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It is known that power series expansion of certain functions such assech   ⁡   (  x  )diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation ofsech   ⁡   (  x  )that is convergent for all    x   . The convergent series is a sum of the Taylor series ofsech   ⁡   (  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  +    x    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.

international journal of differential equations

Convergent Power Series of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">sech</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math> and Solutions to Nonlinear Differential Equations

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