Solution to a Damped Duffing Equation Using He's Frequency Approach

In this paper, we generalize He's frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt–Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques.


Introduction
e deep understanding of the mechanism of nonlinear oscillations has an effective role in interpreting the ambiguities of many natural, physical, and engineering phenomena in various fields of science. Accordingly, many researchers have been able to give correct scientific explanations about their scientific experiences based on a deep understanding of the characteristics of these phenomena after the clarity of the ambiguity about the phenomenon under study. In the framework of nonlinear dynamics, there is no doubt that the scenario of dynamic mechanism of the pendulum motion is one of the objects that have deserved more attention in modeling all kind of phenomena related to oscillations, bifurcations, and chaos. e simple pendulum has been used as a physical model to solve problems such as nonlinear plasma oscillations, Duffing oscillators, Helmholtz oscillations, rigid plates that satisfy the Johanessen performance criteria, transverse vibrations nonlinear of a plate carrying a concentrated mass, a beam supported by a double periodic axial oscillating mount, cracks subjected to concentrated forces, surface waves in a plasma column, coupled modes of nonlinear bending vibrations of a circular ring, double spin spacecraft, motion of spacecraft over slowly rotating asteroids, nonlinear vibration of clasped beams, the nonlinear equation of wave, and nonlinear mathematical models of DNA.
It is known that the main objective of the numerical approaches is to find some numerical solutions to various realistic physical, engineering, and natural problems, especially when exact solutions are unavailable or extremely difficult to determine. ere are many numerical approaches that were used for analyzing the family of the Duffing oscillator and Duffing-Helmholtz oscillator with constant coefficients. It is known that this family is integrable, i.e., its exact solution is available in the absence of the damping effect. On the other hand, if the damping effect and some other friction forces are taken into account, we get a nonintegral differential equation, i.e., its exact solution is not available. e nonlinear oscillators have many applications in science and engineering. One of such oscillators is the Duffing equation. George Duffing, a German engineer, wrote a comprehensive book about this in 1918. Since then, there has been a tremendous amount of work done on this equation, including the development of solution methods (both analytical and numerical), and the use of these methods to investigate the dynamic behavior of physical systems that are described by the various forms of the Duffing equation.
In this paper, we consider the damped and unforced Duffing equation. We solve it using an extended version of He's frequency approach for the damped case. is oscillator was solved in [11] using a generalized elliptic functions. A simplification of the solution in [11] may be obtained by approximating the elliptic functions by means of trigonometric functions [12].

The Damped Duffing Oscillator
Let us consider the i.v.p.
e damped oscillator (1) is integrable only when α � 8/9ε 2 . Equation (1) represents a damped Duffing oscillator. In the case when ε � 0, we have an undamped Duffing oscillator which has an exact solution for any given arbitrary initial conditions. More precisely, the exact solution to the i.v.p.
is given by where e solution is periodic with period where We may use the following approximation formulas for evaluating the elliptic integral K(m) (see Table 1).
See Table 2 for the approximation of 1/K(m). For example, where Let us consider the i.v.p.
A very good approximate analytical solution is given by where Now, observe that the function x(t) � cn(t, m) obeys the Duffing equation We have the following approximations for 0 ≤ m ≤ 0.5 e approximations above may be used to give trigonometric solution to the i.v.p. (9) whose exact solution reads For example, x(t) � Acos 1 − 10 33 e frequency-amplitude formulation for this solution is given by We obtain several frequency-amplitude formulations using formulas. e He's frequency-amplitude formulation for the Duffing equation € x + αx + βx 3 � 0 establishes that

He's Approach
Using He's frequency approach, an approximate analytical solution in the absence of damping (ε � 0) may be obtained using Let Observe that w(0) � 0. He's idea for the undamped case is based on the following fact: Following this idea, for the damped case, we will replace e approximate analytical solution for the damped Duffing equation (1) will then be From (22) it follows that e number ρ is a free parameter that is chosen in order to get as small residual error as possible. e default value is ρ � ε.

Homotopy Perturbation Method
We seek a solution in the ansatz form: e homotopy is defined as follows:

e Scientific World Journal
We have We now equate to zero the coefficients of p j (j � 0, 1, 2, 3, . . .) and then we obtain an ode system. We solve these odes so that the functions v j (t)(j � 1, 2, 3, . . .) do not contain secularity terms. e solutions are

First Approach. Assume the ansatz
We will choose the function ω � ω(t) so that e Scientific World Journal 5 then, where Observe that In other words, when ε � ρ � 0, this corresponds to the He's frequency formulation for the Duffing equation: e number ρ is chosen in order to minimize the residual error. A default value for ρ is obtained by eliminating A from the system: is last condition gives the sextic: A root to this sextic near ρ � ε may be evaluated using the following approximation: e value of A is given by

Second Approach.
e exact solution to the undamped Duffing equation € x + αx + βx 3 � 0 is given by where w(t) � (43) Let us replace α with α − 2ρε + ρ 2 and c 0 with c 0 exp(− ρt) so that For the damped Duffing equation we will define the solution in the form: e constants c 0 and c 1 are determined from the initial conditions. On the other hand, e numbers ρ and λ are free parameters that we choose in order to get as small residual error as possible. e default ρ value is ρ � (2/3ε) and the default value for λ is λ � 1. Observe that this ansatz will give the exact solution for the integrable case, i.e., when ρ � (2/3ε) and α � (8/9ε 2 ).

Lindstedt-Poincaré Method
We seek a solution in the ansatz form: then.
We now equate to zero the coefficients of β j (j � 0, 1, 2, 3, . . .) and then we obtain an ode system. We solve these odes so that the functions u j (t)(j � 1, 2, 3, . . .) do not contain secularity terms. e solutions are e Scientific World Journal 7 e constants c 0 and c 1 are obtained from the initial conditions.

Numerical Solution
We make use of the following backward finite differences formulas for the first and second derivatives: e discretized ode reads

Data Availability
No data were used to support this paper.

Conflicts of Interest
e authors declare no conflicts of interest with regard to any individual or organization.