APPROXIMATION OF SOLUTIONS OF NONLINEAR HEAT TRANSFER PROBLEMS

We develop a generalized approximation method (GAM) to obtain solution of a steady state one-dimensional nonlinear convective-radiative-conduction equation. The GAM generates a bounded monotone sequence of solutions of linear problems. The sequence of approximants converges monotonically and rapidly to a solution of the original problem. We present some numerical simulation to illustrate and confirm our results.


Introduction
Most metallic materials have variable thermal properties, usually depending on temperature.The governing equations describing the temperature distribution along such surfaces are nonlinear.In consequence, exact analytic solutions of such nonlinear problems are not available in general.Scientists use some approximation techniques for example, perturbation method [7], [25], homotopy perturbation method [1], [3], [5], [12], [13], [14] , [24], to approximate solutions of the nonlinear problems.However, these methods have the drawback that the series solutions may not always converge to a solution of the problem and, in some cases, produce inaccurate and meaningless results.
In this paper, we develop the generalized approximation method (GAM), [2], [6], [15], [16], [17], to approximate solutions of nonlinear problems.This method produces excellent result and is independent of the choice of a small parameter.It generates a bounded monotone sequence of solutions of linear problems which converges uniformly and rapidly to a solution Acknowledgement: The author is thankful to the referee for valuable comments and suggestions that improve the manuscript.
Research is supported by HEC, Pakistan.
EJQTDE, 2009 No. 52, p. 1 of the original problem.Hence it can be applied to a much larger class of nonlinear boundary value problems.Moreover, we show that our results are consistent and accurately represent the actual solution of the problem for any value of the parameter.For the numerical simulation, we use the computer programme, Mathematica.For computational purposes, the linear iteration is important.The generalized approximation method which uses linear problems is a particular version of the well studied quasilinearization method [8,9,10,18,19,20,21].
At each iteration, we are dealing with linear problems and obtain a monotone sequence of solutions of linear problems which converges to a solution of the original nonlinear problem.

HEAT TRANSFER PROBLEM: INTEGRAL FORMULATION
Consider a straight fin of length L made of materials with temperature dependent thermal conductivity k = k(T ).The fin is attached to a base surface of temperature T b extended into a fluid of temperature T a with T b > T a and its tip is insolated.Assume that the thermal conductivity k vary linearly with temperature, that is, where η is constant and k a is the thermal conductivity at temperature T a .Choose the tip of the fin as origin x = 0 and the base of the fin at position x = L.The fin surface transfers heat through convection, conduction and radiation.Assume that the emissivity coefficient of the surface E g is constant and the convective heat transfer coefficient h depends on the temperature.The convective heat transfer coefficient h usually varies as a power law of the type, see [23], where h b is the heat transfer coefficient at the base temperature and the number n depends on the heat transfer mode.For example, for laminar film boiling or condensation n = −1/4, for laminar natural convection n = 1/4, for turbulent natural convection n = 1/3, for nucleate boiling n = 2 and for radiation n = 3.Here, we restrict our study to the case n > 1 and T ≥ T a .Our results are also valid for the case n ≤ 1 but with possibly different upper and lower solutions.EJQTDE, 2009 No. 52, p. 2 The energy equation describing one dimensional steady state temperature distribution is given by see [11] and [22], where A is the cross-sectional area and p is a parameter of the fin.In view of (2.1) and (2.2), the boundary value problem (2.3) can be rewritten as follows Introducing the dimensionless quantities θ = T −Ta T b −Ta , y = x L , we obtain where, . From the definition of θ, we have θ ≥ 0. From the differential equation in (2.4), we obtain d dy [(1 + ǫ 1 θ) dθ dy ] ≥ 0, which implies that the function (1 + ǫ 1 θ) dθ dy in nondecreasing on [0, 1].Hence using the boundary condition at 0, it follows that dθ dy ≥ 0, that is, the function θ is monotonically increasing on [0, 1].Hence, 0 ≤ θ(y) ≤ θ(1) = 1, y ∈ [0, 1] and these provide bounds for the possible solutions of the BVP (2.4).Moreover, from (2.4), we obtain 1+ǫ 1 θ , which implies that the function dθ dy may not be monotone on [0, 1].Now, for simplicity, we write the problem (2.4) as follows which can be written as an equivalent integral equation and is the Green's function.Clearly, G(y, s) > 0 on (0, 1) × (0, 1).
Recall the concept of lower and upper solutions corresponding to the BVP (2.5).
Definition 2.1.A function α ∈ C 1 (I) is called a lower solution of the BVP (2.5), if it satisfies the following inequalities, An upper solution β ∈ C 1 (I) of the BVP (2.5) is defined similarly by reversing the inequalities.
For example, α = 0 and β = 1 are lower and upper solutions of the BVP (2.5) respectively as they satisfy the inequalities: For broad variety of nonlinear boundary value problems, it is possible to find a solution between the lower and the upper solutions.To give an estimate of the derivative u ′ of a possible solution, we recall the concept of Nagumo function.Using the relation . In particular, we may choose C = [e 2ǫ 1 + (e 2ǫ 1 − 1) N +ǫ 2 ǫ 1 ] We note that the BVP (2.4) satisfies the conditions of Theorem 2.3 with α = 0 and β = 1 as lower and upper solutions. 3 ,

GENERALIZED APPROXIMATION METHOD (GAM)
Hence, the quadratic form where where F = f + φ.In our case, we choose φ(θ) = m(z) 2 θ 2 , where m(z) = 2ǫ 2 z 2 (6 + 8ǫ 1 z + 3ǫ 2 1 z 2 ) + n(3n + 1)N ≥ R(z).Hence, which implies that where, Hence, Clearly, g is continuous and satisfies the following relations We note that for every θ, z ∈ [0, 1] and z ′ ∈ some compact subset of R, g satisfies a Nagumo condition relative to α, β.Hence, there exists a constant C 1 such that any solution θ of the linear BVP with the property that α ≤ θ ≤ β on I, must satisfies |θ ′ | < C 1 on I.We note that the linear problem (3.9) can be solved analytically.
To develop the iterative scheme, we choose w 0 (y) = α(y) = 0 as an initial approximation and consider the following linear BVP . In general, using (3.8) and the definition of lower and upper solutions, we obtain which imply that w 0 and β are lower and upper solutions of (3.10).Hence, by Theorem 2.3, there exists a solution w 1 of (3.10) such that w 0 ≤ w 1 ≤ β, |w ′ 1 | < C 1 on I. Using (3.8) and the fact that w 1 is a solution of (3.10), we obtain which implies that w 1 is a lower solution of (2.4).Similarly, we can show that w 1 and β are lower and upper solutions of Hence, there exists a solution w 2 of (3.12) 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 (3.13) w n (y) = 1 + The sequence of functions w n is is uniformly bounded and equicontinuous.The monotonicity and uniform boundedness of the sequence {w n } implies the existence of a pointwise limit w on I. From the boundary conditions, we have 0 = w ′ n (0) → w ′ (0) and 1 = w n (1) → w(1).
Hence w satisfy the boundary conditions.Moreover, by the dominated convergence theorem, for any y ∈ I, Passing to the limit as n → ∞, we obtain that is, w is a solution of (2.4).
Hence, the sequence of approximants {w n } converges to the unique solution of the nonlinear BVP (2.4).Moreover, the convergence is quadratic, see [16], [17].The fact that the sequence converges rapidly to the solution of the problem can also be seen from the numerical experiment.

NUMERICAL RESULTS FOR GAM, HPM and VIM
Starting with the initial approximation w 0 = 0 and set N = 0.5, n = 1.1, results obtained via GAM for (ǫ 1 = 0.2ǫ 2 = 0.3), ǫ 1 = 0.5ǫ 2 = 0.5 and ǫ 1 = 1ǫ 2 = 1 are shown in Tables (Table 1, Table 2 Table 3 respectively) and also graphically in Fig. 1, Fig. 2 and Fig. 3. Form the tables and graphs, it is clear that with only few iterations it is possible to obtain good approximations of the exact solution.Moreover, the convergence is very fast.Even for larger values of N, n, the GAM produces excellent results and fast convergence, see for example, 2009 No. 52, p. 4 which implies that f satisfies a Nagumo condition with ω(s) = ǫ 1 s 2 + N + ǫ 2 as a Nagumo function.Hence by Theorem 1.4.1 of [4] (page 14) , there exists a constant C > λ such that any solution θ of the BVP (2.4) which satisfies α ≤ θ ≤ β on I, must satisfies |θ ′ | ≤ C on I.

Fig. 5 ,
Fig.5, Fig 6 and Fig.7.In fact, the GAM accurately approximate the actual solution of the problem independent of the choice of the parameters ǫ 1 and ǫ 2 involved, Fig. 4 and Fig. 8. EJQTDE, 2009 No. 52, p. 8