In order to assess the accuracy of VIM for solving nonlinear equations and to compare it with HPM and perturbation method, we will consider the following examples.

### Cooling of a Lumped System by Combined Convection and Radiation

Consider the problem of combined convective–radiative cooling of lumped system (Aziz & Na, 1984). Let the system have volume , surface area , density , specific heat emissivity and the initial temperature . At , the system is exposed to an environment with convective heat transfer with the coefficient of and the temperature . The system also loses heat through radiation and the effective sink temperature is . The cooling equation and the initial conditions are as follows:

*(1)**(2)*To solve the equation, we do the following changes of parameters:

*(3)*After the parameter change, the heat transfer equation will result in the following form:

*(4)**(5)*For simplicity, we assume the case . So we have:

*(6)**(7)*
Eqs. (6) and (7) are solved though PM, HPM and VIM as follows:

#### Variational Iteration Method

To solve Eqs. (6) and (7), we can construct the following correction functional:

*(8)*Its stationary conditions can be obtained as follows:

*(9)*The Lagrangian multiplier can therefore be identified as:

*(10)*As a result, we obtain the following iteration formula:

*(11)*Now we start with an arbitrary initial approximation that satisfies the initial condition:

*(12)*Using the above variational formula (11), we have:

*(13)*Substituting Eq. (12) into Eq. (13) and after some simplifications, we have:

*(14)*In the same way, we obtain as follows:

*(15)* and so on. In the same way the rest of the components of the iteration formula can be obtained.

#### Homotopy Perturbation Method

After separating the linear and nonlinear parts of the equation, we apply HPM to Eqs. (6) and (7), as follows:

*16) * where

is the linear part of the equation and

is the initial approximation (Rajabi, Ganji & Taherian, 2006).We consider

as:

*(17)*Substituting Eq. (17) into Eq. (16) and rearranging based on powers of terms, we have:

*(18) **(19) * and

*(20)**(21)* and finally,

*(22)**(23) *Solving Eqs. (18)- (23) results in When , we have:

*(24)*#### Perturbation Method

For very small , let us assume a regular perturbation expansion and calculate the first three terms (Nayfeh, 1973), thus we assume:

*(25)*Substituting Eq. (25) into Eq. (6) and after expansion and rearranging based on coefficients of terms, we have:

*(26)**(27)**(28)*The solutions of Eqs. (26)- (28) are:

*(29)*The three-term expansion in Eq. (25) now becomes:

*(30)*