1. Introduction

Consider the initial value problem

(1)

With the help of lacunary spline functions of type (0, 3, 5, 7) see) ^[8], by using that, and that it satisfies the Lipchitz continuous

(2)

Also initial value problems are satisfied, and for all and for all real from ^[5]. These conditions ensure the existence of unique solution of the problem (1).

Many phenomena in physics, engineering, and other sciences can be described very successfully by model using Mathematical tools from interpolation polynomials. The theory of interpolations polynomial and their applications are relatively recent development, classes of spline functions possess many nice structural properties as well as excellent approximation powers, since they are easy to store and the lacunary interpolation can be designed of curves and surfaces see ^{[3, 4, 7]}. Many researchers used different degree of spline functions of the type cubic, quadratic, quantic, and sixtic for different constructions, and also they obtained the error bounds for each case ^{[1, 6, 9]}. The purpose of this paper is continuous of the work ^[8], that he used new technique for ninth degree spline but in the article for seven degree spline.

2. Description of the Method

We present a ninth spline interpolation approximate for one dimensional and for a given sufficiently smooth define on the interval, and , denote the uniform partition of with knots , where and is the length of each subintervals, and d the ninth spline is denoted by and defined on as:

(3)

On the subinterval where are unknowns to be determined. Let as examine subintervals . By taking into account the interpolating conditions, form ^[8] provided that construction has been unique and the expression, for in the follow form:

(4)

Where , which are determined. Now we define the new approximate polynomial on the subinterval , as

(5)

Form the above boundary conditions, and ^[8] found the coefficients of on , as follows

and

The difference between polynomials and obtain the new polynomial denoted by and defined on the interval , putting the value of and where j=4,6,8 and 9 in and , n=1,2,…,9.Also for on the interval , and satisfy the boundary conditions, we obtain the following theorem:

Theorem1. Let be the approximate values defined before. Then the following estimates of the spline function are valid:

,, whereand denote the constants dependent of, and is the modulus continuity.

Proof The first construction polynomial from ^[8] and (3), in the first interval, we have

(6)

Where constant is depend of, similarly form equation (3), we have

and

Where constant is depend of .

Where constant is depend of ,

Where constant is depend of ,

Where constant is depend of ,

Where constant is depend of ,

Where constant is depend of,

Where constant is depend of, similarly on the interval can obtain the following:

and for the other derivatives can be find as follows

where constant is depend of,

and finally

where are constants depend of h.

Theorem 2: Consider is the exact solution of problem (1) andbe the approximate value of the ninth degree spline function approximation then

whereand denote the difference constants dependent of , and .

Proof: since

From theorem 2 of ^[8], the following estimates are valid

(7)

Using equation (7) and estimate in theorem1, we have

Where is a constant depending of h.

Theorem 3: If the function in initial value problem (1) satisfies conditions (2) and (3), then the following inequalities are hold:

where is constants dependent of h, and .

where is constants dependent of h, and .

where is constants dependent of h,and.

Proof: Using condition (1), (2) and (3), we have

Where

Similarly for each the intervals can be proving it.

3. Conclusion

A new approximate polynomial is constructed which converts a errors estimations to its interpolation by a ninth spline model with error bound. The principal difference between the two spline interpolations showed slight superiority over the ninth spline model, the continuity of derivatives across element edges improves convergence for all coefficients. In this construct of approximate polynomial is established that reduces the total errors and order convergence also compared with that developed by ^[1], ^[2] and ^[9], the new methods enable us to the optimal minimize errors with exact solution.

References

[1]	Faraidun K. Hama-Salh, Karwan H. F. Jwamer, Cauchy problem and Modified Lacunary Interpolations for Solving Initial Value Problems, Int. J. Open Problems Comp. Math., Vol. 4, No. 1, pp. 172-183, 2011.
	In article
[2]	Gyovari, J., Cauchy problem and Modified Lacunary Spline functions, Constructive Theory of Functions.Vol.84, pp. 392-396, 1984.
	In article
[3]	Kendall Atkinson, Weimin Han, Theoretical Numerical Analysis, A Functional Analysis Framework, Third Edition, 2009.
	In article
[4]	Klaus Ritter, Average-Case Analysis of Numerical Problems, Springer-Verlag Berlin Heidelberg New York, 2000.
	In article		CrossRef
[5]	Lyman M. Kells, Elementary Differential Equations. Sixth Edition, (1960).
	In article
[6]	Rana, S. S. and Dubey, Y. P., Best error bounds for deficient quartic spline interpolation, Indian J. Pure Appl. Math., 30 (4), 385-393, 1999.
	In article
[7]	Richard L. Burden, J. Douglas Faires, Numerical Analysis, 9th Edition, Brooks/Cole, Cengage Learning, 2011.
	In article
[8]	Rostam K. Saeed, Faraidun K. Hamasalh and Gulnar W. Sadiq, Convergence of Ninth Spline Function to the Solution of a System of Initial Value Problems, World Applied Sciences Journal 16(10):1360-1367, 2012.
	In article
[9]	Saxena, A., Solution of Cauchy's problem by deficient lacunary spline interpolations, Studia Univ. BABES-BOLYAI MATHEMATICA, Vol. XXXII, No.2, 60-70, 1987.
	In article