Boubaker Wavelets Functions: Properties and Applications

This paper is concerned with introducing an explicit expression for orthogonal Boubaker polynomial functions with some important properties. Taking advantage of the interesting properties of Boubaker polynomials, the definition of Boubaker wavelets on interval [0,1) is achieved. These basic functions are orthonormal and have compact support. Wavelets have many advantages and applications in the theoretical and applied fields, and they are applied with the orthogonal polynomials to propose a new method for treating several problems in sciences, and engineering that is wavelet method, which is computationally more attractive in the various fields. A novel property of Boubaker wavelet function derivative in terms of Boubaker wavelet themselves is also obtained. This Boubaker wavelet is utilized along with a collocation method to obtain an approximate numerical solution of singular linear type of Lane-Emden equations. LaneEmden equations describe several important phenomena in mathematical science and astrophysics such as thermal explosions and stellar structure. It is one of the cases of singular initial value problem in the form of second order nonlinear ordinary differential equation. The suggested method converts Lane-Emden equation into a system of linear differential equations, which can be performed easily on computer. Consequently, the numerical solution concurs with the exact solution even with a small number of Boubaker wavelets used in estimation. An estimation of error bound for the present method is also proved in this work. Three examples of Lane-Emden type equations are included to demonstrate the applicability of the proposed method. The exact known solutions against the obtained approximate results are illustrated in figures for comparison.


Introduction:
Wavelet theory is an emerging area in mathematical research and it has a wide range application in engineering discipline, singular analysis, and time frequency analysis (1)(2)(3)(4)(5). It permits the accurate representation of different functions and operators. Furthermore, wavelets functions construct a connection with variety numerical techniques. Many authors have increasingly considered the application of Chebyshev wavelets and shifted Chebyshev wavelets. For example, in (6), the modified Chebyshev wavelets have been applied for solving this film of non-Newtonian fluid problem while Chebyshev wavelets utilized for fractional differential equations by (7) have been shifted, also Chebyshev wavelets have been used for solving problems in mathematics and physics. It is well known that there are other types of wavelet functions and all of them have been applied for solving many practical problems arising in numerous branches of science and engineering, that require solving singular initial value problems and boundary value problems of partial differential equations, linear and nonlinear fractional differential equations. Wavelet functions have been previously applied for obtaining approximate solutions for some of these problems. Authors in (8) have constructed a fast algorithm for some linear and nonlinear wave equations using Taylor wavelets. In the two papers (9,10), the authors treated weakly kernel integral equation of the second kind and fractional delay differential equation respectively using Hermit wavelets. In (11)(12)(13), Legendre wavelets method have been introduced for solving respectively, optimal control problem, fractional differential equations and partial differential equations. In this paper, first a new form of polynomials is introduced, the orthogonal Boubaker polynomials and derive many interest and useful properties, then present the definition of Boubaker wavelet depending on the orthogonal Boubaker polynomials. The main goals of the present work are  Introducing Boubaker wavelets functions and deriving explicit forms of their derivatives.  Presenting the bounded of Boubaker wavelet coefficients.  Using Boubaker wavelets functions together with their properties to solve Lane-Emden type equation with the aid of collocation method. Lane Emden equation is singular initial value problem and many authors have studied them. The solution of Lane Emden equation is numerically found, for example (14)(15)(16)(17).
The paper is organized as follows: In section two, a new explicit formula for defining orthogonal Boubaker polynomial is introduced with some important properties. Then Boubaker wavelets are constructed in section 3. Boubaker coefficients discussion is given in section 4. In section 5 the derivative of Boubaker wavelets in terms of Boubaker wavelets is given while the introduced Boubaker wavelets with the new property is applied for solving Lane-Emden equation using collocation method is included, in section 6. Some conclusion remarks illustrating the validity of the suggested basis functions are listed in section 7.

New Explicit Definition for Orthogonal Boubaker Polynomials
The Boubaker polynomials were deals with in physics studies that get a thermal model of the spray pyrolysis disposal (18). They are merged from an attempt to obtain a solution to heat equation in a determination step through resolution process (19). Definition 1 (20) The expression of Boubaker polynomials ( )is described as (1) and it can be defined with the recurrence relation given below with 0 ( ) = 1, 1 ( ) = and 2 ( ) = 2 + 2 Boubaker polynomials are not orthogonal but when applying the Gram-Schmit process on sets of Boubaker polynomials one can obtain orthogonal Boubaker polynomials ( ). Orthogonal Boubaker polynomials are generated using In order to construct higher order orthogonal Boubaker polynomial, the following recurrence relation is used

Definition 2
The A recursive definition also can be used to generate orthogonal Boubaker polynomials over the interval [0,1] The other property of the orthogonal Boubaker polynomials is

Boubaker Wavelet
Wavelet functions are constructed from dilation and translation of a definite function, named mother wavelet. Wavelet functions may be defined as where a and b are dilation and translation parameters respectively while is the normalized time.
Then the Boubaker wavelets can be defined as below In Eq. 7, (τ) describes orthogonal Boubaker polynomial of order m. Therefore, the total Boubaker wavelet approximation can be presented as below , ( ) (8) By truncating the infinite series in Eq. 8, then the result can be written as , in which (0 , 0) denoted the inner product in 2 [0,1]. Eq. 9 can be written in a matrix form as, and

Bounded of Boubaker Wavelet Coefficients Theorem 1
Let ( ) be a continuous function defined on [0,1] and * ( ) be the approximation of ( ) by applying Boubaker wavelets. Also suppose that ( ) is bounded by a positive constant, that is This is the required result.

The Derivative of Boubaker Wavelet in Terms of Boubaker Wavelets
In the next theorem, a relation between orthogonal Boubaker polynomials and their derivatives is derived, which is very important in deriving the derivative of Boubaker wavelets. Let ( ) be the Boubaker wavelet into [0, 1], then the following relation is satisfied

Application of Solutions in the Boubaker Wavelets Basis
In this section, the solution of Lane-Emden equations are obtained by applying Boubaker wavelets collocation method based on making use of the previously introduced derivative of Boubaker wavelet. Consider an approximate solution to Eq. 16 which is given in terms of Boubaker wavelets as Then one can obtain the following residual after substituting of Eq. 18 into Eq. 16 Using the collocation method yields ( ) = 0, = 1,2,3, … , 2 ( + 1) − 2 , Moreover, uses of initial conditions Eq. 17 give

Example 2
The second test Lane-Emden differential equation is ̈( ) + 2̇( ) + ( ) = 0 , 0 < ≤ 1 with  The comparison between the approximate solution and the exact solution can be seen in Table  2. The values of exact solution and approximate solution at some points are reported in Table 3 with = 6, = 7 and = 1. In addition, the maximum absolute error has been listed in Table 4 for = 5, 6 7 and = 1.  Table 3 that only a small number of Boubaker wavelets basis functions are needed to obtain the approximate solution, which agrees with the actual one. This problem is solved with Boubaker wavelets using = 5, and = 1 the linear system of 5equations is obtained The following unknown parameters are obtained a 0 =1.052083235111342, a 1 = 0.930977360202020, a 2 = 0.363361161797005, a 3 = 0.047245598838191, a 4 = 0.000000000195274.
As it can be shown in Table 5 that only a few number of Boubaker wavelets basis functions are utilized to reach the approximate solution with a satisfying result.

Conclusion:
In this paper, has presented for the first time the exact expression for orthogonal Boubaker polynomials and then defined the Boubaker wavelet. The basic shapes of the first six orthogonal Boubaker polynomials are plotted in Fig.1. These polynomials can be used to present complicated functions. In addition, some important properties are derived and employed for obtaining the approximate solution of Lane-Emden equations. Only a few number of Boubaker wavelet basis is needed to achieve the high accuracy. The approximate solutions obtained using the collocation Boubaker wavelets are compared with the exact solution and the agreement between them is obtained. This method has reasonably shown good performance for all of the Lane -Emden type equations.