**1. Introduction**

Both the ordinary and partial differential equations have an important role in explaining many phenomena that occur in nature or in medical engineering, biotechnology, economic, ocean, plasma physics, etc. [1, 2]. Duffing equation is considered one of the most important differential equations due to its ability for demonstrating the scenario and mechanism of various nonlinear phenomena that occur in nonlinear dynamic systems [3–11]. It is one of the most common models for analyzing and modeling many nonlinear phenomena in various fields of science such as the mechanical engineering [12], electrical engineering [13], plasma physics [14, 15], etc. Mathematically, the Duffing oscillator is a second-order ordinary differential equation with a nonlinear restoring force of odd power

$$\begin{cases} \ddot{\mathfrak{x}} + f(\mathfrak{x}) = \mathbf{0}, \\ f(\mathfrak{x}) = \sum\_{i=1}^{\infty} K\_i \mathfrak{x}^{2i-1}, \end{cases} \tag{1}$$

where *f*ð Þ¼� �*x f x*ð Þ is a continuous function on some interval ½ � �*A*, *A* with *f*ð Þ¼ 0 0, *Ki* is a physical coefficient related to the physical problem under study, and *i* ¼ 1, 2, 3, ⋯∞. It is clear from Eq. (1) that there is no any friction/dissipation (this force arises either as a result of taking viscosity into account or the collisions between the oscillator and any other particle, etc.), and this only occurs in standardized systems such as superfluid (fluid with zero viscosity which it flows without losing any part from its kinetic energy *sometimes* like Bose–Einstein condensation) or the systems isolated from all the external force that resist the motion of the oscillator. The undamped Duffing equation [9] is considered one of the effective and good models for explaining many nonlinear phenomena that are created and propagated in optical fiber, Ocean, water tank, the laboratory and space collisionless and warm plasma (we will demonstrate this point below). As well known in fluid mechanics and in the fluid theory of plasma physics; the basic fluid equations of any plasma model can be reduced to a diverse series of evolution equations that can describe all phenomena that create and propagate in these physical models. For example, we can mention some of the most famous evolution equations that have been used to explain several phenomena in plasma physics and other fields of sciences; the family of one dimensional (1 � *D*) korteweg–de Vries equation (KdV) and it is higher-orders, including the KdV, KdV-Burgers (KdVB), modified KdV (mKdV), mKdV-Burgers (mKdVB), Gardner equation or called Extended KdV (EKdV), EKdV-Burgers (EKdVB), KdV-type equation with higherorder nonlinearity. All the above mentioned equations are partial differential equations and by using an appropriate transformation, we can convert them into ordinary differential equations of the second orders. If the frictional force is neglected, some of these equations can be converted into the undamped Duffing equation with *f x*ð Þ≈*P x*ð Þ¼ *<sup>K</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>K</sup>*2*x*<sup>3</sup> like the mKdV equation, the KdV equation can be transformed to the undamped Helmholtz equation with *f x*ð Þ≈*P x*ð Þ¼ *<sup>K</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>K</sup>*2*x*<sup>2</sup> [16], the Gardner equation can be converted into the undamped H-D equation for *f x*ð Þ≈*P x*ð Þ¼ *<sup>K</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>K</sup>*2*x*<sup>2</sup> <sup>þ</sup> *<sup>K</sup>*3*x*<sup>3</sup> [17, 18], and so on the other mentioned equations.

However, these undamped models (without friction/dissipation) do not exist much in reality except under harsh conditions. In order to describe and simulate the natural phenomena that arise in many realistic physical models and dynamic systems, the friction/dissipation forces must be taken into account, as is the case in many plasma models and electronic systems. Accordingly, the following damped (non-conservative) Duffing equation will be devoted for this purpose

$$
\ddot{\boldsymbol{x}} + 2\boldsymbol{\epsilon}\,\dot{\boldsymbol{x}} + f(\boldsymbol{x}) = \mathbf{0}.\tag{2}
$$

If the frictional force does not neglect, so that all PDEs that have "Burgers � *∂*2 *<sup>x</sup>*ð Þ� " term like KdVB-, mKdVB-, EKdVB-, KPB-, mKPB-, EKPB-, ZKB-, mZKB, EZKB-Eq. [1, 2], etc. can be transformed to damped Duffing equation (*x*€ <sup>þ</sup> <sup>2</sup>*<sup>ε</sup> <sup>x</sup>*\_ <sup>þ</sup> *<sup>K</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>K</sup>*2*x*<sup>3</sup> <sup>¼</sup> 0), damped Helmholtz equation (*x*€ <sup>þ</sup> <sup>2</sup>*<sup>ε</sup> <sup>x</sup>*\_ <sup>þ</sup> *<sup>K</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>K</sup>*2*x*<sup>2</sup> <sup>¼</sup> 0), and damped Duffing-Helmholtz equation (*x*€ <sup>þ</sup> <sup>2</sup>*<sup>ε</sup> <sup>x</sup>*\_ <sup>þ</sup> *<sup>K</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>K</sup>*2*x*<sup>2</sup> <sup>þ</sup> *<sup>K</sup>*2*x*<sup>3</sup> <sup>¼</sup> 0). Eq. (2) without [19] and with [7, 20, 21] including damping term (2*<sup>ε</sup> <sup>x</sup>*\_ ) for *f x*ð Þ<sup>≈</sup> *P x*ð Þ¼ *<sup>K</sup>*1*<sup>x</sup>* <sup>þ</sup> *<sup>K</sup>*2*x*<sup>3</sup> has been investigated and solved analytically and numerically by many authors using different approaches in order to understand its physical characters [22–28].

Many authors investigated the (un)damped Duffing equation, (un)damped Helmholtz Eq. [16, 29–31], and undamped H-D equation. On the contrary, there is a *Analytical Solutions of Some Strong Nonlinear Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97677*

few numbers of published papers about damped Duffing-Helmholtz equation [32, 33]. For example, Zúñiga [32] derived a semi-analytical solution to the damped Duffing-Helmholtz equation in the form of Jacobian elliptic functions, but he putted some restrictions on the coefficient of the linear term, and then obtained a solution that gives good results compared to numerical solutions. Also, it is noticed that Zúñiga solution [32] is very sensitive to the initial conditions. Gusso and Pimentel [33] obtained obtain improved approximate analytical solution to the forced and damped Duffing-Helmholtz in the form of a truncated Fourier series utilizing the harmonic balance method.

In this chapter, we display some novel semi-analytical (approximate analytical) solutions to the strong higher-order nonlinear damped oscillators of the following initial value problem (i.v.p)

$$\begin{cases} \ddot{\boldsymbol{x}} + 2\boldsymbol{\varepsilon}\,\dot{\boldsymbol{x}} + p\boldsymbol{x} + q\boldsymbol{x}^{3} + r\boldsymbol{x}^{5} = \mathcal{F}(t),\\ \boldsymbol{x}(\mathbf{0}) = \boldsymbol{x}\_{0} \,\boldsymbol{\mathfrak{E}}\,\boldsymbol{x}'(\mathbf{0}) = \dot{\boldsymbol{x}}\_{0}, \end{cases} \tag{3}$$

and its family (*ε* ¼ 0 or *r* ¼ 0 or *ε* ¼ *r* ¼ 0).

Our new semi-analytical solution to Eq. (3) is derived in terms of Weierstrass and Jacobian elliptic functions. Also, we will solve Eq. (3) numerically using Runge–Kutta 4*th* (RK4) and make a comparison between both the semi-analytical and numerical solutions. Moreover, as some realistic physical application to the problem (3) and its family will be investigated.
