Advertisement

Heuristic computational intelligence approach to solve nonlinear multiple singularity problem of sixth Painlev́e equation

  • Iftikhar Ahmad
  • Abdul Rehman
  • Fayyaz Ahmad
  • Muhammad Asif Zahoor Raja
Original Article

Abstract

The present study investigate the numerical solution of nonlinear singular system represented with sixth Painlev́e equation by the strength of artificial intelligence using feed-forward artificial neural networks (ANNs) optimized with genetic algorithms (GAs), interior point technique (IPT), sequential quadratic programming (SQP), and their hybrids. The ANN provided a compatible method for finding nature-inspired mathematical model based on unsupervised error for sixth Painlev́e equation and adaptation of weights of these networks is carried out globally by the competency of GA aided with IPT or SQP algorithms. Moreover, a hybrid approach has been adopted for better proposed numerical results. An extensive statistical analysis has been performed through several independent runs of algorithms to validate the accuracy, convergence, and exactness of the proposed scheme.

Keywords

Painlev́e ANN Activation function GA AI 

Notes

Compliance with ethical standards

Conflict of interests

There is no conflict of interest among all the authors.

References

  1. 1.
    Painlev P (1902) Sur les quations diffrentielles du second ordre et d’ordre sup rieure dont l’intgrale gnrale est uniforme. Acta Math 25:185MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gambier B (1910) Sur les quations diffrentielles du second ordre et du premier degr dont l’intgrale gnrale est points critiques fixes. Acta Math 33:155CrossRefGoogle Scholar
  3. 3.
    Picard E (1887) Sur une classe d’équations diffrentielles. CR Acad Sci Paris 104:41–43MATHGoogle Scholar
  4. 4.
    Ince EL (1956) Ordinary differential equations. Dover, New YorkGoogle Scholar
  5. 5.
    Ablowitz MJ, Clarkson PA (1997) Solitons, nonlinear evolution equations and inverse scattering, vol 149Google Scholar
  6. 6.
    Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering trans-form. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  7. 7.
    Its AR, Novokshenov VY (1986) The isomonodromic deformation method in the theory of Painleve equations, vol 1191Google Scholar
  8. 8.
    Lukashevich NA (1967) Theory of the fourth Painleve equation. Diff Equat 3:395–399MATHGoogle Scholar
  9. 9.
    Lukashevich NA (1968) Solution of fifth equation of Painleve. Diff Urav 4:732–735MATHGoogle Scholar
  10. 10.
    Lukashevich NA, Yablonskii AI (1967) On a class of solutions of the sixth Painleve equations. Diff Urav 3:264MathSciNetGoogle Scholar
  11. 11.
    Fokas AS, Ablowitz MJ (1982) On a unified approach to transformations and elementary solutions of Painlev equations. J Math Phy 23:2033–2042CrossRefMATHGoogle Scholar
  12. 12.
    Gromak VI, Laine I, Shimomura S (2002) Painleve differential equations in the complex plane. Walter de Gruyter, Berlin, New YorkGoogle Scholar
  13. 13.
    Okamoto K, Conte R (1999) The Painleve property, one century laterGoogle Scholar
  14. 14.
    Okamoto K (1986) Studies on Painleve equation III, second and fourth Painleve equation, PII and PIV. Math Ann 275:221–255MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Painleve P (1900) Memoire sur les equations differentielles dont 1’integrale generale est uniforme. Bull Soc Math Fr 28:201– 261CrossRefMATHGoogle Scholar
  16. 16.
    Painleve P (1902) Memoire Sur les equations differentielles du second ordre et d’ordre superieur dont 1’integrale generale est uniforme. Acta Math 25:1–85MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gambier B (1910) Sur les equations differentielles du second ordre et du premier degre dont 1’integrale generale est a points critiques fixes. Acta Math 33:1–55MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Chazy J (1911) Sur les equations differentielles du troisieme ordre et d’ordre suprieur dont 1’intgrale generale a ses points critiques fixes. Acta Math 34:317–385MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Garnier R (1912) Sur des equations differentielles du troisieme ordre dont l’integrale generale est uniforme et sur une classe d’equations nouvelles d’ordre superieur. Ann Sci Ecole Normale Sup 48:1–126CrossRefMATHGoogle Scholar
  20. 20.
    Exton H (1973) Non-linear ordinary differential equations with fixed critical points. Rend Mat (6):419–462Google Scholar
  21. 21.
    Martynov IP (1985) Analytic properties of solutions of a third-order differential equation. Differents Uravn 21:764–771MathSciNetMATHGoogle Scholar
  22. 22.
    Martynov IP (1985) Third-order equations without moving critical singularities. Differents Uravn 21:937–946MathSciNetGoogle Scholar
  23. 23.
    Bureau FJ (1964) Differential equations with fixed critical points. Ann Mat Pura Appl (IV) 66:1–116MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ablowitz MJ, Segur H (1977) Exact linearization of a Painleve transcendent. Phys Rev Lett 38:1103–1106MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ablowitz MJ, Ramani A, Segur H (1980) A connection between nonlinear evolution equations and ordinary differential equations of P-type. I. J Math Phys 21:715–721MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Airault H (1979) Rational solutions of Painleve equations. Stud Appl Math 61:31–53MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kudryashov NA (1997) The first and second Painleve equations of higher order and some relations between them. Phys Lett A 224:353–360MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Gordoa PR, Pickering A (1999) Non-isospectral scattering problems: a key to integrable hierarchies. J Math Phys 40:5749–5786MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Gordoa PR, Pickering A (2000) On a new non-isospectral variant of the Boussinesq hierarchy. J Phys A 33:557–567MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Gordoa PR, Joshi N, Pickering A (2001) On a generalized 2 + 1 dispersive water wave hierarchy. Publ Res Inst Math Sci (Kyoto) 37:327–347MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Pickering A (2002) Coalescence limits for higher order Painleve equations. Phys Lett A 301:275–280MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kawai T, Koike T, Nishikawa Y, Takei Y (2004) On the stokes geometry of higher order Painleve equations, Analyse complexe, systemes dynamiques, sommabilite des series divergentes et theories galoisiennes. II. Astrisque No 297:117–166Google Scholar
  33. 33.
    Gordoa PR, Joshi N, Pickering A (2006) Second and fourth Painlev hierarchies and Jimbo-Miwa linear problems. J Math Phys 47:073504MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Koike T (2007) On the Hamiltonian structures of the second and the fourth Painlev hierarchies, and the degenerate Garnier systems, algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlev hierarchies. RIMS Kkyroku Bessatsu, B2, Res Inst Math Sci (RIMS), Kyoto pp 99–127Google Scholar
  35. 35.
    Sakka AH (2009) On special solutions of second and fourth Painleve hierarchies. Phys Lett A 373:611–615MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mugan U, Jrad F (1999) Painleve test and the first Painleve hierarchy. J Phys A 32:7933–7952MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Mugan U, Jrad F (2002) Painleve test and higher order differential equations. J Nonlinear Math Phys 9:282–310MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mugan U, Jrad F (2004) Non-polynomial third order equations which pass the Painleve test. Z Naturforsch A 59:163–180CrossRefGoogle Scholar
  39. 39.
    Cosgrove CM (2000) Higher-order Painleve equations in the polynomial class. I. Bureau symbol P2. Stud Appl Math 104:1–65MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Cosgrove CM (2006) Higher-order Painleve equations in the polynomial class. II. Bureau symbol P1. Stud Appl Math 116:321– 413MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Dai D, Zhang L (2010) On tronque solutions of the first Painlev hierarchy. J Math Anal Appl 368:393–399MathSciNetCrossRefGoogle Scholar
  42. 42.
    Lebeau G, Lochak P (1987) On the second painleve equation: the connection formula via a Riemann-Hilbert problem and other results. Journal of Uiffekential Equations 68:344–372MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Kajiwara K (2003) On a q-difference Painleve III equation: II. rational solutions. J Nonlinear Math Phys 10 (3):282–303MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Yoshikatsu S (2007) Value distribution of the fifth Painlev transcendentsin sectorial domains. J Math Anal 330:817–828MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Contea R, Musetteb M (2002) New contiguity relation of the sixth Painlev equation from a truncation. A Physica D 161:129– 141CrossRefGoogle Scholar
  46. 46.
    Ahmad I, Raja MAZ, Bilal M, Ashraf F (2016) Neural network methods to solve the Lane-Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput and Applic 1–16. doi: 10.1007/s00521-016-2400-y
  47. 47.
    Ahmad I, Mukhtar A (2015) Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model, Appl. Math Comput 261:360MathSciNetGoogle Scholar
  48. 48.
    Ahmad I, Ahmad S, Bilal M, Anwar N (2016) Stochastic numerical treatment for solving Falkner-Skan equations using feed forward neural networks. Neural Comput and Applic 1–14. doi: 10.1007/s00521-016-2427-0
  49. 49.
    He JH (1999) Homotopy perturbation technique. Comput Math Appl Mech Engy pp 178–257Google Scholar
  50. 50.
    Abu Arqub O (2016) Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm. Fundamenta Informaticae 146:231–254MathSciNetCrossRefGoogle Scholar
  51. 51.
    Abu Arqub O, Maayah B (2016) Solutions of Bagley-Torvik and Painlev equations of fractional order using iterative reproducing kernel algorithm. Neural Comput and Applic. doi: 10.1007/s00521-016-2484-4

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  • Iftikhar Ahmad
    • 1
  • Abdul Rehman
    • 1
  • Fayyaz Ahmad
    • 2
  • Muhammad Asif Zahoor Raja
    • 3
  1. 1.Department of MathematicsUniversity of GujratGujratPakistan
  2. 2.Department of StatisticsUniversity of GujratGujratPakistan
  3. 3.Department of Electrical EngineeringCIIT AttockAttockPakistan

Personalised recommendations