In fluid mechanics and heat transfer many engineering problems are basically nonlinear. The majority of these problems do not have analytical solutions. Using numerical techniques, some of them can be solved and some can be solved analytically. Stability and convergence should be considered so that the divergence can be avoided in results obtained by numerical method. To find the analytic solution of these non linear equations, we need other methods such as perturbation method [1] . But in perturbation method, a small parameter is needed that is to be inserted in the equation. The main deficiency of perturbation method is to find that small parameter which is exerted it into the equation.

1992 [2] [3] , Liao introduced the homotopy analysis method (HAM) which doesn’t need such small/large parameter and then he further improved and developed this method. The most significant feature of this method is that region of convergence can be controlled and adjusted, in comparison to other methods. It should be necessary to mention that the Homotopy Perturbation Method (HPM) brought forth in 1998, is only a particular case of HAM [4] [5] .

HAM based on introduction of homotopy in topology combined with the traditional perturbation method for the solution of non linear equations, but contrary to the traditional perturbation methods, HAM doesn’t need a small perturbation parameter in the equation. A homotopy is constructed with an embedding parameter p. Values are given to p from zero to one. The problem under consideration takes a convenient simple form which presents a closed form of analytical solution as p = 0. Similarly if the value of p increases, that finally takes the value one. Then at this stage, the solution of the original problem is obtained. The reliability of HAM also depends on two other auxiliary parameters, first is the parameter and the other one is function. The choice of that function is to be practiced with in order to find out the optimal solution. With the help of -curve, the convergence of the analytic solution is ensured. HAM is a novel technique [6] [7] , which has been used by many researchers for solving non linear ordinary differential equation. Recently, HAM has applied by many researchers to find the solution of different problems in science and engineering. Ayub [8] has considered the problem of steady, third grade fluid flowing past an infinite porous plate and for exact analytical solution of the governing non-linear differential equation, he uses HAM. On the basis of HAM, Wang and Pop [9] also proposed exact analytic solutions for flow within a non-Newtonian fluid film whose motion is caused solely by the unsteady stretching of a horizontal elastic surface. Wang [10] applied HAM to find out the explicit analytic solution of the Volterra equation. The comparison of HAM and HPM through a linear partial differential equation has been made by Liang and Jeffrey [11] . To numerically approximate the Eigen values of the fractional Sturm-Liouvile problems, Abbasbandy and Shirzadi [12] used HAM. The application of HAM was considered by Nassar [13] , as he suggested that by using HAM the solution of the nonlinear Poisson-Boltzmann equation for semiconductor devices was extremely good analytical approximations. Zaman [14] studied the series solution of stagnation point flow with mass transfer along an accelerated vertical porous plate with suction by means of HAM and he found out an exact analytical solution. Also Zaman, H. [15] proposed exact analytic series solution for heat transfer from a continuous surface in a parallel free stream of viscoelastic fluid. The above discussion shows that HAM is more flexible, valid and effective for the solution of non linear ordinary differential equations arises in science and engineering.

In the present study, we use the homotopy analysis method for the solution of two non linear ordinary differential equations introduced by Mandal [16] . The distribution of the paper is as follows. In Section 2, the methematical formulation of the problem is presented. The basic idea of HAM and solution by means of HAM is discussed in Section 3. The convergence of the obtained series solutions is carefully analyzed in Section 4. The graphical results, table and discusion are presented in Section 5. The conclution is presented in Section 6.

For HAM solution, the rate of convergence of approximation strongly depends on auxiliary parameter ħ, which has big effect on the region of convergence. Due to the region of convergence and rate of convergence of a series are essentially determined by the base functions, its convergence is guaranteed.

We observe that the series solutions (24) and (25) contain the non-zero auxilary parameter and. The suitable values for and can be compute by -curve. Figure 2(a) shows the variation of with using the 12^th order of HAM approximation. Figure 2(a) describes that convergent result can be find out by taking a value of from the range. To find the proper value for, we take the same case as discussed by Mandal [16] for permeability parameter and the three values of suction/injection parameter S = −1, 0, 1 in Figure 3(a). Also for exponentail parameter N = 1 and for the same three values of suction/injection parameter S = −1, 0, 1 in Figure 3(b). We tried different values of in the range

for Figure 3(a) and Figure 3(b) and compared with results obtained by Mandal [16] , which shows that the best value for is −0.01. Figure 2(b). represents the range for the admisible values for is also. For

finding the suitable value of, we consider the same case as considered by Mandal [16] shown in Figure 4(b), Figure 5(b), Figure 6(b), Figure 6(d) and Figure 7. We tried different values of in the range and compared with results obtained by Mandal [16] , which shows that the best value for is −0.01 ,-0.1 and −0.11.

In this paper, we consider the same study as discussed by Mandal [16] . Solution of Equations (8) and (9) with the boundary conditions (10) and (11) are find out by means of HAM. For the varification of accuracy of the results obtained by HAM a comparison is made with the results, obtained by Mandal [16] , Magyari and Keller [26] , Bidin and Nazar [23] , El-Aziz [29] and Ishak [25] , for prescribed surface temperature (PST) instead of variable surface heat flux. For this comparison, we also consider the boundary conditions for temprature as follows:

At and as where is the temprature at the sheet.

At last, the boundary conditions take the given form:

At and as (PST).

A comparison is made for the obtained results for precribed surface temprature (PST) coresponding to the values of heat transfer coefficient for different values of parandal number with N = 1, , N = 1 and S = 0 (i.e. for non-porous medium and in the absence of suction/blowing at the boundary) with the available published results of Magyari and Keller [26] , Bidin and Nazar [23] , El-Aziz [29] , Ishak [25] and Mandal [16] are presented in Table 1. The obtained results show an excellent agreement. These results are computed by different values of and from their intervals of convergence respectively, for different orders of approximations.

The analytical solution obtained by HAM has a high order of accuracy with a few iterations.

In order to analyze the effect of various parameters on the flow and temperature profile, analytical computations have been carried out for variable surface heat flux(VHF) using the HAM consider the same case as described by Mandal [16] . To see the effect of different parameters of interest on the velocity and tempprature profile, we have plotted the Figures 3(b)-7.

Let us first discuus with the existence of suction at the wall the effect of exponential parameter N on velocity and temprature profile.

We consider the same case as discussed by Mandal [16] . Figure 4(a) and Figure 4(b) present the effect of exponential parameter N with variable surface heat flux on velocity and temprature profile. Both velocity and temprature decreases with increasing in N due to decreasing behaviour of the momentum and thermal boundary layer thickness. From these figures, it is intresting to mention that the wall temprature decreses for positive value of Nthroughout in the boundary layer.

for exponentially streaching sheet, to see the effect of suction/blowing parameter S on velocity and temprature profile are presented in Figure 5(a) and Figure 5(b) respectively. It is noticed that by increasing suction velocity dereases significantly whereas with increse in blowing fluid velocity is found to increase (Figure 5(a)). It is notice that for the wall suction (S > 0) the boundary layer thikness decreases and the velocity field is reduced. The case of non-porous sretching sheet is represent by S = 0. Noted opposite behaviour for blowing (S < 0). If stronger blowing is consideredthe heated fluid is pushed far from the wall where the flow is accelerated due to less influence of the viscosity. This behaviour increases maximum velocity in the boundary layer. In case of suction the same rule is working but in opposite direction. Figure 5(b) represents the temprature profile for variable suction/blowing parameter S with surface heat flux. By increasing suction it is observerd that decreases whereas temprature increases due to blowing (Figure 5(b)). Actually, the effect of suction more uniform within the boundary layer. At the surface, imposition of fluid suction has the tendency to reduce both the thermal thickness and hydrodynamic of the boundary layer where viscous effect domminate. Due to this effect both the fluid velocity and temnprature are reduced. Alternatively, with injection the thermal boundary layer thickness increase due to which rate of heat transfer decreases.

For the case S = 0, the influence of permeabikity parameter on velocity and temprature are exhibited in Figure 6(a) and Figure 6(b) repectivelly. It is clear that in the presence of a porous medium the fluid flow hashigher restrictionthat, in turn, slows in motion. consequently, at the surface the shear stress increases [20] . Therefore, with increases in permiability parameter, increases the resistance to the fluid motion. With this effect the fluid velocity decreases (Figure 6(a)) and due to which in the boundary layer there is rise in temprature (Figure 6(b)) which implyes that the heat transfer rate improves by Darcian body force. It can thus be infered that an increase in permiability parameter derease the boundary layer thickniss and consequenctly increases in the rate of heat transfer.

In the presence of suction, Figure 4(c) and Figure 4(d) show the effect of the permiability parameter on the velocity and temprature profiles respectivelly. In the presnce of suction, fluid velocity is suppressed a bit more (compare to S = 0 case) with increasing permiiability parameter (Figure 6(c)). In this case, a little bit temprature is found to increase (Figure 6(d)) due to combined effect of permiability parameter and suction.

Figure 7 presents the effect of prandtl number Pr on the temprature profile. In presence of variable heat flux, the temprature decreases with the Pr. The thermal bundary layer thicness is reduced due to increase in Pr. In heat transfer problem, the relative thickneing of the momentum and thermul boundary layeris controlled by the prandtl number Pr. With a small Prandtl number Pr, heat diffuses fastly compared to the velocity ( momentum), that is for liquid metals, the thickness of the boundary layer is much bigger than momentum boundary layer. That fluids whichhave higher thermal conductivity with lower Prandtl number (and thicker thermal boundary layer structures) can diffuse from the sheet faster as compared to that fluid which have higher Pr fluids (thinner htermal boundary layers). Thus, Prandtl increases the rate of cooling in conducting flows [20] .

Figure 6(a) shows the behaviour of skin-friction coefficient with exponential parameter N for three different values of suction/blowing parameter S. It is observed that skin-friction coefficient increase with N. The skin-friction coefficient it is higher for suction than that of blowing.

It is observed from = Figure 3(a) that at the wall stress is negative, negative sign of physically indicates that surface exerts a drugging force on the fluid and positive sign indicates the opposite.

Figure 3(b) diplays against the permeability parameter the nature of skin-friction coefficient. By increasing permiability parameter the Skin-friction coefficient increase.Due to increase of permiabi-

lity parameter the skin-friction alsoincrease. An additional shear stress on the boundary is introduced by the permeability parameter.

In this paper, the homotopy analysis method is used to obtain the analytical solutions of a non linear Ordinary differential equations related to the boundary layer flow and heat transfer flow of a viscous and incompressible fluid towards an exponentially stretching porous sheet with surface heat flux in porous medium. The convergence of the HAM solution is discussed in detail. Definitely, the HAM gives us a simple way to control the convergence of series. That is the fundamental difference between the HAM and other analytical methods. The effect of the emerging parameters is discussed and the results are presented graphically. The results obtained by HAM are compared with the numerical results as discussed in the literature and with other stated available results. The comparision shows acceptable agreement between analytical and numerical solutions.