity at origin x = 0 but also cope with boundary condition at infinity using Adomian decomposition method [8–11]. The problem of divergence in a large bounded range gives rise to Pade´ & Aneela Zameer aneelaz@pieas.edu.pk Muhammad Asif Zahoor Raja rasifzahoor@yahoo.com; muhammad.asif@ciit-attock.edu.pk Junaid Ali Khan junaid.ali@hitecuni.edu.pk Najeeb Alam Khan najeeb@uok.edu.pk Muhammad Anwaar Manzar anwaar.manzar@hamdard.edu.pk 1 Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, Pakistan 2 Department of Computer Science and Engineering, HITEC University Texila, Rawalpindi, Pakistan 3 Department of Computer and Information Science, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan 4 Department of Mathematical Sciences, University of Karachi, Karachi 75270, Pakistan 5 Hamdard Institute of Engineering and Technology, Hamdard University, Islamabad, Pakistan 123 Neural Comput & Applic DOI 10.1007/s00521-017-3193-3