### Abstract

In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drábek, R. Manásevich and M. Ôtani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be dened by systems of rst order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can be used to obtain analytic solutions to the equation of a nonlinear spring-mass system.

Original language | English |
---|---|

Pages (from-to) | 6053-6068 |

Number of pages | 16 |

Journal | Applied Mathematical Sciences |

Volume | 6 |

Issue number | 121-124 |

Publication status | Published - 2012 |

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### Keywords

- Analytic solution
- Generalized sine
- Hamilton system
- nonlinear spring
- Vibration

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Applied Mathematical Sciences*,

*6*(121-124), 6053-6068.

**Some generalized trigonometric sine functions and their applications.** / Wei, Dongming; Liu, Yu; Elgindi, Mohamed.

Research output: Contribution to journal › Article

*Applied Mathematical Sciences*, vol. 6, no. 121-124, pp. 6053-6068.

}

TY - JOUR

T1 - Some generalized trigonometric sine functions and their applications

AU - Wei, Dongming

AU - Liu, Yu

AU - Elgindi, Mohamed

PY - 2012

Y1 - 2012

N2 - In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drábek, R. Manásevich and M. Ôtani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be dened by systems of rst order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can be used to obtain analytic solutions to the equation of a nonlinear spring-mass system.

AB - In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drábek, R. Manásevich and M. Ôtani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be dened by systems of rst order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can be used to obtain analytic solutions to the equation of a nonlinear spring-mass system.

KW - Analytic solution

KW - Generalized sine

KW - Hamilton system

KW - nonlinear spring

KW - Vibration

UR - http://www.scopus.com/inward/record.url?scp=84867320442&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867320442&partnerID=8YFLogxK

M3 - Article

VL - 6

SP - 6053

EP - 6068

JO - Applied Mathematical Sciences

JF - Applied Mathematical Sciences

SN - 1312-885X

IS - 121-124

ER -