Keywords: Galerkin Wavelet, Adomian Decomposition method, Lane-Emden equation, integral equations
American Journal of Applied Mathematics and Statistics, 2013 1 (5),
pp 83-86.
DOI: 10.12691/ajams-1-5-1
Received August 09, 2013; Revised September 21, 2013; Accepted October 08, 2013
Copyright © 2013 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 Lane-Emden 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 Lane-Emden 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, time-frequency 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 Multi-resolution 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 so-called 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 Lane-Emden 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 matrix-vector 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 Wavelet-Galerkin 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 Lane-Emden 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 Lane-Emden 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 Lane-Emden 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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