APPLYING THE DOMINANT BALANCE METHOD TO SOLVE VARIABLE COEFFICIENTS 2ND ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION AROUND IRREGULAR SINGULAR POINT
Keywords:
Asymptotic solutions, approximation, dominant balance, irregular singular point, second order differential equationsAbstract
Ordinary differential equations with variable coefficients are, in general, difficult to solve, specifically finding exact solutions to such type of equations. Thus, approximation solutions are excellent alternatives to exact solutions. There are many types of approximation methods that are available to applied mathematicians to apply. Among these approximation methods are asymptotic methods which include Dominant Balance Method (DBM). In this paper, we apply the Dominant Balance Method to solve 2nd order variable coefficients ordinary differential equations when Frobenius method fails, that happens when we have irregular singular point. The method has been discussed in details and applied to equations and the solutions obtained by the asymptotic method are represented by the first few terms of asymptotic sequence. The reader has to be aware of the perturbation theory and its symbols because they are keys to understand the steps of application.
References
N. G. de Brujin, "Asymptotic Methods in Analysis", Dover publications, Inc. New York, 1981.
C. M. Bender, S. A. Orszag, "Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic methods and perturbation theory", Springer science & Business Media, 2013.
C. J. Chapman, H. P. Wynn, "Fractional power series and the method of dominant balance", the royal society publishing, 2021.
W. Wasow, "Asymptotic Expansion for ordinary differential equations", Dover publications, Inc. New York, 1993.
J. Callaham, J. V. Koch, B. W. Brunton, J. N. Kutz, S. L. Brunton, "Learning dominant physical processes with data-driven balance models", nature communications, 2021.
E. Fan, H. Zhang, "A note of the homogenous balance method", physics letters A, vol. 246, issue 5, pp.403-406, 1998.
S. A. El-Wakil, E. M. Abulwafa, A. Elhanbaly, M. A. Abdou, "The extended homogenous balance method and its applications for a class of nonlinear evolution equations", Chaos, solitons & fractals, vol. 33, issue 5, pp. 1512-1522, 2007.
M. Wang, Y. Zhou, Z. Li, "Application of a homogenous balance method to exact solutions of nonlinear equation in mathematical physics", physics letters A, vol. 216, issue 5, pp. 67-75, 1996.
G. D. Birkhoff, "On a simple type of irregular singular point", Transactions of the American Mathematical Society, vol. 14, issue 4, pp. 462-476, 1913.
A. H. Nayfeh, "Asymptotic solutions of linear equations" in "Perturbation methods", USA: John Wiley-VCH Verlag GmbH & Co. KGaA, 2004,pp.309-312 .
A. H. Nayfeh, "Linear Equations with variable coefficients" in "Problems in perturbation". USA: John Wiley & Sons Inc. 1985,pp.399-410.
A. H. Nayfeh, "Introduction to Perturbation Techniques", New York,USA: John Wiley & Sons , 2004.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Journal of Basic Sciences
This work is licensed under a Creative Commons Attribution 4.0 International License.
