Electronic Journal of Qualitative Theory of Differential Equations 29, No. 2, -5; http://www.math.u-szeged.hu/ejqtde/ THE GENERALIZED APPROXIMATION METHOD AND NONLINEAR HEAT TRANSFER EQUATIONS RAHMAT ALI KHAN Abstract. 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 obtained 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 obtained 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 Key words and phrases. keywords; Heat transfer equation, Upper and lower solutions, Generalized approximation Acknowledgement: Research is supported by HEC, Pakistan, Project 2-3(5)/PDFP/HEC/28/ The Author is thankful to the reviewer for his/her valuable comments and suggestions that lead to improve the original manuscript. EJQTDE, 29 No. 2, p.
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 [], [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. 2. 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, 29 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 and T 2 with T > T 2. The governing equation describing the temperature distribution (2.) d dx (kdt ) =, x [, L], dx T() = T, T(L) = T 2. see [8]. The thermal conductivity k is assumed to vary linearly with temperature, that is, k = k 2 [ + η(t T 2 )], where η is a constant and k 2 is the thermal conductivity at temperature T 2. After introducing the dimensionless quantities θ = T T 2 T T 2, y = x L, ɛ = η(t T 2 ) = k k 2 k 2, where k is the thermal conductivity at temperature T, the problem (2.) reduces to (2.2) d2 θ dy = ɛ(dθ dy )2 2 ( + ɛθ) = f(θ, θ ), y [, ] = I, θ() =, θ() =. Three term expansion of the approximate solution of (2.2) by homotopy perturbation method is given by (2.3) θ(y) = y + ɛ 2 (y y2 ) + ɛ 2 (y 2 y3 2 y 2 ), y I see [8]. Results obtained for different values of ɛ via HPM (2.3) are presented in Table and Fig.. Clearly, for small value for ɛ (ɛ ), (2.3) is a good approximation to the solution. However, as ɛ increases, (2.3) deviates from the actual solution of the problem (2.2) and produce inaccurate results. EJQTDE, 29 No. 2, p. 3
y..2.3.4.5.6.7.8.9 ɛ =.5.92375.824.73425.642.546875.448.344625.236.2375 ɛ =.8.98.8234.73696.64992.56.46528.36384.25376.332 ɛ =.945.86.735.648.5625.472.3735.264.45 ɛ =.5.876375.776.69225.68.546875.472.386625.284.57375 ɛ = 2.828.74.66.552.5.448.384.296.72 ɛ = 2.5.759375.6.5325.45.42875.4.365625.3.84375 ɛ = 3.675.464.3535.32.325.328.335.296.945 ɛ = 3.5.56375.296.6725.38.7875.232.28625.284.22375 ɛ = 4.432.96 -.56 -.72..2.26.264.28 ɛ = 4.5.282375 -.36 -.35875 -.38 -.2325 -.32.34625.236.2375 Table -Approximate solutions of (2.2) via HPM for different values of ɛ.8.6.4.2 -.2.2.4.6.8 y axis Fig.; Graphical results obtained via HPM for different values of ɛ. 3. HEAT TRANSFER PROBLEM: INTEGRAL FORMULATION We write (2.2) as an equivalent integral equation, (3.) θ(y) = ( y) + where, G(y, s)f(θ(s), θ (s))ds = ( y) + G(y, s) = G(y, s) ɛ(θ ) 2 ( + ɛθ) ds, { ( s)y, y s, ( y)s, s y, EJQTDE, 29 No. 2, p. 4
is the Green s function. Clearly, G(y, s) > on (, ) (, ) and since ɛ(θ ) 2, hence, any solution θ of the BVP (2.2) is positive on I. We (+ɛθ) recall the concept of lower and upper solutions. Definition 3.. A function α is called a lower solution of the BVP (2.2), if α C (I) and satisfies α (y) f(α(y), α (y)), y (, ) α(), α(). An upper solution β C (I) of the BVP (2.2) is defined similarly by reversing the inequalities. For example, α = y and β = 2 y2 2 of the BVP (2.2). are lower and upper solutions Definition 3.2. A continuous function h : (, ) (, ) is called a Nagumo function if sds λ h(s) =, for λ = max{ α() β(), α() β() }. We say that f C[R R] satisfies a Nagumo condition relative to α, β if for y [min α, maxβ] = [, 2], there exists a Nagumo function h such that f(y, y ) h( y ). Clearly, and since λ f(θ, θ ) = ɛ(dθ dy )2 ( + ɛθ) ɛ θ 2 = h( θ ) for θ [, 2] sds = sds =, where λ = 2 in this case. Hence f h(s) λ ɛs 2 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, ] for more general problems. Theorem 3.3. Assume that there exist lower and upper solutions α, β C (I) of the BVP (2.2) such that α β on I. Assume that f : R R (, ) is continuous, satisfies a Nagumo condition and is nonincreasing with respect to θ. Then the BVP (2.2) has a unique C (I) EJQTDE, 29 No. 2, p. 5
positive solution θ such that α(y) θ(y) β(y), y I. Moreover, there exists a constant C depending on α, β and h such that θ (y) C. Using the relation C sds maxβ min α = 2, we obtain C λ h(s) 2e2ɛ. In particular, we choose C = 2e 2ɛ. 4. Heat Transfer Problem: Generalized Approximation Observe that (4.) f θθ (θ(s), θ (s)) = 2ɛ3 θ 2 Method (GAM) ( + ɛθ) 3, f θ θ (θ(s), θ (s)) = 2ɛ + ɛθ, f θθ (θ(s), θ (s)) = 2ɛ2 θ ( + ɛθ) and f θθf 2 θ θ = 4ɛ4 θ 2 ( + ɛθ) = (f 4 θθ )2. Hence, the quadratic form v T H(f)v = (θ z) 2 f θθ + 2(θ z)(θ z )f θθ + (θ z ) 2 f θ θ (4.2) ( ɛ = (θ z) 3 θ 2 ( + ɛθ) 3 (θ z 2ɛ ) 2 ), ( + ɛθ) ( ) ( ) fθθ f where H(f) = θθ θ z is the Hessian matrix and v = f θθ f θ θ θ z. Consequently, (4.3) f(θ, θ ) f(z, z ) + f θ (z, z )(θ z) + f θ (z, z )(θ z ). Define g : R 4 R by (4.4) g(θ, θ ; z, z ) = f(z, z ) + f θ (z, z )(θ z) + f θ (z, z )(θ z ), then g is continuous and satisfies the following relations { f(θ, θ ) g(θ, θ ; z, z ), (4.5) f(θ, θ ) = g(θ, θ ; θ, θ ). 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. Hence, there exists a constant C such that any solution θ of the linear BVP θ (y) = g(θ, θ ; z, z ), y I, θ() =, θ() =, EJQTDE, 29 No. 2, p. 6
with the property that α θ β on I, must satisfies θ < C on I. To develop the iterative scheme, we choose w = α as an initial approximation and consider the linear BVP (4.6) θ (y) = g(θ, θ ; w, w ), y I, θ() =, θ() =. In view of (4.5) and the definition of lower and upper solutions, we obtain g(w, w ; w, w ) = f(w, w ) w, g(β, β ; w, w ) f(β, β ) β, on I, which imply that w and β are lower and upper solutions of (4.6). Hence, by Theorem 3.3, there exists a solution w of (4.6) such that w w β, w < C on I. Using (4.5) and the fact that w is a solution of (4.6), we obtain (4.7) w (y) = g(w, w ; w, w ) f(w, w ) which implies that w is a lower solution of (2.2). Similarly, we can show that w and β are lower and upper solutions of (4.8) θ (y) = g(θ, θ ; w, w ), y I, θ() =, θ() =. Hence, there exists a solution w 2 of (4.8) such that w w 2 β, w 2 < C on I. Continuing this process we obtain a monotone sequence {w n } of solutions satisfying α = w w w 2 w 3... w n w n β, w n < C on I, where w n is a solution of the linear problem and is given by (4.9) w n (y) = ( y)+ θ (y) = g(θ, θ ; w n, w n ), y I θ() =, θ() = G(y, s)g(w n (s), w n(s); w n (s), w n (s))ds, y I. The sequence is uniformly bounded and equicontinuous. The monotonicity and uniform boundedness of the sequence {w n } implies the EJQTDE, 29 No. 2, p. 7
existence of a pointwise limit w on I. From the boundary conditions, we have = w n () w() and = w n () w(). Hence w satisfies the boundary conditions. Moreover, by the dominated convergence theorem, for any y I, G(y, s)g(w n (s), w n (s); w n (s), w n (s))ds G(y, s)f(w(s), w (s))ds. Passing to the limit as n, we obtain w(y) = ( y) + G(y, s)f(w(s), w (s))ds, y I, that is, w is a solution of (2.2). Since α = y, β = 2 y2 are lower and upper solutions of the 2 problem (2.2). Hence, any solution θ of the problem satisfies y θ 2 y2, y I. In other words, any solution of the problem is 2 positive and is bounded by 2. 5. Convergence Analysis Define e n = w w n on I. Then, e n C (I), e n on I and from the boundary conditions, we have e n () = = e n (). In view of (4.5), we obtain e n(t) = f(w(t), w (t)) g(w n (t), w n(t); w n (t), w n (t)), t I, which implies that e n is concave on I and there exists t (, ) such that (5.) e n(t ) =, e n(t) on [, t ] and e n(t) on [t, ]. EJQTDE, 29 No. 2, p. 8
Using the definition of g and the non-increasing property of f(θ, θ ) with respect to θ, we have e n(t) = f(w(t), w (t)) g(w n (t), w n(t); w n (t), w n (t)), t I where = f(w n (t), w n (t)) + f θ(w n (t), w n (t))(w(t) w n (t)) + f θ (w n (t), w n (t))(w (t) w n (t)) + 2 vt H(f)v g(w n (t), w n (t); w n (t), w n (t)) = f θ (w n (t), w n (t))(w(t) w n(t))+ f θ (w n (t), w n (t))(w (t) w n (t)) + 2 vt H(f)v v T H(f)v = f θ (w n (t), w n (t))e n(t) + 2 vt H(f)v, ( ɛ (w w n ) 3 ξ2 2 ( + ɛξ ) 3 (w w n ) 2ɛ ) 2, ( + ɛξ ) where w n ξ w and ξ 2 lies between w n and w. ( v T H(f)v e n C 2 ɛ ) 2 ɛ + e n 2ɛ ɛ(c2 ɛ + 2) 2 e n 2 = d e n 2, where d = ɛ(c 2 ɛ+ 2) 2, C 2 = max{c, C } and e n = max{ e n, e n } is the C norm. Hence, e n (t) f θ (w n (t), w n (t))e n (t) + d 2 e n 2, t I, which implies that (5.2) (c (t)e n (t)) dc (t) e n 2 2, t I, where c (t) = e f θ (w n (t),w n (t))dt = ( + ɛw n (t)) 2, t I. Clearly c (t) (+ɛ) 2 on I. Integrating (5.2) from t to t (t t ), using e n (t ) =, we obtain (5.3) e n(t) d t c t (s)ds e n 2 2c (t), t [, t ] EJQTDE, 29 No. 2, p. 9
and integrating (5.2) from t to t, we obtain t (5.4) e n (t) d t c (s)ds e n 2 2c (t), t [t, ]. From (5.3) and (5.4) together with (5.), it follows that (5.5) e n (t) d t c t (s)ds e n 2 2c (t) d c (s)ds e n 2, t [, ] 2c (t) and (5.6) e n(t) d t t c (s)ds e n 2, d c (s)ds e n 2 2c (t) 2c (t) m, t [, ]. Hence, (5.7) e n d e n 2, where d = max{ d c (s)ds 2c : t I}. Integrating (5.5) from to t, using (t) the boundary condition e n() = and taking the maximum over I, we obtain (5.8) e n d e n 2, t I. From (5.7) and (5.8), it follows that e n d e n 2 which shows quadratic convergence. 6. NUMERICAL RESULTS FOR THE GAM Starting with the initial approximation w = y, results obtained via GAM for ɛ =.5,.8 and, are given in the Tables (Table, 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 ɛ, the GAM produces excellent results, see for example, Fig.3 and Fig.4 for (ɛ = 2),(ɛ = 2), (ɛ = 3), (ɛ = 4) respectively. In fact, the GAM accurately approximate the actual solution of the problem independent EJQTDE, 29 No. 2, p.
of the choice of the parameters ɛ involved, see Fig.5. y..2.3.4.5.6.7.8.9 w.926897.85445.7762.685.594355.496892.3976.274894.4573 w 2.927888.85243.7794.68693.59678.4986.392748.276598.4728 w 3.9279.85285.77998.686995.596245.498664.3928.276637.47228 w 4.9279.85286.772.686997.596247.498666.39282.276638.47229 Table ; Results obtained via GAM for ɛ =.5 y..2.3.4.5.6.7.8.9 w.923726.844367.76495.67454.582739.48577.389.266234.4679 w 2.92498.846597.764396.67788.5863.48832.3836.268796.4258 w 3.9258.84679.764665.6783.586448.488658.38346.26928.4265 w 4.9259.84689.764692.67863.586484.488693.38349.2695.42664 Table 2; Results obtained via GAM for ɛ =.8 y..2.3.4.5.6.7.8.9 w.92729.8454.7569.667965.575454.477674.373374.2682.3753 w 2.923296.843436.765.67256.58267.482379.377637.26437.39835 w 3.9235.843836.76579.67395.582.4834.378284.26488.422 w 4.923533.843898.76666.6733.587.48328.378386.264896.467 Table 3; Results obtained via GAM for ɛ = y..2.3.4.5.6.7.8.9 w.93445.824728.733929.64986.545655.447457.345576.238689.24668 w.96286.8327.74648.6533.555869.457667.354822.245968.2894 w.97358.83232.74465.653932.559799.46569.35837.248672.3483 w.97797.8337.745834.65542.5642.46368.359745.249774.39 Table 4; Results obtained via GAM for ɛ = 2 EJQTDE, 29 No. 2, p.
.8.6.4.2.2.4.6.8.8.6.4.2.2.4.6.8.8.6.4.2 Fig.2. Results obtained by the GAM for ɛ =.5 (left graph) and ɛ =.8 (right graph)..8.6.4.2.2.4.6.8.2.4.6.8.8.6.4.2 Fig.3. Results obtained by the GAM for ɛ = (left graph) and ɛ = 2 (right graph)..8.6.4.2.2.4.6.8.2.4.6.8 Fig.4. Results obtained by the GAM for ɛ = 3 (left graph) and ɛ = 4 (right graph). EJQTDE, 29 No. 2, p. 2
.8.6.4.2.2.4.6.8 Fig. 5; Graph of the results obtained by the GAM for ɛ =.5,.8,, 2, 3, 4.8.6.4.2 7. Comparison with homotopy perturbation method 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 ɛ. Clearly, GAM accurately approximate the solution for any value of ɛ, while for larger value of ɛ, the HPM diverges. This fact is also evident from Fig. 8..8.6.4.2.2.4.6.8.2.4.6.8 Fig.6. GAM and HPM for ɛ =.5 (left graph) and ɛ =.8 (right graph). EJQTDE, 29 No. 2, p. 3
.8.6.4.2.2.4.6.8.8.6.4.2.2.4.6.8 Fig.7. GAM and HPM for ɛ = (left graph) and ɛ = 2 (right graph)..7.6.8.5.6.4.3.4.2.2..2.4.6.8.2.4.6.8 Fig.8. GAM and HPM for ɛ = 3 (left graph) and ɛ = 4 (right graph). References [] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Annotated Edition, Parabolic Press, Standford, CA, (975). [2] A. H. Nayfeh, Perturbation Methods, Wiley, New York, (973). [3] J. H. He, Homotopy perturbation technique, J. Comput. Methods Appl. Mech. Eng., 78(999)(257), 3 4. [4] J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Nonlinear Mech., 35(2)(37). [5] J. H. He, Homotopy perturbation technique, J. Comput. Methods Appl. Mech. Eng., 7(999)(8), 257 262. [6] D. D. Gunji, The application of He s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letter A, 355(26), 337 34. EJQTDE, 29 No. 2, p. 4
[7] D. D. Gunji and A. Rajabi, Assessment of homotopyperturbation and perturbation methods in heat radiation equations, Int. Commun. in Heat and Mass Transfer, 33(26), 39 4. [8] A. Rajabi, D. D. Gunji and H. Taherian, Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Physics Letter A, 36(27), 57 573. [9] R. A. Khan and M. Rafique, Existence and multiplicity results for some threepoint boundary value problems, Nonlinear Anal., 66 (27), 686 697. [] R. A. Khan, Generalized approximations and rapid convergence of solutions of m-point boundary value problems, Appl. Math. Comput., 88 (27), 878 89. (Received August 7, 28) Centre for Advanced Mathematics and Physics, National University of Sciences and Technology(NUST), Campus of College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan, E-mail address: rahmat alipk@yahoo.com EJQTDE, 29 No. 2, p. 5