Wavelet-Galerkin Quasilinearization Method for Nonlinear Boundary Value Problems

and Applied Analysis 3 where T is a square matrix of order 2N − 3; that is, T k:l = ∑ N−1 m=0 p m p l−2k+m , where indices k and l vary from 1 to 2N−3. We use the substitution [13] y = 2x, C l = 2c l , throughout our work. This substitution provides a way of calculating the connection coefficients. Here, j and l are integers and 2 is scaling factor. It corresponds to either the expansion (j > 0) or the contraction (j < 0) of the scaling or wavelet function. Define connection coefficients as Ω 0,d k = 2 dj ∫φ (0) (y) φ d (y − k) dy. (11) Similarly, we can obtain 1 2 ⋀ 0,d = T⋀ 0,d . (12) It is homogeneous system and thus does not have a unique nonzero solution. In order to make the system inhomogeneous, one equation is addedwhich is derived from the moment equation of the scaling function [3] as follows:


Introduction
The Galerkin method [1] is a very well-known method for finding the numerical solutions of differential equations.According to wavelet-Galerkin method, connection coefficients are the inner products of Daubechies scaling functions and their derivatives, because we are taking Daubechies scaling functions as a Galerkin basis.The exact and explicit representations of the differential operators in orthonormal bases of compactly supported wavelets are described by Beylkin [2] and he also discussed the sparse representations of shift operators in orthonormal bases of compactly supported wavelets.Latto et al. [3] gave the connection coefficients for zeroth level of resolution; these connection coefficients are essentially based on an unbounded domain.Chen et al. [4] provided the way of calculating the connection coefficients on a bounded interval and these finite integrals play a vital role in the wavelet-Galerkin approximation of differential equations.Restrepo and Leaf [5] reviewed the inner products of Daubechies wavelets and their derivatives and used the connection coefficients for approximation of differential operators.Amaratunga et al. [6] implemented the wavelet-Galerkin technique for solving the one-dimensional counterpart of Helmholtz's equation.Mishra and Sabina [7] used the wavelet-Galerkin method for solving linear, homogeneous boundary value problems with constant coefficients and compared the obtained solution with the exact solution by using a family of Daubechies wavelets and at different levels of resolution.Daubechies scaling functions as the Galerkin bases were used by Jianhua et al. [8] and the authors implemented the wavelet-Galerkin method for differential equations with a boundary layer.In order to implement the wavelet-Galerkin method, they considered the linear, inhomogeneous second order boundary value problem with constant coefficients.In [9][10][11][12][13] several applications of wavelet-Galerkin method are done and these applications are for linear boundary value problems.Motivated by the work of authors [6][7][8][9][10][11][12][13], we extended the wavelet-Galerkin method for the solution of nonlinear boundary value problems.
The quasilinearization approach was introduced by Kalaba and Bellman [14,15] as a generalization of the Newton-Raphson method [16] to solve individual or systems of nonlinear ordinary and partial differential equations.The quasilinearization approach is suitable for a general nonlinear ordinary or partial differential equations of any order.
Jiwari [19] used a uniform Haar wavelet method with quasilinearization technique for the approximate solution of Burgers' equation and compared the results with the solutions obtained by the other numerical methods and the exact solution.The same approach was used by Kaur et al. [20] for the solutions of nonlinear boundary value

Daubechies' Wavelets
Daubechies [22,23] constructed a family of compactly supported orthonormal wavelets.A wavelet system consists of a scaling function () and a wavelet function ().There are two important relations in wavelet theory, which we called two-scale relation.Consider and the equation where   (⋅) := (⋅ − ).Relations (1) and ( 2) are known as refinement relations.The coefficients   are called the wavelet filter coefficients.Relations (1) and ( 2) are also called refinement relations and  (an even integer) is the number of wavelet filter coefficients in the refinement relations.The supports of the scaling function () and wavelet function () are [0,  − 1] and [1 − /2, /2], respectively.Daubechies [22] constructed wavelet filter coefficients   to satisfy the certain conditions.These conditions are linked with certain properties of scaling and wavelet functions [4].Consider for any integer , Relation (3) shows that scaling functions have unit area and relations (4) and ( 5) indicate the orthonormality of  and orthogonality of  and , respectively.Relation (6) shows that mth moment of  is zero; that is, it has  vanishing moments, which implies that we can express the elements of the set 1, , . . .,  /2 − 1 as a linear combination of ( − ), integer translate of ().
Daubechies wavelet has no explicit expressions for the scaling function () and the wavelet function () at arbitrary .
The simplest Daubechies wavelet [24] is the Haar wavelet, which has explicit expression for calculating the scaling function ().It is also called D2, which means the Daubechies wavelet having two filter coefficients.It is also called db1, which means the Daubechies wavelet having one vanishing moment.db2 (D4) has four wavelet coefficients, that is,  = 4, and two vanishing moments.Similarly, DN has coefficients and /2-vanishing moments.

Two-Term Connection Coefficients
In the present work, we are dealing with the second order nonlinear boundary value problems.We will be concerned with the 2-term connection coefficients.Two-term connection coefficients are defined as Take  times derivative of the Daubechies scaling function (1), by assuming that it is  times differentiable, to obtain Use ( 8) in (7) and changing variables, to obtain [3] Let ⋀  1 , 2 be a column vector with 2 − 3 components which are connection coefficients: ] =1:2−3 .
Equation (9) gives a system of linear equations with ⋀  1 , 2 as unknown vector; we can write (9) in vector form as where  is a square matrix of order 2 − 3; that is,  : = ∑ −1 =0    −2+ , where indices  and  vary from 1 to 2 − 3. We use the substitution [13]  = 2  ,   = 2 /2   , throughout our work.This substitution provides a way of calculating the connection coefficients.Here,  and  are integers and 2  is scaling factor.It corresponds to either the expansion ( > 0) or the contraction ( < 0) of the scaling or wavelet function.
Define connection coefficients as Similarly, we can obtain It is homogeneous system and thus does not have a unique nonzero solution.In order to make the system inhomogeneous, one equation is added which is derived from the moment equation of the scaling function [3] as follows: where    is the dth moment of  , () := (2   − ) and we can compute it by considering the orthonormality of () that is Considering the substitution  = 2   in (13), we arrive at Differentiate ( 15)  times to get Taking inner product on both sides of ( 16) with ( − ), or Equation ( 14) implies where    = ∫     () is the dth moment of   and  0 0 = 1 implies the unit area under .Equation (19) shows that dth moment of  , () is equal to the 2 −−/2 times dth moment of ( − ).
Latto et al. [3] derive an explicit formula to compute the moments of ().Consider where   are Daubechies wavelet coefficients.Finally, we get the system for the calculation of connection coefficients.Consider where   is a row vector with all the    .

Implementation of Wavelet-Galerkin Method
Consider the following form of boundary value problem: where , , , and  are real constants.A trial solution for ( 22) is Use (23) in (22) to obtain For simplicity, use substitution  = 2  ,   = 2 /2   .Also, we have Multiplying   (), on both sides of (25), and integrating, we get where () = ∑  =0     , is a polynomial of degree  in .The orthonormality of Daubechies wavelets implies Now use ( 11) and ( 27) in ( 26); we have or Treatment of the boundary conditions [25] is as follows.

Quasilinearization
The quasilinearization approach is a generalized Newton-Raphson technique for functional equations [26,27].It converges quadratically to the exact solution, if there is convergence at all, and it has monotone convergence.
Assume that the problem converges and continue the procedure for obtaining desired accuracy.Recurrence relation is of the form where   () is known and can be used for obtaining  +1 ().Equation ( 37) is always a linear differential equation and boundary conditions are Now consider the nonlinear second order differential equation of the form [26]   () =  (  () ,  () , ) .
Here, the first derivative   () can be considered as another function and (39) implies with the same boundary conditions Similarly, one can follow the same procedure for higher order nonlinear differential equations to obtain the recurrence relation where  is order of the differential equation.Equation ( 42) is always a linear differential equation and can be solved recursively, where   () is known and one can use it to get  +1 ().
In order to test the wavelet-Galerkin method with quasilinearization technique, four different nonlinear problems are considered.

Applications
In this section, we solve some nonlinear ordinary differential equations by the wavelet-Galerkin method along with quasilinearization technique and compare the results with those obtained by other methods and exact solution.
We solved (43) by using 16 and fixed the level of resolution  = 13.Figure 1 shows the exact solution and approximate solution by proposed method at first and second iteration; that is,  0 () is the initial approximation, and by using  0 () we get  1 (), that is, the solution at first iteration, and then  1 () is used to get  2 (), which is the solution of (43) at second iteration.Table 1 is used to compare the approximate solution by proposed method at second iteration for  = 13 and exact solution.We may get more accurate results while increasing level of resolution at higher iteration.Example 2. We consider the nonlinear Bratu's boundary value problem as follows: −  () =  () , (0) = 0,  (1) = 0. (47) The quasilinear form of (47) is with the boundary conditions  +1 (0) = 0,  +1 (1) = 0. Wavelet-Galerkin method for (48) implies and, from boundary conditions, we have with the initial approximation  0 () = 0. Bratu's boundary value problem is solved by using 12 as Galerkin bases and at level of resolution  = 13.Decomposition method [17] is already implemented on (47).We compared our results with the results obtained by decomposition method and exact solution.Our results are more accurate as compared to decomposition method [17] at  = 1 and  = 2 as shown in Tables 2 and 3, respectively.We used the MATLAB command of one-dimensional data interpolation using spline to get the values at  = 0.1,  = 0.2, . . .,  = 0.9 and plot the exact and approximate solutions at these points for  = 1 and  = 2 as shown in Figure 2.
The quasilinearized form of (51) is where 0 ≤  ≤ 1, with the boundary conditions  +1 (0) = 0,  +1 (1) = 1.Implementation of wavelet-Galerkin method to (52) implies and boundary conditions lead to with the initial approximation  0 () = 0. Tables 4 and 5 and Figure 3 represent the solution of (51) at second iteration.We use 8 as Galerkin bases to find the solution of (51) at different level of resolutions and at  = 0.5 and  = 1 as shown in Tables 4 and 5, respectively.Solutions by proposed method are compared with variational iteration method [18] and with exact solution.Our results are in high agreement with exact solution and better than variational iteration method [18].

Conclusion
It is shown that the proposed method, wavelet-Galerkin method with quasilinearization technique, gives stable and accurate results when applied to different nonlinear boundary value problems.The proposed method provides better and more accurate results as compared to variational iteration method and decomposition method, as shown in Tables 2-5.Also results are in good agreement with exact solutions.Figures 1 and 4 show that approximate solution converges to the exact solution while iterations are increased and absolute error goes down.The main advantage of the proposed method is that the different type of nonlinearities can be easily handled.

Figure 1 :
Figure 1: Comparison of exact solution and solution by wavelet-Galerkin method with quasilinearization technique at  = 13, for different iterations, and we used 16.

Figure 2 :
Figure 2: Comparison of exact solution and solution by wavelet-Galerkin method with quasilinearization technique at level of resolution  = 13 and we use 12 as Galerkin bases and at  = 1 and  = 2, respectively.

Figure 3 :
Figure 3: Comparison of exact solution and solution by wavelet-Galerkin method with quasilinearization technique at level of resolution  = 13 and 8 is used as Galerkin bases and at  = 0.5 and  = 1, respectively.

Figure 4 :
Figure 4: Comparison of exact solution and solution by wavelet-Galerkin method with quasilinearization technique at  = 13, for different iterations, and we use 10 as Galerkin bases.

Table 1 :
Comparison of exact solution  exact and solution by the wavelet-Galerkin method with quasilinearization technique  NEW at second iteration: level of resolution  = 13 and 16 is used as Galerkin bases.

Table 2 :
Comparison of exact solution  exact , solution by decomposition method  DM , and solution by the wavelet-Galerkin method with quasilinearization technique  NEW at second iteration and 12 is used as Galerkin bases.

Table 3 :
Comparison of exact solution  exact , solution by decomposition method  DM , and solution by the wavelet-Galerkin method with quasilinearization technique  NEW at second iteration and 12 is used as Galerkin bases.

Table 4 :
Comparison of exact solution  exact , solution by variational iteration method  VIM , and solutions by the wavelet-Galerkin method with quasilinearization technique  NEW at second iteration, at different level of resolutions, and we used 8 as Galerkin bases.

Table 5 :
Comparison of exact solution  exact , solution by variational iteration method  VIM , and solutions by the wavelet-Galerkin method with quasilinearization technique  NEW at second iteration, at different level of resolutions, and we used 8 as Galerkin bases.

Table 6 :
Comparison of exact solution  exact and solution by the wavelet-Galerkin method with quasilinearization technique  NEW at second iteration: level of resolutions  = 13 and 10 is used as Galerkin bases.