**5. Method of solution by homotopy perturbation method**

The comparative advantages and the provision of acceptable analytical results with convenient convergence and stability coupled with total analytic procedures of homotopy perturbation method compel us to consider the method for solving the system of nonlinear differential equations in Eqs. (40) and (41) with the boundary conditions in Eq. (42).

### **5.1 The basic idea of homotopy perturbation method**

In order to establish the basic idea behind homotopy perturbation method, consider a system of nonlinear differential equations given as

$$A(U) - f(r) = \mathbf{0}, \quad r \in \Omega,\tag{44}$$

with the boundary conditions

$$B\left(u, \frac{\partial u}{\partial \eta}\right) = 0, \qquad r \in \Gamma,\tag{45}$$

where *A* is a general differential operator, *B* is a boundary operator, *f r*ð Þ a known analytical function and Γ is the boundary of the domain Ω.

The operator *A* can be divided into two parts, which are *L* and *N*, where *L* is a linear operator, *N* is a non-linear operator. Eq. (44) can be therefore rewritten as follows

$$L(u) + N(u) - f(r) = 0.\tag{46}$$

By the homotopy technique, a homotopy *U r*ð Þ , *p* : Ω � ½ �! 0, 1 *R* can be constructed, which satisfies

$$H(U, p) = (1 - p)[L(U) - L(U\_o)] + p[A(U) - f(r)] = 0, \quad p \in [0, 1], \tag{47}$$

or

$$H(U, p) = L(U) - L(U\_o) + pL(U\_o) + p[N(U) - f(r)] = \mathbf{0}.\tag{48}$$

In the above Eqs. (47) and (48), *p* ∈½ � 0, 1 is an embedding parameter, *uo* is an initial approximation of equation of Eq. (44), which satisfies the boundary conditions.

Also, from Eq. (47) and Eq. (48), one has

$$H(U, \mathbf{0}) = L(U) - L(U\_o) = \mathbf{0},\tag{49}$$

or

$$H(U,0) = A(U) - f(r) = 0.\tag{50}$$

The changing process of *p* from zero to unity is just that of *U r*ð Þ , *p* from *uo*ð Þ*r* to *u r*ð Þ. This is referred to homotopy in topology. Using the embedding parameter *p* as a small parameter, the solution of Eqs. (47) and Eq. (48) can be assumed to be written as a power series in p as given in Eq. (51)

$$U = U\_o + pU\_1 + p^2U\_2 + \dots \tag{51}$$

It should be pointed out that of all the values of *p* between 0 and 1, *p=1* produces the best result. Therefore, setting *p* ¼ 1*,* results in the approximation solution of Eq. (42)

$$\mu = \lim\_{p \to 1} U = U\_o + U\_1 + U\_2 + \dots \tag{52}$$

The basic idea expressed above is a combination of homotopy and perturbation method. Hence, the method is called homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantages of the traditional perturbation techniques. The series Eq. (29) is convergent for most cases.
