THE GENERALIZED APPROXIMATION METHOD AND NONLINEAR HEAT TRANSFER EQUATIONS

Generalized approximation technique for a solution of one-dimensional steady state heat transfer problem in a slab made of a material with temperature dependent thermal conductivity, is developed. The results obtained by the generalized approximation method (GAM) are compared with those studied via homotopy perturbation method (HPM). For this problem, the results ob- tained by the GAM are more accurate as compared to the HPM. Moreover, our (GAM) generate a sequence of solutions of linear problems that converges monotonically and rapidly to a solution of the original nonlinear problem. Each approximate solution is ob- tained as the solution of a linear problem. We present numerical simulations to illustrate and confirm the theoretical results.


Introduction
Fins are extended surfaces and are frequently used in various industrial engineering applications to enhance the heat transfer between a solid surface and its convective, radiative environment.For surfaces with constant heat transfer coefficient and constant thermal conductivity, the governing equation describing temperature distribution along the surfaces are linear and can be easily solved analytically.But most metallic materials have variable thermal properties, usually, depending on temperature.The governing equations for the temperature distribution along the surfaces are nonlinear.In consequence, exact analytic solutions of such nonlinear problems are not available in general and scientists use some approximation techniques such as perturbation method [1], [2], homotopy perturbation method [3], [4], [5] etc., to approximate the solutions of nonlinear equations as a series solution.These methods have the drawback that the series solution may not always converges to the solution of the problem and hence produce inaccurate and meaningless results.

HEAT TRANSFER PROBLEM: HPM METHODS
When using perturbation methods, small parameter should be exerted into the equation to produce accurate results.But the exertion of a small parameter in to the equation means that the nonlinear effect is small and almost negligible.Hence, the perturbation method can be applied to a restrictive class of nonlinear problems and is not valid for general nonlinear problems.
It is claimed that the homoptopy perturbation method does not require the existence of a small parameter and gives excellent results compared to the perturbation method for all values of the parameter, see for example [6,7,8].In these papers, the authors discussed the solutions of temperature distributions in a slab with variable thermal conductivity and the two methods are compared in the field of heat transfer.
However, the claim that the homoptopy perturbation method is independent of the choice of a parameter and gives excellent results compared to the perturbation method for all values of the parameter, is not true.In fact, the solution obtained by the homotopy perturbation method may not converge to the solution of the problem in some cases.
In this paper, we introduce a new analytical method (GAM -Generalized approximation method) for the solution of nonlinear heat flow problems that produce excellent results and is independent of the choice of a parameter.Hence our method can be applied to a much larger class of nonlinear boundary value problems.This method generates a bounded monotone sequence of solutions of linear problems that converges uniformly and rapidly to the solution of the original problem.The results obtained via GAM are compared to those via HPM.For this problem, it is found that GAM produces excellent results compare EJQTDE, 2009 No. 2, p. 2 to homotopy perturbation.We use the computer programme, Mathematica.
Consider one-dimensional conduction in a slab of thickness L made of a material with temperature dependent thermal conductivity k = k(T ).The two faces are maintained at uniform temperatures T 1 and T 2 with T 1 > T 2 .The governing equation describing the temperature distribution (2.1) see [8].The thermal conductivity k is assumed to vary linearly with temperature, that is, where η is a constant and k 2 is the thermal conductivity at temperature T 2 .After introducing the dimensionless quantities where k 1 is the thermal conductivity at temperature T 1 , the problem (2.1) reduces to Three term expansion of the approximate solution of (2.2) by homotopy perturbation method is given by

Definition 3.2. A continuous function
and since , where λ = 2 in this case.Hence f satisfies a Nagomo condition.Existence of solution to the BVP (2.2) is guaranteed by the following theorem.The proof is on the same line as given in [9,10] for more general problems.Observe that Hence, the quadratic form where Consequently, then g is continuous and satisfies the following relations We note that for every θ, z ∈ [min y∈I α, max y∈I β] and z ∈ some compact subset of R, g satisfies a Nagumo condition relative to α, β.In view of (4.5) and the definition of lower and upper solutions, we obtain g(w 0 , w 0 ; w 0 , w 0 ) = f (w 0 , w 0 ) ≥ −w 0 , g(β, β ; w 0 , w 0 ) ≤ f (β, β ) ≤ −β , on I, which imply that w 0 and β are lower and upper solutions of (4.6).Hence, by Theorem 3.3, there exists a solution w 1 of (4.6) such that w 0 ≤ w 1 ≤ β, |w 1 | < C 1 on I. Using (4.5) and the fact that w 1 is a solution of (4.6), we obtain (4.7) which implies that w 1 is a lower solution of (2.2).Similarly, we can show that w 1 and β are lower and upper solutions of −θ (y) = g(θ, θ ; w 1 , w 1 ), y ∈ I, θ(0) = 1, θ(1) = 0. (4.8) Hence, there exists a solution w 2 of (4.8) such that w 1 ≤ w 2 ≤ β, |w 2 | < C 1 on I. Continuing this process we obtain a monotone sequence {w n } of solutions satisfying where w n is a solution of the linear problem and is given by (4.9) The sequence is uniformly bounded and equicontinuous.The monotonicity and uniform boundedness of the sequence {w n } implies the EJQTDE, 2009 No. 2, p. 7 existence of a pointwise limit w on I. From the boundary conditions, we have 1 = w n (0) → w(0) and 0 = w n (1) → w(1).
Hence w satisfies the boundary conditions.Moreover, by the dominated convergence theorem, for any y ∈ I, G(y, s)f (w(s), w (s))ds.
Passing to the limit as n → ∞, we obtain 2 are lower and upper solutions of the problem (2.2).Hence, any solution θ of the problem satisfies 1 − y ≤ θ ≤ 2 − y 2 2 , y ∈ I.In other words, any solution of the problem is positive and is bounded by 2.

Convergence Analysis
Define e n = w − w n on I.Then, e n ∈ C 1 (I), e n ≥ 0 on I and from the boundary conditions, we have e n (0) = 0 = e n (1).In view of (4.5), we obtain which implies that e n is concave on I and there exists t 1 ∈ (0, 1) such that (5.1) e n (t 1 ) = 0, e n (t) ≥ 0 on [0, t 1 ] and e n (t) ≤ 0 on [t Using the definition of g and the non-increasing property of f (θ, θ ) with respect to θ, we have where w n−1 ≤ ξ 1 ≤ w and ξ 2 lies between w n−1 and w . , where Clearly 1 ≤ c 1 (t) ≤ (1 + ) 2 on I. Integrating (5.2) from t to t 1 (t ≤ t 1 ), using e n (t 1 ) = 0, we obtain and integrating (5.2) from t 1 to t, we obtain From (5.3) and (5.4) together with (5.1), it follows that (5.5) e n (t) ≤ d where d 1 = max{ : t ∈ I}.Integrating (5.5) from 0 to t, using the boundary condition e n (0) = 0 and taking the maximum over I, we obtain which shows quadratic convergence.

NUMERICAL RESULTS FOR THE GAM
Starting with the initial approximation w 0 = 1 − y, results obtained via GAM for = 0.5, 0.8 and 1, are given in the Tables (Table 1, Table 2 and Table 3 respectively) and also graphically in Fig. 2. Form the tables and graphs, it is clear that with only a few iterations it is possible to obtain good approximations of the exact solution of the problem.Moreover, the convergence is very fast.Even for larger values of , the GAM produces excellent results, see for example, Fig. 3 and Fig. 4 for ( = 2),( = 2), ( = 3), ( = 4) respectively.In fact, the GAM accurately approximate the actual solution of the problem independent EJQTDE, 2009 No.    Finally, we compare results via GAM (Red) to the corresponding results via HPM (Green), Fig. 6, Fig. 7 and Fig. 8 for different values of .Clearly, GAM accurately approximate the solution for any value of , while for larger value of , the HPM diverges.This fact is also evident from Fig. 8.

Fig. 1 ;
Fig.1; Graphical results obtained via HPM for different values of .

Table 1 -
Approximate solutions of (2.2) via HPM for different values of

Table 1 ;
Results obtained via GAM for = 0.5

Table 2 ;
Results obtained via GAM for = 0.8

Table 3 ;
Results obtained via GAM for = 1