Abstract
Newton polygons corresponding to nonlinear ordinary differential equations of polynomial form help visually determine some properties of differential equations. In particular, the Newton polygons are used at finding asymptotic and exact solutions of nonlinear differential equations. In this report we present the algorithm of the ACNP (automatic construction of Newton polygons) program for the automatic construction of the Newton polygons corresponding to ordinary differential equations. The program has been written in Maple symbolic computing environment. The input to the program is a polynomial ordinary differential equation. The output is a set of points on the plane corresponding, according to a certain rule, to the monomials of the differential equation, the Newton polygon and the pole order of the solution for the differential equation. The application of the ACNP program has been demonstrated for studying the integrability property and for finding exact and asymptotic solutions of nonlinear differential equations.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Newton, I., Colson, J.: The Method of Fluxions and Infinite Series; With Its Application to the Geometry of Curve-Lines. Printed by Henry Woodfall; and Sold by John Nourse, London (1736)
Puiseux, V.: Recherche sur les fonctions algébriques. Journal de mathématiques pures et appliquées 15, 365–480 (1850)
Bruno, A.D.: Asymptotics of solutions of nonlinear systems of differential equations (Asimptotika reshenij nelinejnyh sistem differencialnyh uravnenij) (In Russian). DAN USSR 143(4), 763–766 (1962)
Demina, M.V., Kudryashov, N.A., Sinel’shchikov, D.I.: The polygonal method for constructing exact solutions to certain nonlinear differential equations describing water waves. Comput. Math. Math. Phys. 48(12), 2182–2193 (2008)
Kudryashov, N.A., Demina, M.V.: Polygons of differential equations for finding exact solutions. Chaos Solitons Fractals 33(5), 1480–1496 (2007)
Kudryashov, N.A., Sinelshchikov, D.I.: Power and nonpower asymptotics of solutions of the generalization of the second and third Painleve equations (Stepennye i nestepennye asimptotiki reshenij obobshcheniya vtorogo i tret’ego uravnenij Penleve) (In Russian). Vestnik Natsional’nogo Issledovatel’skogo Yadernogo Universiteta “MIFI” 2(2), 152–160 (2013)
Bruno, A.D.: Asymptotics and expansions of solutions of an ordinary differential equation (Asimptotiki i razlozheniya reshenij obyknovennogo differencialnogo uravneniya) (In Russian). Uspekhi matematicheskikh. nauk 59(3), 31–80 (2004)
Kudryashov, N.A.: First integrals and general solution of the Fokas–Lenells equation. Optik (Stuttg) 195, 163135 (2019)
Kudryashov, N.A.: Construction of nonlinear differential equations for description of propagation pulses in optical fiber. Optik (Stuttg) 192, 162964 (2019)
Kudryashov, N.A., Kutukov, A.A.: The program for constructing Newton polygons corresponding to ordinary differential equations of polynomial type. The certificate of state registration of software № 2019617572, Moscow, Russia (2019)
Kudryashov, N.A.: Methods of nonlinear mathematical physics (Metody nelinejnoj matematicheskoj fiziki) (In Russian). Izdatelskij dom Intellekt, Dolgoprudny. Russia (2010)
Kudryashov, N.A., Kutukov, A.A.: Application of the Painleve test for a fourth-order nonlinear equation in the description of dislocations (Primenenie testa Penleve dlya nelinejnogo uravneniya chetvertogo poryadka pri opisanii dislokacij) (In Russian). Vestnik Natsional’nogo Issledovatel’skogo Yadernogo Universiteta “MIFI” 7(3), 249–252 (2018)
Liang, S., Jeffrey, D.J.: Automatic computation of the travelling wave solutions to nonlinear PDEs. Comput. Phys. Commun. 178(9), 700–712 (2008)
Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals 24(5), 1217–1231 (2005)
Kudryashov, N.A.: Polynomials in logistic function and solitary waves of nonlinear differential equations. Appl. Math. Comput. 219, 9245–9253 (2013)
Kuramoto, Y., Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progress Theoret. Phys. 55(2), 356–369 (1976)
Sivashinsky, G.I.: Instabilities, pattern formation, and turbulence in flames. Annu. Rev. Fluid Mech. 15, 179–199 (1983)
Acknowledgements
The study was carried out with a grant from the Russian Science Foundation (project No. 18-11-00209).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Kudryashov, N.A., Kutukov, A.A. (2020). Application of a Computer Algebra System for Constructing Newton Polygons for Ordinary Differential Equations. In: Misyurin, S., Arakelian, V., Avetisyan, A. (eds) Advanced Technologies in Robotics and Intelligent Systems. Mechanisms and Machine Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-030-33491-8_44
Download citation
DOI: https://doi.org/10.1007/978-3-030-33491-8_44
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-33490-1
Online ISBN: 978-3-030-33491-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)