Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

Received Sep 25, 2017; Revised Dec 7, 2017; Accepted Jan 8, 2018

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1. Introduction

It is well-known that the Taylor series of some functions diverge beyond a finite radius of convergence [1]. For instance, by way of example not exhaustive enumeration, the Taylor series of $\mathrm{sech}(x)$ and $\mathrm{1}/(\mathrm{1}+x^{\mathrm{2}})$ diverge for $x\ge \pi /\mathrm{2}$ and $x\ge \mathrm{1}$, respectively. Increasing the number of terms in the power series does not increase the radius of convergence; it only makes the divergence sharper. The radius of convergence can be increased only slightly via some functional transforms [2]. Among the many different methods of solving nonlinear differential equations [3–9], the power series is the most straightforward and efficient [10]. It has been used as a powerful numerical scheme for many problems [11–19] including chaotic systems [20–23]. Many numerical algorithms and codes have been developed based on this method [10–12, 20–24]. However, the above-mentioned finiteness of radius of convergence is a serious problem that hinders the use of this method to wide class of differential equations, in particular the nonlinear ones. For instance, the nonlinear Schrödinger equation (NLSE) with cubic nonlinearity has the $\mathrm{sech}(x)$ as a solution. Using the power series method to solve this equation produces the power series of a $\mathrm{sech}(x)$, which is valid only for $x<\pi /\mathrm{2}$.

A review of the literature reveals that the power series expansion was exploited by several researchers [10–12, 20–24] to develop powerful numerical methods for solving nonlinear differential equations. Therefore, this paper is motivated by a desire to extend these attempts to a develop a numerical scheme with systematic control on the accuracy and error. Specifically, two main advances are presented in this paper: $(1)$ a method of constructing a convergent power series representation of a given function with an arbitrarily large radius of convergence and $(2)$ a method of obtaining analytic power series solution of a given nonlinear...

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