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

Abstract

It is known that power series expansion of certain functions such as sech( ) diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of sech( ) that is convergent for all . The convergent series is a sum of the Taylor series of sech( ) and a complementary series that cancels the divergence of the Taylor series for ≥ /2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1 + 2 ). A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

High accuracy power series method for solving scalar, vector, and inhomogeneous nonlinear Schrödinger equations

“…[Eqs. (23), (24), and similar equations for the rest of coefficients.] 4) Time evolution and update:…”

“…Here, we present a method that can systematically increase the accuracy in both the spacial and temporal axes. For the temporal evolution, we use an iterative power series method that we have developed previously for ordinary differential equations [24] and applied later to fluid flow [25]. The accuracy in the time evolution increases with the maximum power in the time power series, s. For the spacial part, we use a p-point stencil to discretise the second derivative, where p ≥ 3 is a positive odd integer.…”

Analytic Methods for Solving Higher Order Ordinary Differential Equations

“…In general, solving higher order DEs is complex and numerical methods are usually needed to solve these equations with initial or boundary conditions. For instance, Khawaja et al (2018) [3] used iterative power series of sech(x) to solve nonlinear ordinary differential equations (ODEs) . Vitoriano (2016) [4] also used the finite element method to solve partial differential equations (PDEs).…”