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 On the Analytical Solutions of the Forced Damping Duffing Equation in the Form of Weierstrass Elliptic Function and its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
S. A. El-Tantawy ◽  
Alvaro H. Salas ◽  
M. R. Alharthi
Keyword(s):  
Analytical Solution ◽  
Elliptic Function ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Duffing Equation ◽  
Weierstrass Elliptic Function ◽  
Pendulum Equation ◽  
Rlc Circuit ◽  
Linear Damping ◽  
Forced Term

In this study, a novel analytical solution to the integrable undamping Duffing equation with constant forced term is obtained. Also, a new approximate analytical (semianalytical) solution for the nonintegrable linear damping Duffing oscillator with constant forced term is reported. The analytical solution is given in terms of the Weierstrass elliptic function with arbitrary initial conditions. With respect to it, the semianalytical solution is constructed depending on a new ansatz and the exact solution of the standard Duffing equation (in the absence of both damping and forced terms). A comparison between the obtained solutions and the Runge–Kutta fourth-order (RK4) is carried out. Moreover, some complicated oscillator equations such as the constant forced damping pendulum equation, forced damping cubic-quintic Duffing equation, and constant forced damping Helmholtz–Duffing equation are reduced to the forced damping Duffing oscillator, in which its solution is known. As a practical application, the proposed techniques are applied to investigate the characteristics behavior of the signal oscillations arising in the RLC circuit with externally applied voltage.

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.

scholarly journals A New Approach for Solving the Complex Cubic-Quintic Duffing Oscillator Equation for Given Arbitrary Initial Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Alvaro H. Salas ◽  
Simeon Casanova Trujillo
Keyword(s):  
Differential Equation ◽  
Analytic Solution ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Integrable Case ◽  
New Approach ◽  
One Dimensional ◽  
Weierstrass Elliptic Function ◽  
Quintic Equation ◽  
The One

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.

scholarly journals PARALLEL COMPUTATIONS IN STUDIES OF DEPENDENCE OF GAS DYNAMIC PARAMETERS OF UPWARD SWIRLING FLOW OF GAS ON BLOWING VELOCITY

2016 ◽  
pp. 92-97
Author(s):  
R. E. Volkov ◽  
A. G. Obukhov
Keyword(s):  
Stokes Equations ◽  
Swirling Flow ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Gas Dynamic

The rectangular parallelepiped explicit difference schemes for the numerical solution of the complete built system of Navier-Stokes equations. These solutions describe the three-dimensional flow of a compressible viscous heat-conducting gas in a rising swirling flows, provided the forces of gravity and Coriolis. This assumes constancy of the coefficient of viscosity and thermal conductivity. The initial conditions are the features that are the exact analytical solution of the complete Navier-Stokes equations. Propose specific boundary conditions under which the upward flow of gas is modeled by blowing through the square hole in the upper surface of the computational domain. A variant of parallelization algorithm for calculating gas dynamic and energy characteristics. The results of calculations of gasdynamic parameters dependency on the speed of the vertical blowing by the time the flow of a steady state flow.

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