NUMERICAL SOLUTION TO LINEAR SINGULARLY PERTURBED TWO POINT BOUNDARY VALUE PROBLEMS USING B-SPLINE COLLOCATION METHOD

A Recursive form cubic B-spline basis function is used as basis in B-spline collocation method to solve second linear singularly perturbed two point boundary value problem. The performance of the method is tested by considering the numerical examples with different boundary conditions. Results of numerical examples show the robustness of the method when compared with the analytical solution.


INTRODUCTION
The singularly perturbed two point boundary value problems are governing differential equations to represents the process in the fields such as combustion, chemical reactor theory, nuclear engineering, elasticity, fluid mechanics etc.
In recent years, many numerical methods are developed to solve linear singularly perturbed two point boundary value problem. Researchers presented many numerical methods [13][14][15] from 2006 to 2013 to deal with one-dimensional singular perturbation problems. These problems considered include linear, non-linear reaction-diffusion, delay differential equations. The numerical techniques reviewed in this survey include finite-difference methods, spline approximation methods and computational methods for boundary value techniques.
However, it is observed from the recent literature that B-spline basis functions are derived using fixed equidistant space for a particular degree only. If the recursive formulation given by Carl. De boor [12] is applied, the basis function evaluation can be generalized and without fixing of degree of the basis function can be used in collocation method for uniform or non uniform mesh sizes.
In this paper, after defining the B-spline basis function recursively, the B-spline collocation method is described and formulated. The efficiency of the method is demonstrated using the second order singular differential equations with Neumann's boundary conditions. Considering second order linear singularly perturbed two point boundary value problem of the form of the considered second order singular differential equation (1).

1.1.B-splines
In this section, definition and properties of B-spline basis functions [1,2] are given in detail. A zero degree and other than zero degree B-spline basis functions are defined at i x recursively over the knot vector space where p is the degree of the B-spline basis function and x is the parameter belongs to X .When evaluating these functions, ratios of the form 0/0 are defined as zero.

1.2.Derivatives of B-splines
If p=2, we have In the above equations, the basis functions are defined as recursively in terms of previous degree basis function i.e. the p th degree basis function is the combination of ratios of knots and (p-1) degree basis function. Again (p-1) th degree basis function is defined as the combination ratios of knots and (p-2) degree basis function. In a similar way every B-spline basis function of degree up to (p-(p-2)) is expressed as the combination of the ratios of knots and its previous B-spline basis functions.
The B-spline basis functions are defined on knot vectors. Knots are real quantities. Knot vector is a non decreasing set of Real numbers. Knot vectors are classified as non-uniform knot vectors, uniform knot vector and open uniform knot vectors. Uniform knot vector in which difference of any two consecutive knots is constant is used for test problems in this paper. Two knots are required to define the zero degree basis function .In a similar way, a p th degree B-spline basis function at a knot have a domain of influence of (p+2) knots. B-spline basis functions of degree one and degree two over uniform knot vector are shown graphically below in figures (1) and (2).

1.3.B-spline collocation method
Collocation method is widely used in approximation theory particularly to solve differential equations .In collocation method, the assumed approximate solution is made it exact at some nodal points by equating residue zero at that particular node. B-spline basis functions are used as the basis in B-spline collocation method whereas the base functions which are used in normal collocation method are the polynomials vanishes at the boundary values. Residue which is obtained by substituting equation (2) in equation (1) is made equal to zero at nodes in the given domain to determine unknowns in (2).Let ] , [ b a be the domain of the governing differential equation and is partitioned as  Now using all the above equations (5), (6) i.e. (n+1) a square matrix is obtained which is diagonally dominated matrix because every second degree basis function has values other than zeros only in three intervals and zeros in the remaining intervals, it is a continuing process like when one function is ending its effect in its surrounding region than other function starts its effectiveness as parameter value changing. In other words, every parameter has at most under the three (p=2) basis functions. The systems of equations are easily solved for arbitrary constants C i 's. Substuiting these constants in (2), the approximation solution is obtained and used to estimate the values at domain points.

NUMERICAL EXPERIMENTS
Numerical Example 1.
The effectiveness of the present method is demonstrated by considering the various examples Example 1: Consider linear singularly perturbed boundary value problem given Cubic B-spline collocation is applied to the numerical example 1 and presented the values in the Table 1 at some nodes also included in the same Table 1 The present method is tested for the numerical example 2 for the different values of  (=10 -5, 10 -7 , 10 -9 ) with the different boundary conditions. Compared values of present method and exact solution at different various nodes is shown in Table 2.

CONCLUSIONS
The B-spline basis functions defined recursively are incorporated in the collocation method and applied the same to second order linear singularly perturbed two point boundary value problem. The effectiveness of the proposed method is illustrated by considering two numerical examples. The solution is compared with exact solution and found to be in good approximation. This method may be applied to different types of second order linear singularly perturbed two point boundary value problem for its efficiency.