Keywords: Galerkin Wavelet, Adomian Decomposition method, LaneEmden equation, integral equations
American Journal of Applied Mathematics and Statistics, 2013 1 (5),
pp 8386.
DOI: 10.12691/ajams151
Received August 09, 2013; Revised September 21, 2013; Accepted October 08, 2013
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
In recent years, the studies of initial value problems, presented as second order ordinary differential equations (ODEs) have attached the attention of many mathematicians and physicists. One of the known equations of this type is the Lane Emden equation, formulated as the fallowing
 (1) 
With initial conditions
 (2) 
Where are constant, is a continuous real valued function, and Eq. (1) has been used to model several phenomena in mathematical physics and astrophysics such as the theory of staller structure, the thermal behavior of a spherical cloud of gas ^{[1, 2, 3]}.
Several methods, for solving the LaneEmden equation are known. A discussion of formulation of some of these models and the physical structure of the solutions can be found in ^{[4, 5, 6]}. Most algorithms currently in use for handling the LaneEmden type problems are based on series solutions. Wavelet theory is a relatively new and an emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are successfully used in signal analysis for wave form representation and segmentations, timefrequency analysis and fast algorithms for easy implementation ^{[7]}.
2. Wavelets and Galerkin Wavelets
Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter and the translation parameter vary continuously we have the following family of continuous wavelets ^{[8]}:
If we restrict the parameters and to discrete values as where are positive integers, we have the following family of discrete wavelets.
Where a wavelet basis in In particular, if, then forms an orthonormal basis, ^{[8]}.
A Multiresolution analysis of is defined as a sequence of closed subspaces with the following properties, ^{[4]}
1)
2)
3)
4) Are dens in
The existence of a scaling function is required for which the translate generate a basis in each i.e.
With
In the classical case this basis is orthonormal, so that
 (3) 
Where is the usual inner product.
Let denote a subspace complementing the subspace in i.e.
Each element of can be uniquely written as the sum of two elements, one in and the other in which contains the details needed to pass from an approximation at the level to an approximation at the level
Based on the function . One can find the socalled mother wavelet, of which the translates and dilates constitute orthonormal bases of the spaces generated by the following wavelets
Each function can now be expressed as
 (4) 
Where
Of course, in numerical application the summation (4) are truncated, which corresponds to the projection of into a subspace as otherwise by construction,
In addition to (3).
3. Solution of LaneEmden Equation
In the Wavelet Galerkin method, the solution of the equation can be approximated by the th level wavelet series on the interval, by
 (5) 
Therefore, the Galerkin discritization scheme to Eq. (1) gives a nonlinear system of equations that involves the coefficients of connection. For details about the GWM and its application for solving problem, we refer the reader to ^{[9]}.
Introducing Eq. (5) into Eq. (1) we obtain
 (6) 
In the other expression
 (7) 
To determine the coefficient we take the inner product of both sides of Eq. (7) with as
 (8) 
Or
 (9) 
We assume that is a polynomial of degree in
We write the Eq. (9) as
 (10) 
Where
 (11) 
Equation (10) can be further put into the matrixvector form
 (12) 
Where
 (13) 
And
Where denotes the matrix transpose. Now we have a linear system of equations of the unknown coefficients. We can obtain the coefficients of the approximate solution by solving this linear system. The solution gives the coefficients in the WaveletGalerkin approximation of
4. Adomian Decomposition Method
Consider the following equation
 (14) 
Where is a nonlinear operator from a Hilbert space H in to H and f is a given function in H. Now, we are looking for satisfying (14).
At the beginning of the 1980, Adomian developed a very powerful method to solve Equation (14) in which the solution was considered as the sum of decomposition series:
 (15) 
And as the sum of the decomposition series:
 (16) 
The method consist of the following recursion scheme
 (17) 
Where the s is polynomials depending on are called the Adomian polynomials; these are defined as
 (18) 
The Adomian technique is equivalent to determining the sequence:
 (19) 
By using the iterative scheme
 (20) 
Associated with the functional equation
 (21) 
For the study of the numerical resolution (21), Cherruault used fixed point theorem ^{[10, 11, 12]}. In particular, we know that if is a contraction () then the sequence {} defined by (20) converges to the only solution of (21).
Furthermore we have
5. Illustrative Examples
To illustrate the methods two examples are provided
5.1. Example 1Consider the following LaneEmden equation
 (22) 
The presented method is applied for J=6.
Table 5.1. The absolute errors at different times and space locations for example 1
5.2. Example 2Let’s consider the following form of the LaneEmden equation;
 (23) 
The presented method is applied for J=6.
Table 5.2. The absolute errors at different times and space locations for example 2
6. Conclusion
Many excellent properties of wavelet, such as ‘‘locality’’ and vanishing moments, causes the wavelet basis to be a better choice than others, in function approximation.. In other way it is seen that decomposition method can be an alternative way for the solution of partial differential equations that have no exact solutions. These two methods have been applied for LaneEmden equation, successfully. As the results of illustrative examples show the solutions are in a good agreement with the exact ones, even up to 9 digits significant. This agreement of the solutions are so high that one can clams that these methods are a powerful numerical tools for fast and accurate solution, for such kind of differential equations.
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