Convergent Power Series ofand Solutions to Nonlinear Differential Equations

[摘要] It is known that power series expansion of certain functions such asdiverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation ofthat is convergent for all . The convergent series is a sum of the Taylor series ofand a complementary series that cancels the divergence of the Taylor series for . The method is general and can be applied to other functions known to have finite radius of convergence, such as . 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.

[学科分类] 几何与拓扑