**4. Basic idea of new Homotopy perturbation method**

First, following homotopy is constructed for solving heat conduction equation using NHPM

$$p(1-p)\left(\frac{\partial T}{\partial \theta} - U\_0\right) + p\left(\frac{\partial T}{\partial \theta} - \beta \frac{\partial^2 T}{\partial \mathbf{z}^2}\right) = \mathbf{0} \tag{12}$$

Taking *<sup>L</sup>*�<sup>1</sup> <sup>¼</sup> <sup>Ð</sup> *<sup>θ</sup> θ*0 ð Þ*: dθ* i.e. inverse operator on Eq. (12), then

$$T(\mathbf{z}, \theta) = \int\_{\theta\_0}^{\theta} U\_0(\mathbf{z}, \theta) d\theta - p \int\_{\theta\_0}^{\theta} \left( U\_0 - \beta \frac{\partial^2 T}{\partial \mathbf{z}^2} \right) d\theta + T(\mathbf{z}, \theta\_0) \tag{13}$$
 
$$\text{Where } T(\mathbf{z}, \theta\_0) = U(\mathbf{z}, \theta\_0)$$

Let the solution of Eq. (13) is given by

$$T = T\_0 + pT\_1 + p^2T\_2 + p^3T\_3 + \dots \tag{14}$$

where *T*0, *T*1, *T*2, *T*3, … are to be determined.

Suppose solution given by Eq. (14) is the solution of Eq. (13). On comparing the coefficients of powers of *p* and equating to zero and using Eq. (14) in Eq. (13), following are obtained:

$$p^0: T\_0(\mathbf{z}, \theta) = \int\_{\theta\_0}^{\theta} U\_0(\mathbf{z}, \theta) d\theta + T(\mathbf{z}, \theta\_0)$$

$$p^1: T\_1(\mathbf{z}, \theta) = -\int\_{\theta\_0}^{\theta} \left( U\_0(\mathbf{z}, \theta) - \beta \frac{\partial^2 T\_0}{\partial \mathbf{z}^2} \right) d\theta$$

$$p^2: T\_2(\mathbf{z}, \theta) = \int\_{\theta\_0}^{\theta} \left( \beta \frac{\partial^2 T\_1}{\partial \mathbf{z}^2} \right) d\theta$$

$$p^3: T\_3(\mathbf{z}, \theta) = \int\_{\theta\_0}^{\theta} \left( \beta \frac{\partial^2 T\_2}{\partial \mathbf{z}^2} \right) d\theta$$

$$\text{and so on...} \tag{15}$$

Consider the initial approximation of Eq. (1) as

$$U\_0(z, \theta) = \sum\_{n=0}^{\infty} c\_n(z) P\_n(\theta), T(z, \mathbf{0}) = U(z, \mathbf{0}), P\_k(\theta) = \theta^k,\tag{16}$$

where, *P*1ð Þ*θ* , *P*2ð Þ*θ* , *P*3ð Þ*θ* , … and *c*0ð Þ*z* ,*c*1ð Þ*z* ,*c*2ð Þ*z* , … are specified functions and unknown coefficients respectively, depending on the problem.

Using Eq. (16) in (15), following are obtained:

$$T\_0(\mathbf{z}, \theta) = \left(c\_0(\mathbf{z})\theta + c\_1(\mathbf{z})\frac{\theta^2}{2} + c\_2(\mathbf{z})\frac{\theta^3}{3} + c\_3(\mathbf{z})\frac{\theta^4}{4} + \dots\right) + U(\mathbf{z}, \mathbf{0})\theta$$

*Application of Perturbation Theory in Heat Flow Analysis DOI: http://dx.doi.org/10.5772/intechopen.95573*

$$T\_1(\mathbf{z}, \boldsymbol{\theta}) = \left(-c\_0(\mathbf{z}) - \beta \pi^2 \sin \pi \mathbf{z}\right) \boldsymbol{\theta} + \left(-\frac{1}{2}c\_1(\mathbf{z}) + \frac{1}{2}\beta c\_0""(\mathbf{z})\right) \boldsymbol{\theta}^2$$

$$+ \left(-\frac{1}{3}c\_2(\mathbf{z}) + \frac{1}{3}c\_1""(\mathbf{z})\right) \boldsymbol{\theta}^3 + \dots$$

$$\text{and so on } \dots \tag{17}$$

Now solving the above equations in such a manner that, *T*1ð Þ¼ *z*, *θ* 0 . Therefore Eq. (17) reduces to

$$T\_1(z, \theta) = T\_2(z, \theta) = \dots = \mathbf{0}.$$

So *U z*ð Þ¼ , *<sup>θ</sup> <sup>T</sup>*0ð Þ¼ *<sup>z</sup>*, *<sup>θ</sup>* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*cn*ð Þ*<sup>z</sup> Pn*ð Þ*<sup>θ</sup>* is obtained solution which is found to be exactly same as the exact solution obtained through method of separation of variable.

If *U*0ð Þ *z*, *θ* is analytic at *θ* ¼ *θ*0,

*<sup>U</sup>*0ð Þ¼ *<sup>z</sup>*, *<sup>θ</sup>* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*cn*ð Þ*<sup>z</sup>* ð Þ *<sup>θ</sup>* � *<sup>θ</sup>*<sup>0</sup> *<sup>n</sup>* is the taylor series expansion which can be used in Eq. (9).
