1. Introduction

With the advent of computer, splines have gained more importance. Aryan ^[2] solved two point boundary value problem by using ninth degree lacunary spline function of the type (0, 1, 3, 5, 7), while Karwan and Aryan ^[8] have studied second order initial value problem by lacunary spline function of type (0, 2, 4, 5), and Abbas ^[1] in his topic discussed approximation solution by lacunary interpolation of type (0, 2, 4). Lacunary interpolating by deficient spline of type (0, 1, 3, 5) exhibited by Saeed ^[11]. Saeed and Jwamer ^[12] devoted their paper to lacunary interpolation by spline function of type (0, 1, 4). L. L. ^[9] raised the idea of basic theory spline function. Jwamer and Karim ^[7] showed sextet lacunary solution of fourth order initial value problem. J. Karwan ^[6] found approximation solution of second order initial value problem by spline function. Fazal ^[5] investigates numerical solution of fourth order initial value problem. Splines and differentia equation taked by ^[4] Gianluca. De Boor spoke on apractical guid to spline ^[3]. ^[10] M. K. resolved numerical solution of differential equation.

In our paper we are trying to solve second order initial value problems

(1)

Where and that it satisfies the Lipschitz condition

For all , and all reals and , where r=0, 1, 2,.., n-1, and L is lipschitz constant. Constructing lacunary spline of degree seven of the type (0, 1, 6). Existence and uniqueness of spline function of degree seven have discussed, and convergence and error bound have studied. An illustration example used to show the convergence the lacunary spline function to the exact solution, and the error bound numerically calculated .

The lacunary interpolation problem, which we have searched for in this occupation comprised in finding the seven degree spline of deficiency six, interpolating data has given on the function value and one and sixth derivatives in the interval [0,1].

In the following section spline function of degree seven has offered which interpolates the lacunary data(0, 1, 6). The results concerning existence and uniqueness of the spline function of degree seven are coming in section 3. And convergence and error bounds have studied in section 4. Finally in section 5 the demonstration of the convergence of the particular lacunary spline function, and numerical example have been given.

2. Descriptions of the Method

In order to introduce seven degree spline interpolation for one dimensional and given sufficiently smooth function f defined on [0,1]

Let x₀, x₁,…, x_n be n+1 grid points in the interval [0,1] such that x_i=x₀+ih, x₀=0, x_n=1, i=0, 1,…, n; is the distance of each subintervals, so Is the uniform partition of [0,1]

A seven degree spline interpolation for one dimensional on the interval defined as

(2)

where are unknown to be determined. Let a seven degree spline interpolation on subintervals which is denoted by regarded as follows:

(3)

where are unknown to be determined.

3. Existence and Uniqueness of the Spline Function

In this section, we are supplying the existence and uniqueness theorem for lacunary spline function of degree seven of the type (0, 1, 6).

Theorem (3.1):

Given the real numbers , and for then there exist a unique spline of degree seven as given in the equation (2), (3) such that

(4)

Proof:

The spline function is defined as follows:

Where the coefficients of these polynomials are to be determined by the following condition

(5)

(6)

To find uniquely the coefficients in of equation (2) by using the condition (5) where . Let we obtain the following

(7)

(8)

(9)

From (3) we have

And from (5) and (6)

Now since

Then by solving the above equations (7)-(9),

(10)

(11)

(12)

From the boundary condition (6)

(13)

(14)

By substituting these values (11)-(12) in equation (13) and (14) we get

(15)

(16)

Now we are trying to find the coefficients of for which defined in equation (3) so we have,

(17)

(18)

(19)

Since

So above system has the following unique solution:

(20)

(21)

(22)

From the fact that we have

(23)

(24)

Substituting the values of and in (23) and (24) we get

(25)

(26)

So the coefficient matrix of the system of equations (15), (16), (25) and (26) for the unknown is a non-singular matrix and thus the coefficients are specified uniquely, and consequently the coefficients and . Hence the proof of the theorem has achieved.

4. Convergence and Error Bound

This section, includes studying the error bound of the spline function of degree seven which is defined in section (1) and it is a solution of problem (1).

Theorem4.1:

let and be a unique spline function of degree seven which is the solution of the problem (1). Then for , the following error bounds are holds

Where denotes the modulus of continuity of , where

Proof:

Let the Taylor’s expansion formed about for

where

First to find from (2)

(27)

By using (27) and (12)

(28)

Since for so we have

(29)

(30)

To find we need the following

Using Taylor’s series expansion on a bout

(31)

From (2)

(32)

Then from (10)-(12) and(31), (32)we get

(33)

where x₀< < <.

(34)

To find , from (10)-(12), (2) and Taylor's series expansion on about

(35)

To find from equations (10) – (12), (2) and Taylor's series about

(36)

where

Since for , so we have or , and from (36), then

(37)

Also from (4) and (37) we have , then

(38)

Lemma 4.1:

Let , then for where

(39)

depend on the numbers of interval

Proof:

For , then from Taylor’s expansion formula, we have

where and similar expressions for the derivatives for can be used.

Now if , then from equations (15) and (16) we obtain

where

If , then from equations (15)-(17), and using (39) we obtain

By the same way aforementioned above and using the step before we can show that the inequality

Hence the proof have completed.

Lemma 4.2:

Let , then for

where

(40)

and depend on the numbers of intervals.

Proof:

For , then from Taylor’s expansion formula, we have

where and similar expressions for the derivatives for can be used.

Now if , then from equations (16), and (40) we obtain

where then so .

Also if , then from equation (15), (16), and (18) and using (40) we obtain

By same way in above and using the step before we can show that the inequality

for this completes the proof of the lemma (4.2).

Theorem 4.2:

Let and be a unique spline function of degree seven which is a solution of the problem (4). Then for , the following error bounds are holds:

Proof:

Let ,

From equation (2) and Taylor’s expansion formula we get

(41)

, and from (22)

(42)

By (5), , so we have

(43)

To find where about is

(44)

And from (3)

then from (21), (22) and (40) we get

(45)

From lemma (4.1) and (4.2) we obtain

(46)

To find from Taylor’s about we get

and from (3) and (4) , (20) –(22)

hence

Where

(47)

Where .

To find from Taylor’s about we get

and from (3) and (4)

then from (21)-(23)

Where

(48)

where .

To find from Taylor’s about we get

and from (3) and (20)-(22)

Hence

Where .

Hence

(49)

where .

To find from (4) and (49) we have from which we obtain

(50)

To find from (4) and (50) we have from which we obtain

Thus the proof has completed for .

5. Numerical Conclusion

In this section we are performing numerical result to show the convergence of the spline function of degree seven which constructed in section 4 to the second order initial value problem.

Example:

Consider the second order initial value problem

where with the exact solution

Solution:

Let h=0.1 , n=10.

The following are absolute errors for and its derivative.

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References

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