On the polynomial solution of the first Painlevé equation

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    The Painlevé equations and their solutions arises in pure, applied mathematics and theoretical physics. In this manuscript we apply the Optimal Homotopy Asymptotic Method (OHAM) for solving the first Painlevé equation. Our approximation technique is based on the use of polynomial solutions, which are shown to be accurate when compared to the computed numerical solutions, thus providing a very close description of the evolution of the system.


  • Keywords


    Painlevé transcendent, first Painlevé equation, optimization methods.

  • References


      [1] Painlevé P., Sur les équations différentielles du second ordre et d’ordre superieur dont l’intégrale général est uniforme, Acta Math. 25 (1902), 1-85.

      [2] H. Segur and M.J. Ablowitz. Asymptotic solutions of nonlinear evolution equations and a Painlevé transcedent. Physica D, (1993), 3:165.

      [3] M. Adler, T. Shiota, and P. van Moerbeke. Random matrices, vertex operators and the Virasoro algebra. Phys. Lett. A, 208 (1995), 67-78.

      [4] M. Shiroishi, M. Takahashi, and Y. Nishiyama. Emptiness formation probability for the one-dimensional isotropic XY model. J. Phys. Soc. Jpn., 70 (2001), 3535.

      [5] B.M. McCoy, J.H. Perk, and R.E. Shrock. Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separation. Nucl. Phys. B, 220 (2002), 269.

      [6] T. Kanna, K. Sakkaravarthi, C. Senthil, M. Lakshmanan, and M. Wadati. Painlevé singularity structure analysis of three component Gross-Pitaevskii type equations. J. Math. Phys., 50 (2009), 113520.

      [7] E.A. Moskovchenko and V.P. Kotlyarov. Periodic boundary data for an integrable model of stimulated Raman scattering: long-time asymptotic behavior. J. Phys. A: Math. Gen., 43 (2010), 5.

      [8] A.S. Fokas, A.R. Its, and A.V. Kitaev. Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys., 142 (1991), 313.

      [9] M. Adler, T. Shiota, and P. van Moerbeke. Random matrices, vertex operators and the Virasoro algebra. Phys. Lett. A, 208 (1995), 67-78.

      [10] B. Dubrovin. Painlevé trascendents and two-dimensional topological field theory. CRM Math. Phys., (1999) pages 287-412.

      [11] Gariel, G. Marcilhacy, and N.O. Santos. Stationary axisymmetric solutions involving a third order equation irreducible to Painlevé transcendents. J. Math. Phys., 49 (2008), 022501.

      [12] X. Cao and C. Xu. A Backlund transformation for the Burgers hierarchy. Abs. App. Ana., (2010), 241898.

      [13] S.Y. Lee, R. Teodorescub, and P. Wiegmannc. Shocks and nite-time singularities in Hele-Shaw ow. Physica D, 238 (2009), 1113.

      [14] M. Florjanczyk and L. Gagnon. Exact solutions for a higher-order nonlinear Schrodinger equation. Phys. Rev. A, 41 (1990), 4478.

      [15] Aganagic M., Cheng M.C.N., Dijkgraaf R., Krefl D., Vafa C., Quantum geometry of refined topological strings, J. High Energy Phys. (2012), no. 11, 019, 53 pages, arXiv:1105.0630.

      [16] Aganagic M., Dijkgraaf R., Klemm A., Mariño M., Vafa C., Topological strings and integrable hierarchies, Comm. Math. Phys. 261 (2006), 451-516, hep-th/0312085.

      [17] B. Fornberg and J.A.C. Weideman, A numerical methodology for the Painleve equations, Journal of Computational Physics, 230 (2011), pp. 5957-5973.

      [18] K. El-Kamel, Sinc numerical solution for solitons and solitary waves, J. Comput. Appl. Math., 130 (2001), pp. 283-292.

      [19] E. Hesameddini and S. Shekarpaz, Wavelet solutions of the first Painleve equation, 23rd International Conference of Jangjeon Mathematical Society, Shahid Chamran Uinvesity of Ahvaz, Ahvaz, Iran, (2010).

      [20] Marinca V, Herisanu N, Bota C, Marinca B. An optimal homotopy asymptotic method applied to the steady flow of fourth-grade fluid past a porous plate. Appl Math Lett 22 (2009), 245-51.

      [21] Herisanu N, Marinca V. Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method. Comp Math Appl 60 (2010), 1607-15.

      [22] Marinca V, Herisanu N. Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method. Journal of Sound and Vibration 329 (2010), 1450-1459.

      [23] Marinca V, Herisanu N, Nemes I. Optimal homotopy asymptotic method with application to thin film flow. Central European Journal of Physics 6 (2008), 648-53.

      [24] Marinca V, Herisanu N. An optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int Commun Heat Mass Transfer 35 (2008), 710-5.

      [25] Marinca V, Herisanu N. Nonlinear dynamical systems in engineering. Some Approximate Approaches. (2011), Springer, Verlag.

      [26] Islam S, Shah RA, Ali I. Optimal homotopy asymptotic solutions of Couette and Poiseuille flows of a third grade fluid with heat transfer analysis. Int J Nonlinear Sci Numer Simul 11(6) (2010), 389-400.

      [27] Idrees M, Islam S, Tirmizi SIA, Haq S. Application of the optimal homotopy asymptotic method for the solution of the Korteweg–de Vries equation. Mathematical and Computer Modelling 55 (2012),

      1324-33.

      [28] Mohsen S, Hamid RA, Domairry D, Hashim I. Investigation of the laminar viscous flow in a semi-porous channel in the presence of uniform magnetic field using optimal homotopy asymptotic method.

      Sains Malaysiana 41(10), (2012), 1281-5.

      [29] Ghoreishi M, Ismail AIM, Alomari AK. Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth-order integro-differential equation. Math Meth Appl Sci 34 (2011), 1833-42.


 

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Article ID: 6559
 
DOI: 10.14419/ijamr.v6i1.6559




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