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Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

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

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.

Key words

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

CLC number

TP183 O175 

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Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Muhammad Asif Zahoor Raja
    • 1
  • Iftikhar Ahmad
    • 2
  • Imtiaz Khan
    • 3
  • Muhammed Ibrahem Syam
    • 4
  • Abdul Majid Wazwaz
    • 5
  1. 1.Department of Electrical EngineeringCOMSATs Institute of Information TechnologyAttockPakistan
  2. 2.Department of MathematicsUniversity of GujratGujratPakistan
  3. 3.Department of MathematicsPreston University, Islamabad CampusKohat, IslamabadPakistan
  4. 4.Department of Mathematical SciencesUnited Arab Emirates UniversityAl-Ain BoxUAE
  5. 5.Department of MathematicsSaint Xavier UniversityChicagoUSA

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