Assessment of Haar Wavelet-Quasilinearization Technique in Heat Convection-Radiation Equations

Abstract

We showed that solutions by the Haar wavelet-quasilinearization technique for the two problems, namely, (i) temperature distribution equation in lumped system of combined convection-radiation in a slab made of materials with variable thermal conductivity and (ii) cooling of a lumped system by combined convection and radiation are strongly reliable and also more accurate than the other numerical methods and are in good agreement with exact solution. According to the Haar wavelet-quasilinearization technique, we convert the nonlinear heat transfer equation to linear discretized equation with the help of quasilinearization technique and apply the Haar wavelet method at each iteration of quasilinearization technique to get the solution. The main aim of present work is to show the reliability of the Haar wavelet-quasilinearization technique for heat transfer equations.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Francesco Morabito

School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan

Received 31 August 2013; Revised 25 November 2013; Accepted 26 December 2013; 5 February 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Haar wavelet is the lowest member of Daubechies family of wavelets and is convenient for computer implementations due to availability of explicit expression for the Haar scaling and wavelet functions [1]. The quasilinearization approach was introduced by Bellman and Kalaba [2] as a generalization of the Newton-Raphson method to solve the individual or systems of nonlinear ordinary and partial differential equations.

Haar wavelet-quasilinearization technique [3-6] is recently developed method for the nonlinear differential equation, which deals with all types of nonlinearities. Boundary value problems are considerably more difficult to deal with than initial value problems. The Haar wavelet method for boundary value problems is more complicated than for initial value problems. In the present work we deal with both initial and boundary value problems.

In this present work, our purpose to solve the nonlinear equations arising in heat transfer through Haar wavelet-quasilinearization technique and show that it is strongly reliable method for heat transfer problems than the other existing methods. Convergence of Haar wavelet-quasilinearization technique has been given in [6].

We use the cubic spline interpolation [7] to get the solution at grid points for the sake of comparison. For this purpose we use the MATLAB built-in function _{y i} = interp 1 ( x , y , _{x i} , "spline" ) , for one-dimensional data interpolation by cubic spline interpolation.

The paper is arranged as follows: in Section 2 we review basic definition of fractional differentiation and integration, while in Section 3 we describe the Haar wavelets. In Section 4 we present the main features of the quasilinearization approach. In Section 5 we apply the Haar wavelet method with quasilinearization technique to nonlinear heat transfer problems. Finally in Section 6 we conclude our work.

2. Preliminaries

In this section, we review basic definitions of fractional differentiation and fractional integration [8].

2.1. Riemann-Liouville Fractional Integral Operator of Order α

The operator ^{I x α} , defined on _{L 1} [ a , b ] by [figure omitted; refer to PDF] for a ...4; x ...4; b , where α ∈ ^{... +} , is called Riemann-Liouville fractional integral of order α .

2.2. Riemann-Liouville and Caputo Fractional Derivative Operator of Order α

The operator ^{D x α} , defined by [figure omitted; refer to PDF] for a ...4; x ...4; b , where α ∈ ^{... +} and n = [left ceiling] α [right ceiling] , is called Riemann-Liouville fractional derivative of order α .

The Caputo fractional derivative of a function y ∈ _{L 1} [ a , b ] is defined as [figure omitted; refer to PDF] for a ...4; x ...4; b , where α ∈ ^{... +} and n = [left ceiling] α [right ceiling] .

3. The Haar Wavelets

The Haar function contains just one wavelet during some subinterval of time and remains zero elsewhere and is orthogonal. The uniform Haar wavelets are useful for the treatment of solution of differential equations which have no abrupt behavior. The i th uniform Haar wavelet _{h i} ( x ) , x ∈ [ a , b ] , is defined as follows [9]: [figure omitted; refer to PDF] where i = ^{2 j} + k + 1 , j = 0,1 , 2 , ... , J is dilation parameter, m = ^{2 j} , and k = 0,1 , 2 , ... , ^{2 j} - 1 is translation parameter. J is maximal level of resolution and the maximal value of i is 2 M where M = ^{2 J} . In particular, _{h 1} ( x ) : = _{χ [ a , b ]} ( x ) , where _{χ [ a , b ]} ( x ) is characteristic function on interval [ a , b ] , is the Haar scaling function. For the uniform Haar wavelet, the wavelet-collocation method is applied. The collocation points for the Haar wavelets are usually taken as _{x j} = ( j + 0.5 ) / 2 M , where j = 1,2 , ... , 2 M .

3.1. Integral of the Haar Wavelets

Any function y ∈ _{L 2} [ a , b ] can be represented in terms of the Haar series: [figure omitted; refer to PDF] where _{b l} are the Haar wavelet coefficients given as _{b l} = ^{∫ - ∞ ∞} ... y ( x ) _{h l} ( x ) d x .

The Riemann-Liouville fractional integral of the Haar wavelets is given as [figure omitted; refer to PDF]

4. Quasilinearization [2]

The quasilinearization approach is a generalized Newton-Raphson technique for functional equations. It converges quadratically to the exact solution, if there is convergence at all, and it has monotonic convergence.

Let us consider the nonlinear n th order differential equation [figure omitted; refer to PDF] Application of quasilinearization technique to (7) yields [figure omitted; refer to PDF] with the initial/boundary conditions at ( r + 1 ) th iteration, where n is the order of the differential equation. Equation (8) is always a linear differential equation and can be solved recursively, where _{y r} ( x ) is known and one can use it to get _{y r + 1} ( x ) .

5. Applications

5.1. Temperature Distribution Equation in Lumped System of Combined Convection-Radiation in a Slab Made of Materials with Variable Thermal Conductivity

Let the lumped system have volume V , surface area A , density ρ , specific heat c , initial temperature _{T i} , temperature of the convection environment _{T a} , heat transfer coefficient h , and _{c a} which is specific heat at temperature _{T a} . Consider that the mathematical model describing the temperature distribution in lumped system of combined convection-radiation in a slab made of materials with variable thermal conductivity is given by the following nonlinear boundary value problem: [figure omitted; refer to PDF] where y = ( T - _{T a} ) / ( _{T i} - _{T a} ) is dimensionless temperature, x = t / ( ρ V _{c a} / h A ) is dimensionless time, and [straight epsilon] = β ( T - _{T a} ) .

5.1.1. Haar Wavelet-Quasilinearization Technique

Applying the quasilinearization technique to (9), we get [figure omitted; refer to PDF]

Now we implement the Haar wavelet method to (10); we approximate the higher-order derivative term by the Haar wavelet series as [figure omitted; refer to PDF] Lower-order derivatives are obtained by integrating (11) and using the boundary conditions: [figure omitted; refer to PDF] where _{C 2 , l} = ^{∫ 0 1} _{p 2 , l} ( x ) d x .

Substitute (11) and (12) in (10) to obtain [figure omitted; refer to PDF] with the initial approximation _{y 0} ( x ) = 0 .

Figure 1 shows the temperature _{y Haar} by Haar wavelet-quasilinearization technique for different [straight epsilon] at J = 5 and at 4 th iteration. According to Figure 1 and Tables 1 and 2, temperature increases with decreasing [straight epsilon] ; also temperature varies with time x . Tables 1 and 2 show that the obtained solutions are in good agreement with the numerical solution provided by Maple and are better than generalized approximation method _{y GA} [10] and homotopy perturbation method _{y HPM} [10].

Table 1: Numerical results for temperature distribution equation for [straight epsilon] = 0.6 : Haar wavelet-quasilinearization technique at 4th iteration and level of resolutions J = 8 .

4th iteration ( J = 8 )
x	Maple	y GA [10]	y HPM [10]	y Haar
0.0	0.834542	0.963536	0.640000	0.834543
0.2	0.840390	0.964009	0.652096	0.840391
0.4	0.858269	0.965742	0.689536	0.858269
0.6	0.889247	0.969893	0.755776	0.889248
0.8	0.935346	0.979233	0.866576	0.935346

Table 2: Numerical results for temperature distribution equation for [straight epsilon] = 2.0 : Haar wavelet-quasilinearization technique at 4th iteration and level of resolutions J = 8 .

4th iteration ( J = 8 )
x	Maple	y GA [10]	y HPM [10]	y Haar
0.0	0.694318	0.968771	-0.666667	0.694362
0.2	0.703698	0.968804	-0.625600	0.703739
0.4	0.732894	0.969008	-0.489600	0.732927
0.6	0.785488	0.970024	-0.220267	0.785510
0.8	0.869161	0.975059	-0.246400	0.869176

Figure 1: Solutions by Haar wavelet-quasilinearization technique for different [straight epsilon] at J = 5 and n = 4 .

[figure omitted; refer to PDF]

5.2. Cooling of a Lumped System by Combined Convection and Radiation

Consider that the system has volume V , surface area A , density ρ , specific heat c , emissivity E , initial temperature _{T i} , temperature of the convection environment _{T a} , heat transfer coefficient h , and _{c a} which is specific heat at temperature _{T a} . In this case system loses heat through radiation and the effective sink temperature is _{T s} . The mathematical model describing the cooling of a lumped system by combined convection and radiation is given by the following nonlinear initial value problem: [figure omitted; refer to PDF] For the solution of (14), we do the following certain changes in parameters: [figure omitted; refer to PDF] Equation (14) implies after changing the parameters [figure omitted; refer to PDF] For the sake of simplicity we assume that _{y a} = _{y s} = 0 , (15) becomes [figure omitted; refer to PDF]

5.2.1. Haar Wavelet-Quasilinearization Technique

Implementation of the quasilinearization technique to (16) gives [figure omitted; refer to PDF]

According to the Haar wavelet method to (18), approximate the higher-order derivative term by the Haar wavelet series as [figure omitted; refer to PDF] Solution can be obtained by integrating (19) and using the initial condition to yield [figure omitted; refer to PDF] Substituting (19) and (20) in (18), [figure omitted; refer to PDF] with the initial approximation _{y 0} ( x ) = 1 .

To get the solution on large interval, say [ 0,5 ] , we divide the interval [ 0,5 ] into three subintervals [ 0,1.25 ] , [ 1.25,3.75 ] , and [ 3.75,5 ] ; let A = 0 , B = 1.25 , C = 3.75 , and D = 5 ; step-size for each subinterval is [figure omitted; refer to PDF] The coordinates of the grid points are as follows.

: For j = 1,2 , ... , ( M / 2 ) + 1 [figure omitted; refer to PDF]

: For j = 1,2 , ... , M [figure omitted; refer to PDF]

: For j = 1,2 , ... , M / 2 [figure omitted; refer to PDF]

And collocation points are as follows.

: For j = 1,2 , ... , 2 M [figure omitted; refer to PDF]

Temperature _{y Haar} at higher interval, [ 0,5 ] , by Haar wavelet-quasilinearization technique at J = 5 and iteration n = 4 of the cooling equation for different values of [straight epsilon] is shown in Figure 2. It shows that temperature decreases with increasing [straight epsilon] and also shows that temperature reduces to zero when time x is increasing. According to Table 3, we conclude that our results are in good agreement with exact solution and more accurate than variational iteration method _{y VIM} [11] and homotopy perturbation method _{y HPM} [11].

Table 3: Numerical results for cooling equation for different [straight epsilon] and x = 0.5 : Haar wavelet-quasilinearization technique at 4th iteration and level of resolutions J = 8 .

4th iteration ( J = 8 )
[straight epsilon]	Exact	y VIM [11]	y HPM [11]	y Haar
0.0	0.606531	0.606531	0.606531	0.606531
0.1	0.591591	0.591617	0.591638	0.591592
0.2	0.578023	0.578207	0.578371	0.578023
0.3	0.565620	0.566185	0.566732	0.565620
0.4	0.554217	0.555440	0.556720	0.554217
0.5	0.543681	0.545868	0.548335	0.543681
0.6	0.533903	0.537369	0.541576	0.533904
0.7	0.524793	0.529850	0.536445	0.524793
0.8	0.516275	0.523226	0.532940	0.516275
0.9	0.508284	0.517412	0.531062	0.508284
1.0	0.500765	0.512333	0.530812	0.500765

Figure 2: Solutions by Haar wavelet-quasilinearization technique for different [straight epsilon] at J = 5 and n = 4 .

[figure omitted; refer to PDF]

We can get more accurate results while increasing level of resolution J , iteration n , or both, according to convergence analysis [6].

6. Conclusion

It is shown that Haar wavelet method with quasilinearization technique gives excellent results when applied to different nonlinear heat transfer problems. The results obtained from Haar wavelet-quasilinearization technique are better from the results obtained by other methods and are in good agreement with exact solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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