Abstract
This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant.AMS Subject Classification34L30
Highlights
Solving differential equations is an important issue in sciences because many physical phenomena are modelled using such equations
It is has been applied to the improve the accuracy of truncated power obtained by power series method (PSM) (Forsyth 1906; Geddes 1979; Ince 1956; Vazquez-Leal and Guerrero 2013), Adomian Decomposition method (ADM) (Wazwaz 2006; Wang et al 2011), homotopy perturbation method (HPM) (Bararnia et al 2012; Rashidi and Keimanesh 2010; Torabi and Yaghoobi 2011), homotopy analysis method (HAM) (Guerrero et al 2013), differential transform method (DTM) (Rashidi and Keimanesh 2010; Rashidi et al 2010; Rashidi and Pour 2010a, 2010b)
In this work, we propose that the solution of a differential equation can be directly expressed as a rational power series of the independent variable, in other words as a Padé approximant
Summary
Solving differential equations is an important issue in sciences because many physical phenomena are modelled using such equations. The Padé method is a well established resummation method from literature. It can increase the domain of convergence of truncate power series (Bararnia et al 2012; Guerrero et al 2013; Torabi and Yaghoobi 2011; Vazquez-Leal and Guerrero 2013). Note that for a fixed value of L + M + 1, the error (3) is smallest when the numerator and denominator of (2) have the same degree or when the numerator has degree one higher than the denominator
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