**3. Method of solution using regular perturbation method**

It is very difficult to develop closed-form solution for the above non-linear Eq. (10). Therefore, in this work, recourse is made to apply a relatively simple and accurate method approximate analytical method, the perturbation method.

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

Perturbation theory is based on the fact that the equation(s) describing the phenomena or process under investigation contain(s) a small parameter (or several small parameters), explicitly or implicitly. Therefore, the perturbation method is applicable to very small magnitudes of ε where the nonlinearity is slightly effective. Although, it has been shown to have a good accuracy, even for relatively large values of the perturbation parameter, *ε* [1, 2].

In solving Eq. (10), one needs to expand the dimensionless temperature as

$$
\theta = \theta\_0 + \varepsilon \theta\_1 + \varepsilon^2 \theta\_2 + \dots \tag{13}
$$

Substituting Eq. (13) into Eq. (10), up to first order approximate, we have

$$\frac{d^2\theta\_0}{dX^2} - \left(M^2 + Nr + Ha\right)\theta\_0 + \varepsilon \left[\frac{d^2\theta\_1}{dX^2} + \theta\_0 \frac{d^2\theta\_0}{dX^2} + \left(\frac{d\theta\_0}{dX}\right)^2 - \left(M^2 + Nr + Ha\right)\theta\_1\right]$$

$$= \varepsilon^2 \left[\frac{d^2\theta\_2}{dX^2} + \theta\_1 \frac{d^2\theta\_0}{dX^2} + \theta\_0 \frac{d^2\theta\_1}{dX^2} + 2\left(\frac{d\theta\_1}{dX}\right)\left(\frac{d\theta\_0}{dX}\right) - \left(M^2 + Nr + Ha\right)\theta\_2\right] = 0\tag{14}$$

Leading order and first order equations with the appropriate boundary conditions are given as:

**Leading order equation:**

$$\frac{d^2\theta\_o}{dx^2} - \left(M^2 + Nr + Ha\right)\theta\_o = 0\tag{15}$$

Subject to:

$$X = 0, \quad \frac{d\theta\_o}{dX} = -Bi\_{e, \text{eff}}\theta\_o \tag{16}$$

$$X = \mathbf{1}, \quad \frac{d\theta\_o}{dX} = Bi\_{c,\theta\overline{f}}(\theta\_o - \mathbf{1}) \tag{17}$$

#### **First-order equation:**

$$\frac{d^2\theta\_1}{dX^2} - \left(M^2 + Nr + Ha\right)\theta\_1 = -\left(\frac{d\theta\_o}{dX}\right)^2 - \theta\_0 \frac{d^2\theta\_o}{dX^2} \tag{18}$$

Subject to:

$$X = 0, \quad \theta\_0 \frac{d\theta\_0}{dX} + \frac{d\theta\_1}{dX} = -Bi\_{\epsilon, \theta \overline{f}} \theta\_1 \tag{19}$$

$$dX = \mathbf{1}, \quad \theta\_0 \frac{d\theta\_0}{dX} + \frac{d\theta\_1}{dX} = Bi\_{\text{c.eff}} \theta\_1 \tag{20}$$

#### **Second-order equation**

$$\frac{d^2\theta\_2}{dX^2} - \left(M^2 + Nr + Ha\right)\theta\_2 = -\theta\_1 \frac{d^2\theta\_0}{dX^2} - \theta\_0 \frac{d^2\theta\_1}{dX^2} - 2\left(\frac{d\theta\_1}{dX}\right)\left(\frac{d\theta\_0}{dX}\right) \tag{21}$$

The boundary conditions

$$dX = 0, \quad \theta\_1 \frac{d\theta\_o}{dX} + \theta\_0 \frac{d\theta\_1}{dX} + \frac{d\theta\_2}{dX} = -Bi\_{e, \text{eff}} \theta\_2 \tag{22}$$

$$X = 1, \quad \theta\_1 \frac{d\theta\_0}{dX} + \theta\_0 \frac{d\theta\_1}{dX} + \frac{d\theta\_2}{dX} = Bi\_{\text{c,eff}}\theta\_2\tag{23}$$

It can be shown from Eq. (15), (18) and (21) with the corresponding boundary conditions of Eqs. (16), (19) and (22) that the:

Leading order solution for θ<sup>o</sup> is

$$\theta\_{o} = \frac{Bi\_{i}\left\{\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\cosh\left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\mathbf{X}\right) - Bi\_{i}\sinh\left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\mathbf{X}\right)\right\}}{\begin{cases} Bi\_{i}\left\{\left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\right)\cosh\left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\right) - Bi\_{i}\sinh\left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\right)\right\}}\\ + \sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\left\{\mathrm{Bi}\_{i}\cosh\left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\right) - \left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\right)\sinh\left(\sqrt{\left(\mathbf{M}^{2} + N\mathbf{r} + Ha\right)}\right)\right\} \end{cases} \tag{2.4}$$

While the first order solution θ<sup>1</sup> is

The second-order solution θ<sup>2</sup> is too huge to be included in the manuscript. On substituting Eqs. (24) and (25) into Eq. (13) up to the first order (i.e. neglecting the higher orders), one arrives at

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