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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)

Abstract

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.

Keywords

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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References

  1. 1.
    Bassom, A.P., Clarkson, P.A., Hicks, A.C.: Numerical studies of the fourthPainlevè equation. IMA Journal of Applied Mathematics 50, 139–167 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    He, J.H.: Variational iteration method, a kind of non-linear analytical technique, some examples. Internat. J. Nonlin. Mech. 34(4), 699–708 (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    He, J.H.: Some asymptotic methods for strongly nonlinear equation. Int. J. N. Phy. 20(10), 1144–1199 (2006)Google Scholar
  4. 4.
    Wazwaz, A.M.: A first course in integral equations. WSPC, New Jersey (1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    Tajiri, M., Kawamoto, S.H.: Reduction of Kdv and cylindrical KdV equationsto Painlevè equation. Journal of the Physical Society of Japan 51, 1678–1681 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wazwaz, A.M.: Construction of solitary wave solution and rational solutionsfor the KdV equation by ADM. Chaos, Solution and Fractals 12, 2283–2293 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ghorbani, A., Nadjfi, J.S.: He’s homotopy perturbation method for calculating Adomian’s polynomials. Int. J. Nonlin. Sci. Num. Simul. 8(2), 229–332 (2007)CrossRefGoogle Scholar
  8. 8.
    Biazar, J., Babolian, E., Islam, R.: Solution of the system of ordinary differential equations by Adomian decomposition method. Applied Math. Comput. 147(3), 713–719 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    He, J.H.: Variational approach to the Thomas-Fermi equation. Appl. Math. Comput. 143, 533–535 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Golbabai, A., Keramati, B.: Solution of non-linear Fredholm integralequations of the first kind using modified homotopy perturbation. Chaos, Solution and Fractals 5, 2316–2321 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Aarts, L.P., Veer, P.V.D.: Neural Network Method for solving the partial Differential Equations. Neural Processing Letter 14, 261–271 (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Rarisi, D.R., et al.: Solving differential equations with unsupervised neural networks. J. Chemical Engineering and Processing 42, 715–721 (2003)CrossRefGoogle Scholar
  13. 13.
    Raja, M.A.Z., Khan, J.A., Qureshi, I.M.: A new Stochastic Approach for Solution of Riccati Differential Equation of Fractional Order. Ann. Math. Artif. Intell. 60(3-4), 229–250 (2010), doi:10.1007/s10472-010-9222-xMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Raja, M.A.Z., Khan, J.A., Qureshi, I.M.: Solution of fractional order system of Bagley-Torvik equation using Evolutionary computational intelligence. Mathematical Problems in Engineering, 1–18 (2011), doi:10.1155/2011Google Scholar
  15. 15.
    Zahoor Raja, M.A., Khan, J.A., Qureshi, I.M.: Evolutionary Computational Intelligence in Solving the Fractional Differential Equations. In: Nguyen, N.T., Le, M.T., Świątek, J. (eds.) ACIIDS 2010, Part I. LNCS (LNAI), vol. 5990, pp. 231–240. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Raja, M.A.Z., Khan, J.A., Qureshi, I.M.: Swarm Intelligent optimized neural networks for solving fractional differential equations. International Journal of Innovative Computing, Information and Control 7(11), 6301–6318Google Scholar
  17. 17.
    Khan, J.A., Raja, M.A.Z., Qureshi, I.M.: Stochastic Computational Approach for Complex Non-linear Ordinary Differential Equations. Chin. Phys. Lett. 28(2), 020206 (2011), doi:10.1088/0256-307XMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by Simulated Annealing. Science, New Series 220(4598), 671–680 (1983)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kroumov, V., Yu, J., Shibayama, K.: 3D Path Planning for mobile robots using simulated annealing neural network. International Journal of Innovative Computing, Information and Control 6(7), 2885–2899 (2007)Google Scholar
  20. 20.
    Hooke, R., Jeeves, T.A.: Direct search solution of numerical and statistical problems. Journal of the Association for Computing Machinery (ACM) 8(2), 212–229 (1961)CrossRefzbMATHGoogle Scholar
  21. 21.
    Dolan, E.D., Lewis, R.M., Torczon, V.J.: On the local convergence of pattern search. SIAM Journal on Optimization 14(2), 567–583 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Torczon, V.J.: On the convergence of pattern search algorithms. SIAM Journal on Optimization 7(1), 1–25 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Karmarkar, N.: A New Polynomial Time Algorithm for Linear Programming. Combinatoric 4(4), 373–395 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wright, S.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997) ISBN 0-89871-382-XCrossRefzbMATHGoogle Scholar
  25. 25.
    Wright, M.H.: The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull. Amer. Math. Soc. (N.S) 42, 39–56 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    García, P., Mingo, J.D., Carro, P.L., Valdovinos, A.: Efficient Feedforward Linearization Technique Using Genetic Algorithms for OFDM Systems. Journal on Advances in Signal Processing 1(354030), 10 (2010), doi:10.1155/2010/354030Google Scholar
  27. 27.
    Reeves, C.R., Rowe, J.E.: Genetic algorithms principles and perspective: A guide to GA Theory. Kluwer Academic Publishers, Norwell (2003)zbMATHGoogle Scholar
  28. 28.
    Hesameddini, E., Peyrovi, A.: The use of variational iteration method andHomotopy perturbation method for Painlevé equation I. Applied Mathematics Sciences 3, 1861–1871 (2009)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Behzadi, S.S.: Convergence of Iterative Methodsfor Solving Painlevé Equation. Applied Mathematical Science 4(30), 1489–1507 (2010)MathSciNetzbMATHGoogle Scholar

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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