Intidhar Zamil Mushtt, Saad Shakir Mahmood, Dunya Mohamed Hamed
Mustansiriyah University, College of Education, Baghdad, Iraq
Correspondence to: Intidhar Zamil Mushtt, Mustansiriyah University, College of Education, Baghdad, Iraq.
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Copyright © 2020 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
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Abstract
The aim of this paper is to solve the fourth order boundary value problem by using quantic bspline where the rectangular system be solved by using singular value decomposition technique (SVD), and because the solution is not unique we used the least square optimization to optimal the numerical solution of the BVP. Numerical results are reported where we make a comparison between the exact solution and the approximate solution using the new technique.
Keywords:
Quantic bspline, Fourth order BVP, Approximate solution, Exact solution, SVD
Cite this paper: Intidhar Zamil Mushtt, Saad Shakir Mahmood, Dunya Mohamed Hamed, Solving Fourth Order Boundary Value Problem by Using Extended Quantic Bspline Interpolation, Applied Mathematics, Vol. 10 No. 2, 2020, pp. 2833. doi: 10.5923/j.am.20201002.02.
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 Bspline, 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 Bspline for solving boundary value problem we refer the reader [5,6,7,8,9,10].Quantic Bspline interpolation method [11]The interval of domain has been subdivided as To provide the support for the quantic Bspline near the end boundaries, ten additional knots have been introduced as and The basis function of quantic Bspline are as:  (2) 
The set of quantic Bspline form a basis over the region the global approximation defined using quantic Bspline:  (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 (nonnegative real values called singular values) order so that (if is a singular value of it's square is an eigenvalue of ).Extended Quantic Bspline by using SVD and continuous least square errorConsider the Fourth order BVP:On such that , where are a continuous function defined on , and are all constants.The quantic Bspline is defined in equation (2)Then AndLet , 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 Bspline 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 lefthand side and the right –hand side of BVP (1) is minimum.Example 1: Consider the following ODE With boundary conditions and the exact solution isNow let then This system can be written as follows: Such that And Since be the values of the coefficient Table (1). Shows the Extended Bspline with exact solution for the example 
 

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