scholarly journals Numerical efficiency of the elliptic function expansions of the first-order intermediary for general planetary theory

1996 ◽  
Vol 172 ◽  
pp. 101-104 ◽  
Author(s):  
Victor A. Brumberg ◽  
Sergei A. Klioner
Keyword(s):  
Elliptic Function ◽  
Elliptic Functions ◽  
Compact Form ◽  
Planetary Theory ◽  
Numerical Efficiency ◽  
First Order ◽  
Intermediate Orbit ◽  
General Planetary Theory

We compare numerical efficiency of the two kinds of series for the first-order intermediate orbit for general planetary theory: (1) the classical expansion involving mean longitudes of the planets; (2) an expansion resulting from the theory of elliptic functions. We conclude that mutual perturbations of close couples of planets (the ratio of major semi-axes ∼ 1) can be represented in more compact form with the aid of the second kind of series.

Construction of a general planetary theory of the first order

1973 ◽  
Vol 8 (3) ◽  
pp. 335-355 ◽  
Author(s):  
V. A. Brumberg ◽  
J. Chapront
Keyword(s):  
Planetary Theory ◽  
First Order ◽  
General Planetary Theory

ANALYTICAL STUDY ON NONLINEAR DIFFERENTIAL–DIFFERENCE EQUATIONS VIA A NEW METHOD

2010 ◽  
Vol 24 (08) ◽  
pp. 761-773
Author(s):  
HONG ZHAO
Keyword(s):  
Difference Equations ◽  
Elliptic Function ◽  
Analytical Study ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Jacobian Elliptic Function ◽  
Jacobian Elliptic Functions ◽  
Periodic Wave Solutions ◽  
Trigonometric Function Solutions

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

Analytical Solutions of Two Space-Time Fractional Nonlinear Models Using Jacobi Elliptic Function Expansion Method

2021 ◽  
pp. 173-188
Author(s):  
Zillur Rahman ◽  
M. Zulfikar Ali ◽  
Harun-Or-Roshid ◽  
Mohammad Safi Ullah
Keyword(s):  
Elliptic Function ◽  
Nonlinear Models ◽  
Applied Mathematics ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Space Time ◽  
Jacobi Elliptic Function ◽  
Function Expansion ◽  
Partial Models ◽  
Jacobi Elliptic Function Expansion

In this manuscript, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM) models have been investigated which are frequently arises in nonlinear optics, solid states, fluid mechanics and shallow water. Jacobi elliptic function expansion integral technique has been used to build more innovative exact solutions of the s-tfEW and s-tfWBBM nonlinear partial models. In this research, fractional beta-derivatives are applied to convert the partial models to ordinary models. Several types of solutions have been derived for the models and performed some new solitary wave phenomena. The derived solutions have been presented in the form of Jacobi elliptic functions initially. Persevering different conditions on a parameter, we have achieved hyperbolic and trigonometric functions solutions from the Jacobi elliptic function solutions. Besides the scientific derivation of the analytical findings, the results have been illustrated graphically for clear identification of the dynamical properties. It is noticeable that the integral scheme is simplest, conventional and convenient in handling many nonlinear models arising in applied mathematics and the applied physics to derive diverse structural precise solutions.

A METHOD TO CONSTRUCT WEIERSTRASS ELLIPTIC FUNCTION SOLUTION FOR NONLINEAR EQUATIONS

2011 ◽  
Vol 25 (14) ◽  
pp. 1931-1939 ◽  
Author(s):  
LIANG-MA SHI ◽  
LING-FENG ZHANG ◽  
HAO MENG ◽  
HONG-WEI ZHAO ◽  
SHI-PING ZHOU
Keyword(s):  
Nonlinear Equations ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Order Derivative ◽  
First Order ◽  
Function Solution ◽  
Weierstrass Elliptic Function ◽  
Klein Gordon

A method for constructing the solutions of nonlinear evolution equations by using the Weierstrass elliptic function and its first-order derivative was presented. This technique was then applied to Burgers and Klein–Gordon equations which showed its efficiency and validality for exactly some solving nonlinear evolution equations.

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