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

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 alsomore accurate than the other numerical methods and are in good agreement with exact solution. According to theHaar 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.


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][4][5][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 waveletquasilinearization 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   = interp1(, ,   , "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.

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

Riemann-Liouville Fractional Integral Operator of Order 𝛼.
The operator    , defined on  1 [, ] by for  ≤  ≤ , where  ∈ R + , is called Riemann-Liouville fractional integral of order .

Riemann-Liouville and Caputo Fractional Derivative
Operator of Order .The operator    , defined by for  ≤  ≤ , where  ∈ R + and  = ⌈⌉, is called Riemann-Liouville fractional derivative of order .

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 th uniform Haar wavelet ℎ  (),  ∈ [, ], is defined as follows [9]: where  = 2  +  + 1,  = 0, 1, 2, . . .,  is dilation parameter,  = 2  , and  = 0, 1, 2, . . ., 2  − 1 is translation parameter. is maximal level of resolution and the maximal value of  is 2 where  = 2  .In particular, 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   = ( + 0.5)/2, where  = 1, 2, . . ., 2.

Integral of the Haar Wavelets. Any function 𝑦 ∈ 𝐿 2 [𝑎, 𝑏]
can be represented in terms of the Haar series: where   are the Haar wavelet coefficients given as The Riemann-Liouville fractional integral of the Haar wavelets is given as

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 th order differential equation Application of quasilinearization technique to (7) yields =  (  () ,    () , . . .,  −1  () , ) with the initial/boundary conditions at ( + 1)th iteration, where  is the order of the differential equation.Equation ( 8) is always a linear differential equation and can be solved recursively, where   () is known and one can use it to get  +1 ().

Haar Wavelet-Quasilinearization Technique.
Applying the quasilinearization technique to (9), we get Now we implement the Haar wavelet method to (10); we approximate the higher-order derivative term by the Haar wavelet series as Lower-order derivatives are obtained by integrating (11) and using the boundary conditions: where  2, = ∫ 1 0  2, ().Substitute (11) and ( 12) in (10) to obtain with the initial approximation  0 () = 0.  iteration.According to Figure 1 and Tables 1 and 2, temperature increases with decreasing ; also temperature varies with time .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  GA [10] and homotopy perturbation method  HPM [10].

Cooling of a Lumped System by Combined Convection and Radiation
. Consider that the system has volume , surface area , density , specific heat , emissivity , initial temperature   , temperature of the convection environment   , heat transfer coefficient ℎ, and   which is specific heat at temperature   .In this case system loses heat through radiation and the effective sink temperature is   .The mathematical model describing the cooling of a lumped system by combined convection and radiation is given by the following nonlinear initial value problem: For the solution of ( 14), we do the following certain changes in parameters: Equation ( 14) implies after changing the parameters For the sake of simplicity we assume that   =   = 0, ( According to the Haar wavelet method to (18), approximate the higher-order derivative term by the Haar wavelet series as Solution can be obtained by integrating (19) and using the initial condition to yield Substituting ( 19) and (20) in (18), with the initial approximation  0 () = 1.
For  = 1, 2, . . ., 2 Temperature  Haar at higher interval, [0, 5], by Haar wavelet-quasilinearization technique at  = 5 and iteration  = 4 of the cooling equation for different values of  is shown in Figure 2. It shows that temperature decreases with increasing  and also shows that temperature reduces to zero when time  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  VIM [11] and homotopy perturbation method  HPM [11].
We can get more accurate results while increasing level of resolution , iteration , or both, according to convergence analysis [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.

Figure 1
shows the temperature  Haar by Haar waveletquasilinearization technique for different  at  = 5 and at 4th

Table 1 :
Numerical results for temperature distribution equation for  = 0.6: Haar wavelet-quasilinearization technique at 4th iteration and level of resolutions  = 8.
have volume , surface area , density , specific heat , initial temperature   , temperature of the convection environment   , heat transfer coefficient ℎ, and   which is specific heat at temperature   .Consider that the mathematical model describing the temperature distribution in lumped system of combined convection-radiation in a slab made of

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

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