NUMERICAL SOLUTIONS OF SINGULAR INITIAL VALUE PROBLEMS IN THE SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS USING HYBRID ORTHONORMAL BERNSTEIN AND BLOCK-PULSE FUNCTIONS

Orthonormal Bernstein coupled with Block-Pulse Functions for integro-differential form of the singular EmdenFowler initial value problems, is considered. First, we introduce the proposed method then the singular differential equations are transformed to Volterra integro-differential equations. We transform the obtained equations to algebraic system of equations. Solution to the problem is identified by solving this system and the constructed solutions are of approximation form. We compare the results with exact solutions and other methods to show the accuracy of proposed method. The obtained results guarantee the method provides an approximate solution to the generalized singular Lane-Emden type equations.


Introduction
Differential equations are used to describe most phenomena in mathematical physics, astrophysics and nonlinear mechanics which an exact solution in terms of known functions could not found [1][2][3][4].One of these equations characterizing this type of differential equations is the Lane-Emden equation formulated as  The name of this equation is due to the two astrophysicists Jonathan Lane and Robert Emden [1] who were the first to study the equation and this equation used to model several phenomena in the field of theory of stellar structure, the thermal behavior of gas, isothermal gas sphere and theory of thermionic currents well known fields in astrophysics.Lane-Emden equation ( 1) has singularity at the origin x = 0. Ling Xu, Feng Ding [5][6][7] represented a new parameter estimation methods for signal modeling.An effective strategy was established in [8] to convert the Lane-Emden to equivalent Volterra integral equations of any order.The new Volterra integral forms were used in [8], combined with the Adomian decomposition method (ADM), to address the singularity issue.This strategy was proven to be reliable and efficient as confirmed in [8].The physical structure of the solutions of the formulation can be represented in [9][10][11][12].Many researchers focused to give an accurate approximate solution to these equations and many methods were proposed.A general study has been given by Wazwaz [13][14] to construct both exact and series solutions to Lane-Emden equations through ADM.Abdelkrim Bencheikh et al [15] presented Bernstein polynomials method for numerical solutions of integro-differential form of the singular Emden-Fowler initial value problems.In [16] Boubaker Polynomials Expansion Scheme (BPES) approach developed and is applied in order to obtain analytical-numerical solutions to two separate Lane-Emden problems.Dehghan and Shakeri [17] presented the variational iteration method with an exponential transformation is used in order to resolve the difficulty of singular point to solve the Lane-Emden type equation.Parand et al. [18,19] solved the Lane-Emden equations by two numerical techniques, one is depend on the Hermite functions collocation method and the other is depend on the Legendre pseudo-spectral approach.Mukherjee et al. [20] described a differential transform method to find the exact solution for the Lane-Emden type equations.Using Bernstein polynomial operational matrix of integration, Kumar et al. [21,22] presented a numerical method for the Lane-Emden-type equations.Huan Huan Wang et al. [23] presented solutions of fractional Emden-Fowler equations using homotopy analysis method.In this article, we are concerned with the application of Hybrid orthonormal Bernstein and Block-Pulse Functions (HOBB) to the approximation solution of (1).The method consists of converting of Lane-Emden equation to an integro-differential equation and expanding the solution by HOBB with unknown coefficients.The HOBB method converts the Volterra integro-differential equation to a system of algebraic equations which can be solved by any of the usual numerical methods.In this operation there is no iterations is required to remove the non linearity in the Lane-Emden equation.This indeed provides the advantage of proposed method over other method in terms of less computational effort and time for getting good approximate solution to the Lane-Emden type equations .The paper is presented as.In Section 2, we present the properties of Hybrid orthonormal Bernstein and Block-Pulse Functions and approximation of function using it.In Section 3, our main work is to establish Volterra integrodifferential equation equivalent to the singular Lane-Emden equation initial value problems.In Section 4, we use Hybrid orthonormal Bernstein and Block-Pulse Functions method and its convergence analysis for approximating the solution integro-differential form of the singular Lane-Emden equation.The efficiency of the proposed method is demonstrated by solving numerical examples in section 5. Conclusion part given in section 6.

HOBB functions and their properties
The Block-Pulse functions on [0,1) are disjoint, so for , ,...., 2 , 1 , is the order for Block-Pulse functionsis x olynomials and ein pis the order for orthonormal Bernst j , ) (x b i defined on the interval [0,1) as follows:

Function expansion
We can assume that is a Hilbert space with the inner product defined by h x y ), ( since V is a finite-dimensional subspace of H , it is closed and convex.F.Ding et al, represented a decomposition based least squares iterative identification algorithm for multivariate pseudo-linear ARMA systems using the data filtering as in [24][25][26][27][28]. Thus, for , H f  there are a unique approximation out of V such as g, ., where , such that .,. denotes the inner product .
And D is the dual matrix of ) ( HOBB x and is an ) ) , ( with respect to x using Leibniz rule as:

Volterra integro-differential form of the Lane-Emden type differential equation
In this section, we discuss the generalized Lane-Emden equation of shape factor of the form where, ) ( y f can take any linear or nonlinear forms.First we set to convert (8) to an integral form then Differentiating eq. ( 9) twice and using the Leibniz rule, we have x y in (11) we get equation (8).That is the Volterra integrodifferential equation is equivalent to the generalized Lane-Emden equation ( 8) is given by .
For k →1, the integral form of equation ( 9) is Based on this, we set t the Lane-Emden equations in Volterra integral forms are as: Application of HOBB method for Lane-Emden in its integro-differential form Consider equation ( 12) obtained from equation (8).In order to apply the HOBB, the unknown function y(x) is approximated first as First, integrating equation ( 12) and from the condition y(0) = α, one gets .
Then from equations ( 13) and ( 14), we have where Now we collocate the equation ( 16) at the points To apply the Gaussian integration formula for equation (16), we first use the transformation to transfer the interval of integration into the interval [−1, 1].Now, equation ( 16) can be written as By using the Gaussian integration formula, we get  nonlinear algebraic equations with same number of unknowns for coefficient matrix C. By using Newton's method to solve this system, we obtain the values of unknowns for C and hence we get the solution.

Convergence analysis
In this section, we will obtain a comprehensive analysis of the speed of convergence for our numerical method.

Theorem 1
The series solution The above relation possible only if  = ) (x y .so that ) (x y and i  converges to the same value.It provides the convergence guarantee of the proposed HOBB method.

Illustrative numerical examples
In this section, we will study for Lane-Emden initial value examples having singular behavior at x = 0, to demonstrate the high accuracy of the solution obtained by HOBB method and then we compare all results with the exact solution.Example 1: Consider the singular differential equation of Lane-Emden equation [29].
with the exact solution.= n method, we get 16 number algebraic equations with same number of unknowns and these equations are numerically solved by Newton's method with help of maple program, with the initial guess zero, we get the following HOBB coefficients as C =[0,0, 0.02083333333, 0.07812500000, 0.07812500000, 0.1354166667, 0.2291666667, 0.3750000000, 0.3750000000, 0.5208333333, 0.7187500000, 0.9843750000, 0.9843750000, 1.250000000, 1.583333333, 2] So, in this case the approximate of y(x) is the obtained results have been compared with that of by 7-th order Adomian decomposition method (ADM) [17] along with the exact solutions and presented in Table 1.The outcomes reveal that the results by HOBB, with using only a small number of bases, are very promising and superior to ADM and evaluated absolute errors by HOBB for ) (x y will be decreased rapidly in comparison with ADM.n , as M increases, the accuracy increases, and also for a certain value of M , as n increases, the accuracy increases as well and the numerical results obtained for this example are highly agreed with the exact results.(25) This equation can be transformed to integro-differential form as follow

  
We selected our example from [13], in which they solved the homogeneous form of Lane-Emden equation by Adomian decomposition method.Tables 3 and 4, exhibit the Numerical solutions of Table 3 show that, for 0 , 2 = = m k , the obtained results coincides with the exact solution and efficiency of the proposed method described through the absolute error with the exact solution 2 ) ( x e x y = .In Table 5 a comparison between these results with the second derivative multistep method (SDMM) in [30] is also given.It is clear that the results obtained by HOBB, by using 15 , 16 = = n M are very promising and superior to those of SDMM.It is shown that, for a certain value of n , as M increases, the accuracy increases and for a certain value of M , as n increases, the accuracy increases as well.Therefore, HOBB for solving this problem is very effective and more accurate with respect to a second derivative multistep method.


The results obtained by present method are compared with the exact solution as shown in Table 5.For examples 4 we can get very less absolute error by increasing the order of HOBB.

Conclusion
In this paper we presented Volterra integro-differential equations as equivalent to the Lane-Emden type second order singular differential equations.The obtained Volterra integro-differential form of Lane-Emden type equations overcomes the difficulty of the singular behavior at 0 = x .The class of singular equations was generalized, by changing the coefficient of / y and the proposed method was presented.Using this method, the integro-differential forms are reduced to a system of algebraic equations.We obtained a system of equations involving ) 1 ( + n M variables which can be solved using Newton's iterative method through MAPLE program.It was evident that, the proposed method for a certain value of n , as M increased, the accuracy was increased, and also for a certain value of M , as n increased, the accuracy was increased high as well.

,,
Eq. (1) is the standard Lane-Emden equation can be obtained from (1this equation is the Lane-Emden equation.
of orthonormal Bernstein polynomials and Block-Pulse functions where both are complete and orthogonal, then the set is a complete orthogonal system.Hybrid Orthonormal Bernstein and Block-Pulse functions i where the domain D in the xt -plane that contains the region R , are defined functions having continuous derivatives for     x .A global Leibniz rule presented converts Volterra integral equations to differential equations.
and is said the dual matrix of ) idea for the above approximation is the exactness of the Gaussian integration formula for polynomials of degree


be the partial sums with j i  .We need to show   i  is a Cauchy sequence in Hilbert space. is a Cauchy sequence and it converges to for example w (say).We assert that

.
These results agreed with the method in[13].
Table 2 compare the maximum absolute errors of HOBB at different values of M and n .These results have been included to demonstrate the validity and capability of HOBB, for a certain value of

Table 2 :
Maximum absolute errors at different values of M and n for Example 2 via HOBB

Table 3 :
[30]rical solutions for Example.3 when Consider the Lane-Emden type which is of second order homogeneous singular differential equation given in[30].

Table 5 :
Approximate and exact solutions for Example.4.