Haar wavelet collocation method for the approximate solutions of Emden-Fowler type equations

This paper investigates the Haar wavelet collocation method (HWCM) to obtain approximate solution of the linear Emden-Fowler type equations. To show the efficiency and accuracy of the proposed method, some problems are solved and the obtained solutions are compared with the approximate solutions obtained by using the other numerical methods as well as the exact solutions of the problems.

In this paper, Haar collocation method is presented to obtain the approximate solution of initial value problems (IVPs) of generalized Emden-Fowler type equations in the following form where , ,  and  are constants, and (), () and () continuous functions.Also, especially, () is a linear function.
The paper organized as follows.In section 2, we have given some definitions for Haar wavelet collocation method.In section 3, we use Haar wavelet collocation method to obtain approximate solution of the class of equations given by (1).In section 4, by using tables and graphs, some test problems are given to show the abilities of present method.Finally, in section 5, we have completed the paper with a conclusion.
The operational matrix of integration , which is a 2 square matrix, is defined by the equation The elements of the matrices ,  and  can be evaluated Chen and Hsiao [Chen & Hsiao, 1997] showed that the following matrix equation for calculating the matrix  of order  holds where  is a null matrix of order
1 0 If () is piecewise constant by itself, then () will be terminated at finite terms, that is where the coefficients  ()  and the Haar function vector ℎ () () are defined for  = 2  as

Method of solution of linear Emden-Fowler equation
Consider linear Emden-Fowler type equation with initial conditions (0) = ,  ′ (0) = .If terms of equation ( 2) using (1) expand in terms of Haar wavelets, then we obtain and, similarly, () may be expanded by the Haar functions as follows: where  ()  is a known constant vector.Substituting equations ( 3)-( 6) into the equation ( 2), we obtain, In equation ( 7), if we write collocation points   , it is obtained that linear Haar matrix system.When this system is solved, we obtain unknown coefficients   in (1).

Numerical examples
In this section, we present several numerical examples for the Emden-Fowler type equations to show the accuracy of the introduced method.We use Mathematica10 for all calculations.
Example1 Consider linear Emden-Fowler equation given by [Chowdhury & Hashim, 2009, Iqbal & Javed, 2011] in the following form where () =  5 −  4 + 44 2 − 30 and the exact solution of equation ( 8) is () =  4 −  3 .The numerical solutions obtained by using the present method for this problem are presented in Table 1.In Table 2, it is given that the comparisons of the present method with optimal homotopy asymptotic method (OHAM).Additionally, the graphics of the exact and approximate solutions for different values of  for Example1 are given in Figure 1.Example2 Consider linear Emden-Fowler equation given by Ahamed et al. (2017) in the following form; where () =  3 +  2 + 12 + 6 and the exact solution of equation ( 9) is () =  2 +  3 .The numerical solutions obtained by using the present method for this problem are presented in Table 3.Additionally, the graphics of the exact and approximate solutions for different values of  for Example2 are given in Figure 2.

Conclusion
In this paper, Haar wavelet collocation method is applied to linear Emden-Fowler type equations with initial conditions.All computations associated with the examples are done using Mathematica.
The exact and approximate solutions are compared for all the examples.It can be concluded that Haar wavelet collocation method is a quite effective and accurate method.We aim to apply Haar wavelet collocation method to nonlinear Emden-Fowler type equations in a furure study.

Figure 1 .
Figure 1.The graphics of the exact and approximate solutions for different values of  for example1.

Table 1 .
Absolute errors for different values of

Table 3 .
Absolute errors for different values of  Figure 2. The graphics of the exact and approximate solutions for different values of  for example2.