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

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14 Citations (Scopus)

Abstract

It is known that power series expansion of certain functions such as sech⁡(x) diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of sech⁡(x) that is convergent for all x. The convergent series is a sum of the Taylor series of sech⁡(x) and a complementary series that cancels the divergence of the Taylor series for x≥π/2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1+x2). 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.

Original languageEnglish
Article number6043936
JournalInternational Journal of Differential Equations
Volume2018
DOIs
Publication statusPublished - 2018

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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