*6.1.1 Galerkin decomposition method*

With the application of Galerkin decomposition procedure, the governing partial differential equations of motion can be separated into spatial and temporal parts of the lateral displacement function as

$$w(\mathbf{x}, t) = \phi(\mathbf{x}) q(t) \tag{74}$$

Using one-parameter Galerkin decomposition procedure, one arrives at

$$\int\_{0}^{1} R(\varkappa, t)\phi(\varkappa)d\varkappa = 0\tag{75}$$

where *R x*ð Þ , *t* is the governing equation of motion for nanobeam i.e.

$$\begin{split} R(\mathbf{x},t) &= \left[ \mathbf{1} + K\_p \hbar^2 + H a\_m \hbar^2 - a\_t^d \hbar^2 + \frac{\hbar^2}{2} \right] \left( \frac{\partial w}{\partial \mathbf{x}} \right)^2 \mathbf{dx} \left[ \frac{\partial^4 w}{\partial \mathbf{x}^4} + \left[ a\_t^d - K\_w \hbar^2 - K\_P - H a\_m - \frac{1}{2} \right] \left( \frac{\partial w}{\partial \mathbf{x}} \right)^2 \mathbf{dx} \right] \frac{\partial^2 w}{\partial \mathbf{x}^2} \\ &+ K\_u w + \frac{\partial^2 w}{\partial t^2} - \hbar^2 \frac{\partial^4 w}{\partial \mathbf{x}^2 \partial t^2} + K\_2^d \left[ w^2 - \hbar^2 \frac{\partial^2 (w^2)}{\partial \mathbf{x}^2} \right] + K\_3^d \left[ w^3 - \hbar^2 \frac{\partial^2 (w^3)}{\partial \mathbf{x}^2} \right] = \mathbf{0} \end{split} \tag{76}$$

where *ϕ*ð Þ *x* is the basis or trial or comparison function or normal function, which must satisfy the boundary conditions in Eq. (73), and *q t*ð Þ is the temporal part (time-dependent function).

Substituting Eqs. (75) into (74), then multiplying both sides of the resulting equation by *ϕ*ð Þ *x* and integrating it for the domain of (0,1), we have

$$\frac{d^2q(t)}{dt^2} + \lambda\_1 q(t) + \lambda\_2 q^2(t) + \lambda\_3 q^3(t) = 0\tag{77}$$

where

$$
\lambda\_1 = \frac{\overline{\lambda}\_1}{\overline{\lambda}\_0}; \lambda\_2 = \frac{\overline{\lambda}\_2}{\overline{\lambda}\_0}; \lambda\_3 = \frac{\overline{\lambda}\_3}{\overline{\lambda}\_0}; \tag{78}
$$

*Perturbation Methods to Analysis of Thermal, Fluid Flow and Dynamics Behaviors of… DOI: http://dx.doi.org/10.5772/intechopen.96059*

$$\overline{\lambda}\_0 = \int\_0^1 \left(\phi^2 - h^2 \phi \frac{\partial^2 \phi}{\partial \mathbf{x}^2} \right) d\mathbf{x} \tag{79}$$

$$\overline{\lambda}\_1 = \int\_0^1 \left( K\_w \phi^2 + \left( \mathbf{1} + K\_p h^2 + H a\_m h^2 - a\_t^d h^2 \right) \phi \frac{\partial^4 \phi}{\partial x^4} + \left( a\_t^d - K\_w h^2 - K\_p - H a\_m \right) \phi \frac{\partial^2 \phi}{\partial x^2} \right) d\mathbf{x} \tag{80}$$

$$\overline{\lambda}\_2 = \int\_0 K\_2^d \left( \phi^3 - h^2 \phi \frac{\partial^2 \left( \phi^2 \right)}{\partial \mathbf{x}^2} \right) d\mathbf{x} \tag{81}$$

$$\overline{\lambda}\_3 = \int\_0^1 K\_3^d \left( \phi^4 - h^2 \phi \frac{\partial^2 (\phi^4)}{\partial x^2} \right) d\mathbf{x} + \frac{h^2}{2} \int\_0^1 \left( \frac{\partial \phi}{\partial x} \right)^2 d\mathbf{x} \int\_0^1 \phi \frac{\partial^2 \phi}{\partial x^2} d\mathbf{x} - \frac{1}{2} \int\_0^1 \left( \frac{\partial \phi}{\partial x} \right)^2 d\mathbf{x} \int\_0^1 \phi \frac{\partial^4 \phi}{\partial x^4} d\mathbf{x} \tag{82}$$

The initial conditions are given as

$$q(\mathbf{0}) = A, \quad \frac{dq(\mathbf{0})}{dt} = \mathbf{0} \tag{83}$$

A is the maximum vibration amplitude of the structure.

From the initial conditions in Eq. (83), one can write the initial approximation, *uo* as

$$u\_o = A \cos(at) \tag{84}$$

Eq. (22) satisfies the initial conditions in Eq. (83). The homotopy perturbation representation of Eq. (77) is

$$H(q, p) = \left[\frac{d^2q}{dt^2} + \lambda\_1 q\right] - \left[\frac{d^2u\_o}{dt^2} + \lambda\_1 u\_o\right] + p\left[\frac{d^2u\_o}{dt^2} + \lambda\_1 u\_o\right] + p\left(\lambda\_2 q^2 + \lambda\_3 q^3\right) = 0\tag{85}$$

From the procedure of homotopy perturbation method, assuming that the solution of Eq. (77) takes the form of:

$$q = q\_0 + pq\_1 + p^2 q\_2 + p^3 q\_3 + \dots,\tag{86}$$

On substituting Eqs. (86) into the homotopy Eq. (85)

$$H(q, p) = \left[\frac{d^2(q\_0 + pq\_1 + p^2q\_2 + p^3q\_3 + \dots)}{dt^2} + \lambda\_1(q\_0 + pq\_1 + p^2q\_2 + p^3q\_3 + \dots)\right]$$

$$-\left[\frac{d^2u\_o}{dt^2} + \lambda\_1u\_0\right] + p\left[\frac{d^2u\_o}{dt^2} + \lambda\_1u\_0\right] + p\left(\begin{matrix}\lambda\_2(q\_0 + pq\_1 + p^2q\_2 + p^3q\_3 + \dots)^2\\ +\lambda\_3(q\_0 + pq\_1 + p^2q\_2 + p^3q\_3 + \dots)^3\\ \end{matrix}\right) = 0\tag{87}$$

rearranging the coefficients of the terms with identical powers of *p*, one obtains series of linear differential equations as.

#### **Zero-order equation**

$$\left[p^{0}: \left[\frac{d^{2}q\_{0}}{dt^{2}} + \lambda\_{1}q\_{0}\right] - \left[\frac{d^{2}u\_{o}}{dt^{2}} + \lambda\_{1}u\_{o}\right] = 0\tag{88}$$

with the conditions

$$q\_0(\mathbf{0}) = A \text{ and } \frac{dq\_0(\mathbf{0})}{dt} = \mathbf{0} \tag{89}$$
