Numerical Treatment for Painlevé Equation I Using Neural Networks and Stochastic Solvers

  • Muhammad Asif Zahoor Raja
  • Junaid Ali Khan
  • Siraj-ul-Islam Ahmad
  • Ijaz Mansoor Qureshi
Part of the Studies in Computational Intelligence book series (SCI, volume 442)


In this chapter, a new stochastic numerical treatment is presented for solving Painlevé I equation. The mathematical model of the equation is formulated with feed-forward artificial neural networks. Linear combination of the networks defines the unsupervised error for the equation. The error is reduced subject to the availability of appropriate weights of networks. Training of weights is done with genetic algorithm, simulating annealing and pattern search algorithms hybridized with interior point algorithm for rapid local search. The reliability and effectiveness is validated with the help of statistical analysis. Comparison of results is made with standard approximate analytic solvers of the equation. It is found that the proposed results are in a good agreement with their corresponding numerical solutions.


Genetic Algorithm Fractional Differential Equation Pattern Search Homotopy Perturbation Method Variational Iteration Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Berlin Heidelberg 2013

Authors and Affiliations

  • Muhammad Asif Zahoor Raja
    • 1
    • 4
  • Junaid Ali Khan
    • 1
    • 4
  • Siraj-ul-Islam Ahmad
    • 2
  • Ijaz Mansoor Qureshi
    • 3
  1. 1.Department of Electronic EngineeringInternational Islamic University IslamabadIslamabadPakistan
  2. 2.Pakistan Institute of Engineering and Applied SciencesIslamabadPakistan
  3. 3.Department of Electrical EngineeringAir UniversityIslamabadPakistan
  4. 4.Department of Electrical EngineeringCOMSATS Institute of Information TechnologyAttockPakistan

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