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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| Format: | Article |
| Language: | English |
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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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| author | U. Al Khawaja Qasem M. Al-Mdallal |
| author_facet | U. Al Khawaja Qasem M. Al-Mdallal |
| author_sort | U. Al Khawaja |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-1d6d19ab84c54d399b7e8f628f1c8936 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-1d6d19ab84c54d399b7e8f628f1c89362025-02-03T01:27:26ZengWileyInternational Journal of Differential Equations1687-96431687-96512018-01-01201810.1155/2018/60439366043936Convergent Power Series of sech(x) and Solutions to Nonlinear Differential EquationsU. Al Khawaja0Qasem M. Al-Mdallal1Physics Department, United Arab Emirates University, P.O. Box 15551, Al Ain, UAEDepartment of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, UAEIt 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.http://dx.doi.org/10.1155/2018/6043936 |
| spellingShingle | U. Al Khawaja Qasem M. Al-Mdallal Convergent Power Series of sech(x) and Solutions to Nonlinear Differential Equations International Journal of Differential Equations |
| title | Convergent Power Series of sech(x) and Solutions to Nonlinear Differential Equations |
| title_full | Convergent Power Series of sech(x) and Solutions to Nonlinear Differential Equations |
| title_fullStr | Convergent Power Series of sech(x) and Solutions to Nonlinear Differential Equations |
| title_full_unstemmed | Convergent Power Series of sech(x) and Solutions to Nonlinear Differential Equations |
| title_short | Convergent Power Series of sech(x) and Solutions to Nonlinear Differential Equations |
| title_sort | convergent power series of sech x and solutions to nonlinear differential equations |
| url | http://dx.doi.org/10.1155/2018/6043936 |
| work_keys_str_mv | AT ualkhawaja convergentpowerseriesofsechxandsolutionstononlineardifferentialequations AT qasemmalmdallal convergentpowerseriesofsechxandsolutionstononlineardifferentialequations |