Neural Computing and Applications

, Volume 26, Issue 5, pp 1055–1071 | Cite as

Comparison of three unsupervised neural network models for first Painlevé Transcendent

  • Muhammad Asif Zahoor Raja
  • Junaid Ali Khan
  • Syed Muslim Shah
  • Raza Samar
  • Djilali Behloul
Original Article


In this paper, a reliable soft computing framework is presented for the approximate solution of initial value problem (IVP) of first Painlevé equation using three unsupervised neural network models optimized with sequential quadratic programming (SQP). These mathematical models are constructed in the form of feed-forward architecture including log-sigmoid, radial base and tan-sigmoid activation functions in the hidden layers. The optimization of designed parameters for each model is performed with SQP, an efficient constraint optimization problem-solving algorithm. The designed methodology is tested on the IVP, and comparative study is carried out with standard solution based on numerical and analytical solvers. The accuracy, convergence and effectiveness of the schemes are validated on the given benchmark problem by large number of simulations and their comprehensive analysis.


Painlevé Transcendents Artificial neural network Sequential quadratic programming Nonlinear differential equations Activation functions Unsupervised learning 


  1. 1.
    Bassom AP, Clarkson PA, Hicks AC (1993) Numerical studies of the fourth Painlevé equation. IMA J Appl Math 50(2):167–193MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    He JH (1999) Variational iteration method, a kind of non-linear analytical technique, some examples. Int J Nonlin Mech 34(4):699–708MATHCrossRefGoogle Scholar
  3. 3.
    He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int JModern Phys B 20(10):1144–1199Google Scholar
  4. 4.
    Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, Studies in Applied Mathematics 4, PhiladelphiaGoogle Scholar
  5. 5.
    Turcotte DL, Spence DA, Bau HH (1982) Multiple solutions for natural convective flows in an internally heated, vertical channel with viscous dissipation and pressure work. Int J Heat Mass Transfer 25(5):699–706MATHCrossRefGoogle Scholar
  6. 6.
    Haberman R (1979) Slowly varying jump and transition phenomena associated with algebraic bifurcation problems. SIAM J Appl Math 37:69–106MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fokas AS, Its AR, Kitaev AV (1991) Discrete Painlevé equations and their appearance in quantum gravity. Comm Math Phys 142(2):313–344MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Paniak LD, Szabo RJ (2001) Fermionic quantum gravity. Nucl Phys B 593:671–725MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Tajiri M, Kawamoto Sh (1982) Reduction of KdV and cylindrical KdV equations to Painlevé equation. J Phys Soc Jpn 51:1678–1681MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lee S-Y, Teodorescu R, Wiegmann P (2011) Viscous shocks in Hele-Shaw flow and Stokes phenomena of Painlevé I transcendent. Phys D. doi: 10.1016/j.physd.2010-09.017
  11. 11.
    Dai D, Zhang L (2010) On Tronqueé solutions if the first Painlevé Hierarchy. J Math Anal Appl 368(2):393–399MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Olde Daalhuis AB (2005) Hyperasymptotics for nonlinear ODEs II. The first Painlevé equation and a second-order Riccati equation. Proc R Soc A 461:3005–3021Google Scholar
  13. 13.
    Hesameddini E, Peyrovi A (2009) The use of variational iteration method and Homotopy perturbation method for Painlevé equation I. Appl Math Sci 3:1861–1871MATHMathSciNetGoogle Scholar
  14. 14.
    Behzadi SS (2010) Convergence of iterative methods for solving Painlevé Equation. Appl Math Sci 4(30):1489–1507MATHMathSciNetGoogle Scholar
  15. 15.
    Raja MAZ, Khan JA, Haroon T (2014) Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks, particle swarm optimization and sequential quadratic programming. J Chem Inst Taiwan. doi: 10.1016/j.jtice.2014.10.018
  16. 16.
    Khan JA, Raja MAZ, Qureshi IM (2011) Stochastic computational approach for complex non-linear ordinary differential equations. Chin Phys Lett 28(2):020206–020209MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yazdi, HS, Reza P (2010) Unsupervised adaptive neural-fuzzy inference system for solving differential equations. Appl Soft Comput 10.1:267–275Google Scholar
  18. 18.
    Khan JA, Raja MAZ, Qureshi IM (2011) Hybrid evolutionary computational approach: application to van der Pol Oscillator. Int J Phys Sci 6(31):7247–7261. doi: 10.5897/IJPS11.922 Google Scholar
  19. 19.
    Khan JA, Raja MAZ, Qureshi IM (2011) Novel approach for van der Pol Oscillator on the continuous Time Domain. Chin Phys Lett 28:110205. doi: 10.1088/0256-307X/28/11/110205 CrossRefGoogle Scholar
  20. 20.
    Khan JA, Raja MAZ, Qureshi IM (2011) Numerical treatment of Nonlinear Emden-Fowler equation using stochastic technique. Ann Math Artif Intell 63(2):185–207MathSciNetCrossRefGoogle Scholar
  21. 21.
    Monterola Christopher, Saloma Caesar (2001) Solving the nonlinear Schrodinger equation with an unsupervised neural network. Opt Express 9(2):16CrossRefGoogle Scholar
  22. 22.
    Raja MAZ, Samar R (2014) Numerical treatment for nonlinear MHD Jeffery–Hamel problem using neural networks optimized with interior point algorithm. Neurocomputing 124:178–193. doi: 10.1016/j.neucom.2013.07.013 CrossRefGoogle Scholar
  23. 23.
    Raja MAZ, Samar R (2014) Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms. Comput Fluids 91:28–46MathSciNetCrossRefGoogle Scholar
  24. 24.
    Raja MAZ (2014) Solution of one-dimension Bratu equation arising in fuel ignition model using ANN optimized with PSO and SQP. Connect Sci 26(3):195–214. doi: 10.1080/09540091.2014.907555 CrossRefGoogle Scholar
  25. 25.
    Raja MAZ, Ahmad SI (2014) Numerical treatment for solving one-dimensional Bratu Problem using Neural Networks. Neural Comput Appl 24(3–4):549–561. doi: 10.1007/s00521-012-1261-2 CrossRefGoogle Scholar
  26. 26.
    Raja MAZ (2014) Stochastic numerical techniques for solving Troesch’s Problem. Inf Sci 279:860–873. doi: 10.1016/j.ins.2014.04.036 MathSciNetCrossRefGoogle Scholar
  27. 27.
    Raja MAZ (2014) Unsupervised neural networks for solving Troesch’s Problem. Chin Phys B 23(1):018903CrossRefGoogle Scholar
  28. 28.
    Raja MAZ, Ahmad SI, Samar r (2013) Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu Problem. Neural Comput Appl 23(7–8):2199–2210. doi: 10.1007/s00521-012-1170-4 CrossRefGoogle Scholar
  29. 29.
    Raja MAZ, Samar R, Rashidi MM (2014) Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation. Neural Comput Appl. doi: 10.1007/s00521-014-1641-x Google Scholar
  30. 30.
    Raja MAZ, Ahmad S, Samar R (2014) Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming. Neural Comput Appl. doi: 10.1007/s00521-014-1664-3 Google Scholar
  31. 31.
    Raja MAZ (2014) Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl Soft Comput 24:806–821. doi: 10.1016/j.asoc.2014.08.055 CrossRefGoogle Scholar
  32. 32.
    Raja MAZ, Khan JA, Behloul D, Haroon T, Siddiqui AM, Samar R (2014) Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation. Appl Soft Comput 26:244–256. doi: 10.1016/j.asoc.2014.10.009 CrossRefGoogle Scholar
  33. 33.
    Kumar M, Yadav N (2011) Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: a survey. Comput Math Appl 62(10):3796–3811MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Raja MAZ, Khan JA, Qureshi IM (2011) Swarm Intelligent optimized neural networks for solving fractional differential equations. Int J Innov Comput Inf Control 7:11Google Scholar
  35. 35.
    Raja MAZ, Khan JA, Qureshi IM (2010) Evolutionary computational intelligence in solving the fractional differential equations. Lecture Notes in Computer Science, vol 5990, part 1, Springer, ACIIDS Hue City, Vietnam, pp 231–240Google Scholar
  36. 36.
    Raja MAZ, Khan JA, Qureshi IM (2010) Heuristic computational approach using swarm intelligence in solving fractional differential equations. GECCO (Companion) 2010:2023–2026Google Scholar
  37. 37.
    Raja MAZ, Khan JA, Qureshi IM (2011) Solution of fractional order system of Bagley-Torvik equation using evolutionary computational intelligence. Math Probl Eng 2011 Article ID 765075:01–18Google Scholar
  38. 38.
    Raja MAZ, Khan JA, Qureshi IM (2010) A new Stochastic approach for solution of Riccati Differential equation of fractional order. Ann Math Artif Intell 60(3–4):229–250MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Raja MAZ, Khan JA, Ahmad SI, Qureshi IM (2013) Numerical treatment of Painleve equation I using neural networks and stochastic solvers. In: Jordanov I, Jain LC (eds) Innovations in intelligent machines—3, SCI 442. Springer, Berlin, pp 103–117Google Scholar
  40. 40.
    Raja MAZ, Khan JA, Ahmad SI, Qureshi IM (2012) Solution of the Painlevé Equation-I using neural network optimized with swarm intelligence. Comput Intell Neurosci ID 721867:10Google Scholar
  41. 41.
    Beidokhti RS, Malek A (2009) Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques. J Franklin Inst 346(9):898–913MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Aarts LP, Veer PVD (2001) Neural network method for solving the partial differential equations. Neural Process Lett 14:261–271MATHCrossRefGoogle Scholar
  43. 43.
    Rarisi DR et al (2003) Solving differential equations with unsupervised neural networks. J Chem Eng Process 42:715–721CrossRefGoogle Scholar
  44. 44.
    Nocedal J, Wright SJ (1999) Numerical optimization, Springer Series in Operations Research. Springer, BerlinGoogle Scholar
  45. 45.
    Fletcher R (1987) Practical methods of optimization. Wiley, New YorkGoogle Scholar
  46. 46.
    Schittkowski K (1985) NLQPL: a FORTRAN-subroutine solving constrained nonlinear programming problems. Ann Oper Res 5:485–500MathSciNetCrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2014

Authors and Affiliations

  • Muhammad Asif Zahoor Raja
    • 1
  • Junaid Ali Khan
    • 2
  • Syed Muslim Shah
    • 3
  • Raza Samar
    • 3
  • Djilali Behloul
    • 4
  1. 1.Department of Electrical EngineeringCOMSATS Institute of Information TechnologyAttockPakistan
  2. 2.Faculty of Engineering Science and Technology, Islamabad CampusHamdard UniversityKarachiPakistan
  3. 3.Department of Electrical EngineeringMohammad Ali Jinnah UniversityIslamabadPakistan
  4. 4.Department of Computer ScienceUSTHBBab Ezzouar, AlgiersAlgeria

Personalised recommendations