1. Introduction

Differential equations are an important tool in constructing mathematical models for physical phenomena. This modeling allows for a much clearer understanding and interpretation of the particular event. Finding the analytical and approximate solution of these models with boundary conditions thus becomes essential. Many analytical and approximate methods were developed solution of ordinary differential equation with boundary value conditions and a many these method as [1,2,3,4].

In this paper, the approximate solution of fourth order boundary value problem will be determine via Quantic B-spline, and comparisons with current studie will be made in the literature where the extended are given by the continuous least square optimization.

Since the fourth order B.V.P which is in the from:

(1)

Such that , and

Where are all constants, and are all a continuous functions defined on interval .

Hence there has been much research activity concerning B-spline for solving boundary value problem we refer the reader [5,6,7,8,9,10].

Quantic B-spline interpolation method [11]

The interval of domain has been subdivided as

To provide the support for the quantic B-spline near the end boundaries, ten additional knots have been introduced as

and

The basis function of quantic B-spline are as:

(2)

The set of quantic B-spline form a basis over the region the global approximation defined using quantic B-spline:

(3)

By [12], The nodal value of and it's derivatives of to fourth order are given in terms of the parameters from the use of spline (2) and the trial solution (3).

Definition: [13]

Any real number matrix can be decomposed as where is and column orthogonal (it's column are eigenvectors of )

and orthogonal (it's column are eigenvectors of )

diagonal (non-negative real values called singular values)

order so that (if is a singular value of it's square is an eigenvalue of ).

Extended Quantic B-spline by using SVD and continuous least square error

Consider the Fourth order BVP:

On such that , where are a continuous function defined on , and are all constants.

The quantic B-spline is defined in equation (2)

Then

And

Let , be an approximate solution of equation (1) where is unknown real coefficients, let are grid points in the interval and also in order to get a matrix of transactions that is contrary to the matrix of transaction that is contrary to the matrix from behind to make the image on the following form:

This system can be written in matrix form as follows such that

Where

And

The singular decomposition of has the form so

and

So

let then

and

Hence

then we have

be the solution of ODE by Quintic B-spline on the interval

The last equation can be solved by using continuous least square error to obtain the values of the constants with make the difference between the left-hand side and the right –hand side of BVP (1) is minimum.

Example 1: Consider the following ODE

With boundary conditions and the exact solution is

Now let then

This system can be written as follows:-

Such that

And

Since be the values of the coefficient

Table (1). Shows the Extended B-spline with exact solution for the example

2. Conclusions

Fourth order boundary value problem solved by using the extended quantic B-spline with continuous Least Square approximation and singular value decomposition technique. The numerical results showed that the extended quantic B-spline approximations are considered very well.