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IJAMR International Journal of Applied Mathematical Research E-ISSN 2227-4324

Exact parametric solutions for the first Painlevé nonlinear ODE

Authors and Affiliations

  • Dimitrios Panayotounakos
  • Theodoros I Zarmpoutis
  • George Kosotogiannis

About this article

DOI
https://doi.org/10.14419/ijamr.v2i1.535
Received
31-10-2012
Accepted
12-11-2012
Published
30-11-2012
Views
573
Downloads
278
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Abstract

We present a mathematical methodology for constructing the exact parametric solution of the first Painlevé second order nonlinear  ordinary differential equation. Several admissible functional transformations are introduced through an intermediary analysis delivering us from the a priori construction of power series solutions.

Author Biographies

  • Dimitrios Panayotounakos
    Prof School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Street 5, Hellas, GR 15773.
  • Theodoros I Zarmpoutis
    PHD Researcher School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Street 5, Hellas, GR 15773.
  • George Kosotogiannis
    PHD Candidate  School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Street 5, Hellas, GR 15773.

References

[1] H. T. Davis,. Introduction to Nonlinear Differential and Integral Equations. Dover Publ. Inc., New York, 1962.

[2] A.S. Fokas, A.R. Its, A.A. Kaparev and V.Vu, Painlevé transcendents. The Riemman-Hilbert approach. Mathematical Surveys and Monographs, 128 Providence, R.I American Mathematical Society, 2006.

[3] G. Julia Exercices d’Analyse. Bd. 3, Paris, 1933.

[4] E.L. Ince, Ordinary Differential Equations. Dover Reprint, 1927.

[5] E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen. Vol.1, B.G Teubeur, Stuttgart, 1977.

View more references (8)

[6] N. Nishioha, A note on the transcendency of Painlevé’s first transcendent. Nagoya Mathematical Journal . 109 (1988), pp. 63-67

[7] P. Painlevé, Sur le équations differentials du second ordre a d’ ordre superieur dont l’intégrale génerale est uniform. Acta Mathematica, 25 (1902), pp. 1-85.

[8] D. E. Panayotounakos, T. I. Zarmpoutis and C.I. Siettos, On the construction of the general closed-form solutions of standing waves of the cubic nonlinear Schrödinger equation. Arch. of Appl. Mech. DOI 10.1007/s00419-012-0658-9, Vol 82, 10 (2012), pp. 1557-1568.

[9] D.E. Panayotounakos and Th. Zarmpoutis, Construction of exact parametric or closed form solutions of some unsolvable classes of nonlinear ODEs, (Abel’s nonlinear ODEs of the first kind and relative degenerate equations). Int J. of Math. and Math Sciences, DOI 10.1155/2011/ 387429, ID 387429 (2011), pp. 1-11.

[10] I. G. Petrovski and E.M. Landis, On the number of limit cycles of the equation where P and Q are polynomials of the second degree, Math. Sbornik 10 (2) (1955), pp. 209-250.

[11] A.D. Polyanin and V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman &Hall/CRC Press, New York, 2003.

[12] H. Umemura, Painlevé equations and classical functions. Sugaku Expositions. 2 (2),(1998), pp. 231-252.

[13] H. Umemura, On the irreducibility of Painlevé differential equations. Sugaku Expositions. 11 (1),(1989), pp. 77-100.


How to Cite

Panayotounakos, D., Zarmpoutis, T. I., & Kosotogiannis, G. (2012). Exact parametric solutions for the first Painlevé nonlinear ODE. International Journal of Applied Mathematical Research, 2(1), 55-61. https://doi.org/10.14419/ijamr.v2i1.535

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