OCTIC B-SPLINE COLLOCATION SOLUTION WITH NON-UNIFORM LENGTH FOR EIGHTH ORDER LINEAR DIFFERENTIAL EQUATION

Presentation of Numerical solution for eighth order linear boundary value problem using Octic B-spline collocation method with non-uniform length is the subject of this paper. In this approach recursive form of B-spline function is used as basis in collocation method. Numerical examples are considered to show the advantage of recursive of B-spline function particularly in non-fixing the length of subintervals.


Introduction
The general eighth order linear differential equation with the boundary conditions is given as Solving such higher order linear differential equations and getting exact solutions is sometimes difficult.Authors developed methods to obtain numerical methods.Some of the selected numerical methods are mentioned.Finite difference method is used to solve such equations by Boutayeb and Twizell [1] .Vishwanadam and Ballem [2] used Galerkin method with quintic Bspline, Siddiqi and iftikhar [3] solved by using homotopy analysis method.
In this paper, A Octic degree B-spline based collocation method has been elaborated for the solution of linear eighth order differential equation with boundary conditions defined in Eq.( 1) with Eq.( 2).for non-fixed length.

The Numerical Scheme
Let ] , [ b a be the domain of the governing differential equation and is partitioned as } where i C 's are constants to be determined and are the Octic B-spline functions , be the approximate global solution to the exact solution ) (x U of the considered eighth order linear differential equation (1).A zero degree and other than zero degree B-spline basis functions [5,6] are defined at i x recursively over the knot vector space } , 1 .........
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.

Derivatives of B-splines
If p=8, we have 1 1 The s i x ' are known as nodes, the nodes are treated as knots in collocation B-spline method where the B-spline basis functions are defined and these nodes are used to make the residue equal to zero to determine unknowns s i C ' in (2).Eight extra knots are taken into consideration besides the domain of problem to maintain the partition of unity [7] when evaluating the Octic Bspline basis functions at the nodes which are within the considered domain.
Solving the system of equations for constants and substituting these constants in equation ( 3) then assumed solution becomes the known approximation solution for equation ( 1) at corresponding the collocation points.This is implemented using the Matlab programming.

Numerical Experiment
A linear eighth order differential equation with boundary conditions is considered for testing the accuracy of the proposed method.Example The exact solution for the given problem is Absolute relative errors at the collocation points are shown graphically in figure2

Conclusion
In this paper, collocation method using the B-splines as basis function has been developed to approximate the linear eighth-order boundary value problems with variable coefficients.In this method, the assumed approximate solution is directly substituted in given differential equation and evaluated unknown values in that approximate solution with the help of collocation points and boundary conditions.It is observed that the approximate values is almost equal to exact values and consequently approximate values converging to the exact values by increasing the number of collocation points.Also implementing of proposed method is very easy.