Convergent Power Series of sech(x) and Solutions to Nonlinear Differential Equations
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...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2018-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2018/6043936 |
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| Summary: | 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. |
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| ISSN: | 1687-9643 1687-9651 |