scholarly journals Approximation of Elliptic Functions by Means of Trigonometric Functions with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Alvaro H. Salas ◽  
Lorenzo J. H. Martinez ◽  
David L. R. Ocampo R.
Keyword(s):  
Elliptic Function ◽  
General Principle ◽  
Nonlinear Problems ◽  
Elliptic Functions ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Nonlinear Odes ◽  
Weierstrass Elliptic Function ◽  
Weierstrass Functions ◽  
Approximate Expressions

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

On the periods of the exponential and elliptic functions

1973 ◽  
Vol 73 (2) ◽  
pp. 339-350 ◽  
Author(s):  
D. W. Masser
Keyword(s):  
Differential Equation ◽  
Elliptic Function ◽  
Elliptic Functions ◽  
Weierstrass Elliptic Function ◽  
Image Position

Let ℘ be a Weierstrass elliptic function satisfying the differential equationand let ζ(z) be the associated Weierstrass ζ-function satisfying (z) = −℘(z). Corresponding to a pair of fundamental periods ω1, ω2 of ℘(z), there is a pair of quasi-periods η1, η2 of ζ(z) defined byand we have ηi = 2ζ(ωi/2) for i = 1, 2.

scholarly journals An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator

2021 ◽  
Vol 2021 ◽  
pp. 1-8 ◽  
Author(s):  
Alvaro H. Salas ◽  
S. A. El-Tantawy ◽  
Noufe H. Aljahdaly
Keyword(s):  
Elliptic Function ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Duffing Oscillator ◽  
Higher Order ◽  
Strong Nonlinearity ◽  
Classical Dynamics ◽  
Elementary Functions ◽  
Weierstrass Elliptic Function ◽  
Quadratic Damping

The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.

scholarly journals Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-5 ◽  
Author(s):  
Xun Liu ◽  
Lixin Tian ◽  
Yuhai Wu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Traveling Wave ◽  
Arbitrary Order ◽  
Traveling Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Weierstrass Elliptic Function ◽  
Wave Solutions

We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity.

scholarly journals New Solutions for the Generalized BBM Equation in terms of Jacobi and Weierstrass Elliptic Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Alvaro H. Salas ◽  
Lorenzo J. Martinez H ◽  
David L. Ocampo R
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Soliton Solutions ◽  
Elliptic Functions ◽  
Jacobi Elliptic Function ◽  
Series Method ◽  
Power Series Method ◽  
Weierstrass Elliptic Function ◽  
Generalized Bbm Equation ◽  
Bbm Equation

The Jacobi elliptic function method is applied to solve the generalized Benjamin-Bona-Mahony equation (BBM). Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. A power series method is also applied in some particular cases. Some solutions are expressed in terms of the Weierstrass elliptic function.

Elliptic Functions with Disconnected Julia Sets

2016 ◽  
Vol 26 (06) ◽  
pp. 1650095 ◽  
Author(s):  
Lorelei Koss
Keyword(s):  
Fixed Point ◽  
Critical Points ◽  
Elliptic Function ◽  
Julia Set ◽  
Julia Sets ◽  
Elliptic Functions ◽  
Equivalence Classes ◽  
Weierstrass Elliptic Function ◽  
Typical Function ◽  
Rhombic Lattice

In this paper, we investigate elliptic functions of the form [Formula: see text], where [Formula: see text] is the Weierstrass elliptic function on a real rhombic lattice. We show that a typical function in this family has a superattracting fixed point at the origin and five other equivalence classes of critical points. We investigate conditions on the lattice which guarantee that [Formula: see text] has a double toral band, and we show that this family contains the first known examples of elliptic functions for which the Julia set is disconnected but not Cantor.

LOCALLY SIERPINSKI JULIA SETS OF WEIERSTRASS ELLIPTIC ℘ FUNCTIONS

2006 ◽  
Vol 16 (05) ◽  
pp. 1505-1520 ◽  
Author(s):  
JANE M. HAWKINS ◽  
DANIEL M. LOOK
Keyword(s):  
Elliptic Function ◽  
Fundamental Domain ◽  
Julia Set ◽  
Julia Sets ◽  
Sufficient Conditions ◽  
Elliptic Functions ◽  
Weierstrass Elliptic Function ◽  
Naturally Occurring ◽  
Sierpiński Curve ◽  
Sierpinski Curve

We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpinski Julia sets for certain elliptic functions. We give examples satisfying these conditions. We show this results in naturally occurring Sierpinski curves in the plane, sphere and torus as well.

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.

Sign in / Sign up

Export Citation Format

Share Document