Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

Muhammad Asif Zahoor Raja, Iftikhar Ahmad, Imtiaz Khan, Muhammed Ibrahem Syam, Abdul Majid Wazwaz

Research output: Contribution to journalArticlepeer-review

41 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)464-484
Number of pages21
JournalFrontiers of Information Technology and Electronic Engineering
Issue number4
Publication statusPublished - Apr 1 2017


  • Functional differential equations (FDEs)
  • Genetic algorithms (GAs)
  • Initial value problems (IVPs)
  • Interior-point technique (IPT)
  • Neural networks
  • Unsupervised learning

ASJC Scopus subject areas

  • Signal Processing
  • Hardware and Architecture
  • Computer Networks and Communications
  • Electrical and Electronic Engineering


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