Sinc-Collocation Method for Solving Linear and Nonlinear System of Second-Order Boundary Value Problems

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the sinc-collocation method for solving linear and nonlinear system of second order differential equation. The method is then tested on linear and nonlinear examples and a comparison with B-spline method is made. It is shown that the sinc-collocation method yields better results.


Introduction
Numerous problems in physics, chemistry, biology and engineering science are modelled mathematically by systems of ordinary differential equations, e.g.series circuits, mechanical systems with several springs attached in series lead to a system of differential equations (for example see [1,2]).However, many classical numerical methods used with second-order initial value problems cannot be applied to second-order boundary value problems (BVPs).
Most realistic systems of ordinary differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used.There are many publications dealing with the linear system of second-order boundary value problems.They introduced various numerical methods.For instance, a finite difference method has been proposed in recent works [3][4][5][6][7][8].For a nonlinear system of second-order BVPs, there are few valid methods to obtain numerical solutions.Geng et al. have studied the numerical solution of a nonlinear system of second-order boundary value problems in the reproducing kernel space [9].Lu considered the variational iteration method to solve a nonlinear system of second-order boundary value problems [10].Recently, Bataineh et al. [11] represented modified homotopy method for solving systems of second-order boundary value problems.Sinc-collocation method was applied to solve nonlinear systems of second order boundary value problems in [12].
In this paper, we discuss the use of sinc-collocation method for solving a class of linear and non-linear system of differential equations subject the boundary conditions where , ,  f x u u , and


, and , for , are analytic functions.It will always be assumed that (1) possesses a unique solution .0,1, 2 i   J Numerical examples including regular, singular as well as singularly perturbed problems are considered.On the basis of these examples, the results reveal that the method is very effective and convenient.
The paper is organized into five sections.Section 2 contains notation, definitions and some results of sinc function theory.In Section 3, the sinc-collocation method is developed for linear second-order system of differential equation with homogeneous boundary conditions.The method is developed for nonlinear second-order system of differential equation in Section 4. Some numerical examples are presented in Section 5. Finally, Section 6 provides conclusions of the study.

Sinc Function
In recent years, a lot of attention has been devoted to the study of the sinc method to investigate various scientific models.The efficiency of the method has been formally proved by many researchers [13][14][15][16][17][18][19][20][21][22].
A general review of sinc function approximation is given in [23,24].Hence, only properties of the sinc function that are used in the sequel.
If   f x is defined on the real line, then for the Whittaker cardinal expansion of f is given by: where , and the mesh size is given by where is suitably chosen and N  depends on the asymptotic behavior of   f x .The n-th derivative of the function f at the sampling points k x kh  can be approximated using a finite number of terms as: We note that , a n d The interpolation formula for where the basis functions on   , a b are then given by and the transformation function transforms   The n-th derivative of the function f at points k x can be approximated using a finite number of terms as Setting which will be used later.

System of Linear Second Order Equations
Consider a linear, system of linear second order equations of the form We assume that   1 u x and the solutions of ( 11) and ( 2), is approximated by the finite expansion of Sinc basis functions and where is the function for some fixed step size h.If we replace each term of (11) with its corresponding approximation given by the right-hand side of ( 10) and ( 8) we have 14) and applying the collocation to it, we eventually obtain the following theorem.
Theorem: If the assumed approximate solution of problem ( 11) and ( 2) is ( 12) and ( 13) , then the discrete sinc-collocation system for the determination of the unknown coefficients is given by where We now rewrite these equations in matrix form.The system in (16)  , Now we have a linear system of equations of the 4N  4N  unknown coefficients.We can obtain the coefficients of the approximate solution by solving this linear system.The system (17) may be easily solved by a variety of methods.In this paper we used the Q-R method.The solution c gives the coefficients in the approximate sinc-collocation solutions and

System of Non-Linear Second Order Equations
   , we write the above equation in the form Consider a nonlinear, system second-order equations of the form where 1 and 2 are nonlinear functions of 1 and 2 is an analytic function and may be a polynomial or a rational function, or exponential.Due to the large number of different possibilities, our work will be focused mainly on the following forms or any analytic function of which has a power series expansion.We limit our study to the case where is an integer, or a fraction.n We consider next applying of the sinc-collocation method to solve problem ( 18) and (2).
Lemma: The following relation holds where N and h are now dependent on both    .Replacing the terms of (18) with the appropriate representation defined in ( 8), (10) and (19) and applying the collocation to it, we eventually obtain the following theorem.
Theorem: If the assumed approximate solution of problem ( 18) and ( 2) is ( 12) and ( 13), then the discrete sinc-collocation system for the determination of the unknown coefficients is given by , Let be the n c 4N  -vector with j-th component given by n j c .In this notation the system in (20)   equations in the 4N 2   unknown coefficients.We can obtain the coefficients in the approximate solution by solving this nonlinear system by Newton's method.
Starting from an initial estimate , the corrections are made using Here, j c is the current iterate, and is the new iterate.A common numerical practice is to stop the Newton iteration whenever the distance between two iterates is less than a given tolerance, i.e. when

Numerical Examples
In this section, some numerical examples are studied to demonstrate the accuracy of the present method.The results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.Comparison between sinc-collocation and other method shall be presented.
All computations were carried out using Matlab on a personal computer with a machine precision of 32 10  .In all cases, is taken to be d π 2 d  .The selection of a larger yields more accuracy, but at the expense of a lengthier computation.We report absolute error which is defined as Example 1: [3,11] consider the linear system of second order boundary value problems whose exact solutions are Maximum absolute errors for 1 and 2 are tabulated in Table 1 for the sinc-collocation method.

u u
Maximum absolute error are tabulated in Table 2 for sinc-collocation together with the analogous results of N. Caglar and H. Caglar [3].
subject to the boundary conditions whose exact solutions are The computational results are summarized in Table 3. Example 3: Now we turn to a singular problem, subject to the boundary conditions whose exact solutions are Maximum absolute errors for and are tabulated in subject to the boundary conditions whose exact solutions are Table 6 the maximum absolute errors obtained by using the sinc-collocation method for N = 40 d re and sin π .u x x  an diffent  .

C nc o lusions
d an efficient method for solvin der boundary value problems.Our variety of linea no This paper describe system of second-or g the approach was based on the sinc-collocation method.Properties of the sinc-collocation method are utilized to reduce the computation of this problem to some linear or nonlinear algebraic equations.The method is computationally attractive and applications are demonstrated through illustrative examples.Numerical examples including regular, singular as well as singularly perturbed problems are presented.As expected, the accuracy increases as the number of terms N in the sinc expansion increases.The obtained results showed that this approach can solve the problem effectively.
The sinc-collocation method is a simple method with high accuracy for solving a large r and nlinear system of differential equations.So it may be rchers and e s familiar w th systems of partial differential equations offers an excellent opportunity for future research.
takes the matrix form

Table 1 . Maximum absolute error for example 1.
ror

Table 5
for the sinc-collocation me