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Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

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Abstract

We present a neuro-heuristic computing platform for finding the solution for initial value problems (IVPs) of nonlinear pantograph systems based on functional differential equations (P-FDEs) of different orders. In this scheme, the strengths of feed-forward artificial neural networks (ANNs), the evolutionary computing technique mainly based on genetic algorithms (GAs), and the interior-point technique (IPT) are exploited. Two types of mathematical models of the systems are constructed with the help of ANNs by defining an unsupervised error with and without exactly satisfying the initial conditions. The design parameters of ANN models are optimized with a hybrid approach GA–IPT, where GA is used as a tool for effective global search, and IPT is incorporated for rapid local convergence. The proposed scheme is tested on three different types of IVPs of P-FDE with orders 1–3. The correctness of the scheme is established by comparison with the existing exact solutions. The accuracy and convergence of the proposed scheme are further validated through a large number of numerical experiments by taking different numbers of neurons in ANN models.

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Authors and Affiliations

  1. Department of Electrical Engineering, COMSATs Institute of Information Technology, Attock, 43200, Pakistan

    Muhammad Asif Zahoor Raja

  2. Department of Mathematics, University of Gujrat, Gujrat, 50700, Pakistan

    Iftikhar Ahmad

  3. Department of Mathematics, Preston University, Islamabad Campus, Kohat, Islamabad, 44000, Pakistan

    Imtiaz Khan

  4. Department of Mathematical Sciences, United Arab Emirates University, Al-Ain Box, 15551, UAE

    Muhammed Ibrahem Syam

  5. Department of Mathematics, Saint Xavier University, Chicago, IL, 60655, USA

    Abdul Majid Wazwaz

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  1. Muhammad Asif Zahoor Raja
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  2. Iftikhar Ahmad
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Correspondence to Muhammad Asif Zahoor Raja.

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ORCID: Muhammad Asif Zahoor RAJA, http://orcid.org/0000-0001-9953-822X

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Raja, M.A.Z., Ahmad, I., Khan, I. et al. Neuro-heuristic computational intelligence for solving nonlinear pantograph systems. Frontiers Inf Technol Electronic Eng 18, 464–484 (2017). https://doi.org/10.1631/FITEE.1500393

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