A Difference Scheme Using Spline for a Class of Singular Boundary Value Problem

In this paper, a finite difference scheme using cubic splines has been discussed to find the numerical solution for a class of singular two point boundary value problems for certain ordinary differential equations. The cubic spline approximation leads to the tridiagonal system of equations, which can be solved using Newton’s method. Second order convergence of the method has been established for quite general conditions. Two numerical examples are given to demonstrate the method and verify the second order accuracy.


INTRODUCTION
Mathematical research on the cubic spline functions and their application to the numerical solution of the differential equations has developed rapidly in the last few decades. The main advantages of using a cubic spline collocation procedure are that the governing matrix system obtained is always tridiagonal and the requirement of a uniform mesh is not necessary. It appears that the cubic spline approximation possesses some of the advantages of finite element techniques without the disadvantages of high computing cost and complex problem formulation.
Consider the class of singular two-point boundary value problems where , and , finite constants. We assume that ( ) satisfies the following conditions is analytic in * | | +.
Existence-uniqueness of the problem (1) for general non-negative function ( ) satisfying conditions (A)(i)-(iii) with boundary conditions ( ) and ( ) has been established in Ref. [8]. Existence-uniqueness for more general problem has been established in Ref. [6] with non-linear boundary condition at .
In this article, second order spline method described in Ref. [7] are further extended to a class of non-negative functions ( ) satisfying conditions (A)(i)-(iii) with boundary conditions (2). The order of accuracy of the method has been established for general class of functions ( ) and under quite general conditions on ( ). Numerical examples for general functions ( ) are given to illustrate the method and verify the order of accuracy.

DESCRIPION OF THE SPLINE METHOD
For a positive integer , we consider a general non-uniform mesh over , -: .
It is obvious that and I S S N 2277-3061 V o l u m e 1 5 N u m b e r 1 3 Multiplying Equation (3) by ( ) and then integrating from to , we get To get spline approximation in the interval ( ), we divide equation (4) by ( ), integrating from to and then setting interpolating condition ( ) , we get Setting in equation (5) we get the spline in the interval ( ). Now using Taylor's expansion where lies between and , ( ) , ( ) , etc., we approximate , and as follows Substituting expression given in (7), (8) in equation (5) and using continuity conditions for ( ) at mesh point and setting ̅ , ̅ , ̅ , we get the three point finite difference approximation as: where ̅ ⁄ , For the boundary condition ( ) ( ) , we require one more difference equation i.e. for . Using equation (4) for the interval ( ) and the boundary condition at , and setting ̅ , ̅ , we get

Truncation Error
Consider the mesh ratio parameter ⁄ then we have , . Using Taylor expansion given by equation (7) for ( ) in equation (6) and substituting the expression for and then expanding in Taylor series, we get, where ⁄ . Now substituting the expressions for , in , , and expanding in Taylor series, we get the following: Substituting (11)-(18) in equation (9) and simplifying then for difference scheme (9), we get the truncation error Similarly the truncation error in (10) can be written as:

CONVERGENCE OF THE METHOD
In this section, we show that under quite general conditions this method is second order convergent. Let ( ̃) ( ̃ ̃ ) , ̃ ( ̃ ̃ ) and ( ) , then the difference scheme (9)-(10) can be expressed in matrix form as where, where Since , , , for sufficiently small mesh size, it is easy to see that and the matrices and are irreducible, monotone and hence , ( ) exists, is nonnegative and ( ) . . We next obtain the bound for ( ). Since , and for sufficiently small (uniform mesh size) ( )( ) then from , we get, Let there exist constants , ( ) such that | ( ) | , for ( ) , and then for an uniform mesh and for sufficiently small , we get,

Now since
, with the help of , we obtain,

NUMERICAL ILLUSTRATIONS
To illustrate the method and to verify the order of convergence of the method for general non-negative functions ( ), we consider two example of singular boundary value problems.     Tables I and  II respectively for three value of , which show that the method works well and is of second-order accuracy.