**5. An approximate analytic solution of the forced damped Duffing-Helmholtz equation**

Let us define the following i.v.p.

$$\begin{cases} \ddot{\mathbf{x}} + 2\boldsymbol{\epsilon}\,\dot{\mathbf{x}} + p\mathbf{x} + q\mathbf{x}^2 + r\mathbf{x}^3 = F, \\ \quad \boldsymbol{\varkappa}(\mathbf{0}) = \boldsymbol{\varkappa}\_0 \,\boldsymbol{\&}\,\mathbf{x}'(\mathbf{0}) = \dot{\boldsymbol{\varkappa}}\_0. \end{cases} \tag{69}$$

Suppose that

$$\lim\_{\varepsilon \to +\infty} \varkappa(t) = d, \varepsilon > 0,\tag{70}$$

then the first equation in system (69) can be written as

$$pd + qd^2 + rd^3 = F.\tag{71}$$

For solving the i.v.p. (69), the following ansatz is assumed

$$\mathbf{x}(t) = \exp\left(-\rho t\right)\mathbf{y}(f(t)),\tag{72}$$

with

$$f(t) = \frac{1 - \exp\left(-2(\varepsilon - \rho)t\right)}{2(\varepsilon - \rho)},\tag{73}$$

where the function *y* � *y t*ð Þ represents the exact solution to the following i.v.p.

$$\begin{cases} \mathbf{y}''(t) + \left( 3d^2r + 2dq - 2\epsilon\rho + p + \rho^2 \right) \mathbf{y}(t) + \left( 3dr + q \right) \mathbf{y}(t)^2 + r\mathbf{y}(t)^3 = \mathbf{0}, \\\ \mathbf{y}(\mathbf{0}) = \mathbf{x}\_0 - d \ \mathbf{8} \mathbf{y}'(\mathbf{0}) = \dot{\mathbf{x}}\_0 + \rho(\mathbf{x}\_0 - d). \end{cases} \tag{74}$$

Let us define the following residual

$$R(t) \equiv \ddot{\varkappa}(t) + 2\varepsilon \dot{\varkappa}(t) + p\varkappa(t) + q\varkappa^2(t) + r\varkappa^3(t) - F,\tag{75}$$

and by applying the condition *R*<sup>0</sup> ð Þ¼ 0 0, we obtain

$$\begin{aligned} 4\rho^3 - 12\epsilon\rho^2 + \left(3d^2r + 3dq + 3dr\mathbf{x}\_0 + 8\epsilon^2 + 4p + 5q\mathbf{x}\_0 + 6r\mathbf{x}\_0^2\right)\rho \\ -4\epsilon\left(d^2r + dq + dr\mathbf{x}\_0 + p + q\mathbf{x}\_0 + r\mathbf{x}\_0^2\right) = 0. \end{aligned} \tag{76}$$

By solving this equation we can get the value of *ρ*. **Example 7.**

Let

$$\begin{cases} \ddot{\boldsymbol{x}} + \mathbf{0}.02\dot{\boldsymbol{x}} + 5\boldsymbol{\varkappa} + 2\boldsymbol{\varkappa}^2 + \boldsymbol{\varkappa}^3 = \mathbf{1}/2, \\ \boldsymbol{\varkappa}(\mathbf{0}) = \mathbf{0}.1 \& \boldsymbol{\varkappa}'(\mathbf{0}) = \mathbf{0}.1. \end{cases} \tag{77}$$

*Analytical Solutions of Some Strong Nonlinear Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97677*

**Figure 7.** *A comparison between solution (78) and RK4 solution.*

The approximate analytic solution of the i.v.p. (77) reads

$$\mathbf{x}\_{\text{app}}(t) = 0.0961263$$

$$+ e^{-0.009999t} \begin{pmatrix} 0.0429135\\ -\frac{0.252808}{1 + 2.13749\wp(0.631364 - 121944.(1 - e^{-8.2 \times 10^{-4}}); 2.43588, 0.737005)} \end{pmatrix}. \tag{78}$$

The distance error as compared to the RK4 numerical solution is given by

$$\max\_{0 \le t \le 5} \left| \boldsymbol{\kappa}\_{\text{app}}(t) - \boldsymbol{\kappa}\_{\text{RK}}(t) \right| = 0.000944148. \tag{79}$$

Also, the comparison between solution (78) and RK4 solution is presented in **Figure 7**.

**Remark 5**. For the damped and constant forced Helmholtz equation

$$\begin{cases} \ddot{\mathbf{x}} + 2\boldsymbol{\varepsilon}\,\dot{\mathbf{x}} + p\mathbf{x} + q\mathbf{x}^2 = F, \\\\ \boldsymbol{\varkappa}(\mathbf{0}) = \boldsymbol{\varkappa}\_0 \,\boldsymbol{\Re}\,\mathbf{x}'(\mathbf{0}) = \dot{\boldsymbol{\varkappa}}\_0. \end{cases} \tag{80}$$

The value of *<sup>d</sup>* can be determined from: *pd* <sup>þ</sup> *qd*<sup>2</sup> <sup>¼</sup> *<sup>F</sup>:* However, if this equation has no real solutions we can choose *d* ¼ 0.

**Remark 6**. Letting *q* ¼ 0, we obtain the damped and constant forced Duffing equation

$$\begin{cases} \ddot{\boldsymbol{\kappa}} + 2\boldsymbol{\varepsilon}\,\dot{\boldsymbol{\kappa}} + p\boldsymbol{\kappa} + r\boldsymbol{\kappa}^3 = F, \\\\ \boldsymbol{\kappa}(\mathbf{0}) = \boldsymbol{\kappa}\_0 \,\boldsymbol{\Re}\,\boldsymbol{\kappa}'(\mathbf{0}) = \dot{\boldsymbol{\kappa}}\_0. \end{cases} \tag{81}$$

In this case, the number *<sup>d</sup>* must be a root to the cubic *pd* <sup>þ</sup> *rd*<sup>3</sup> <sup>¼</sup> *<sup>F</sup>:*
