by using that and
that it satisfies the Lipschitz continuous:
q = 0,1, …, n1
for all and
all real .
These conditions ensure the existence of the unique solution of the problem
1^{[3]}.
This study is organized as follows: First consider the spline function of degree six is presented which interpolates the lacunary data (0, 3, 5). Some theoretical results about existence and uniqueness of the spline function of degree six are introduced and also convergence analysis is studied. To demonstrate the convergence of the prescribed lacunary spline function, numerical examples presented, finally, we prescribe the conclusion and discussion of the result. MATERIALS AND METHODS
Descriptions of the method: We present for the first time according
to our knowledge a six degree spline (0, 3, 5) interpolation for one dimensional
and given sufficiently smooth function f(x) defined on i = [0,1] and .
Denote the uniform partition of I with knots x_{i} = ih, where i =
0, 1, 2, …, n1. We denote by S^{5}_{m,6}the class of six degree splines S(x) such that:
on the interval [x_{0}, x_{1}] where a_{0,j}, j = 2,4,6 are unknowns to be determined^{[4,5]}. Let us examine now intervals [x_{i}, x_{i+1}], i = 1,2,…, n2. By taking into account the interpolating conditions, we can write the expression, for S_{i}(x) in the following form:
where, are
unknowns we need to determine it.
On the last interval [x_{n1}, x_{n}] we define S_{n}(x) as follows: where, a_{n1,j}, j = 1,2,4,6 are unknowns to be determined. The existence and uniqueness theorem for spline function of degree six which interpolate the lacunary data (0, 3, 5) are presented and examined.
Theorem 1: Existence and uniqueness: Given the real numbers y(x_{i}),
y^{(3)}(x_{i}) and y^{(5)}(x_{i}) for i = 0,
1, 2, …, n, then there exist a unique spline of degree six as given in the Eq.
24 such that:
Proof: Let as define a spline function S(x) as follows:
where the coefficients of these polynomials are to be determined by the following conditions: and To find uniquely the coefficients in S_{0}(x) of Eq. 2 by using the condition (3.2) where i = 0, we obtain the following:
and
From the boundary condition (3.3) we have: Solving these equations to obtain the following:
Substituting these values of a_{0,2}, a_{0,4} and a_{0,6} we get: We shall find the coefficients of S_{i}(x) for i = 1, 2, 3, …, n2. Here we have:
and
Solving the first three equations, we obtain the following:
Substituting the values of a_{i,2}, a_{i,4} and a_{i,6} in the fourth equation, we obtain the following relation between a_{i+1,1} and a_{i,1}, where S_{i}(x) for i = 1,2,…,n2:
The coefficient matrix of the system of Eq. 12 and 16
in the unknown a_{i,1}, i = 1, 2,…,n1 is a nonsingular matrix and
hence the coefficients a_{i,1}, i = 1, 2,…,n1 are determined uniquely
and so are, therefore the coefficients a_{i,2}, a_{i,4} and
a_{i,6}.
Finally, for finding the coefficients of S_{n1}(x), we have:
and
Solving these equations, we see that the coefficients a_{n1,j}; i = 2, 4, 6 are uniquely determined. Hence the proof of Theorem 1 is completed. The error bound of the spline function S(x) which is a solution of the problem (5) is obtained for the uniform partition I by the following theorem:
Theorem 2: Let and
S(x) be a unique spline function of degree six which a solution of the problem
(5). Then for ;
i = 1, 2,…, n1:
where, W_{6}(h) denotes the modules of continuity of y^{(6)},
defined by .
To prove this theorem we need the following lemma:
Lemma 1: Let .
Then, for
i = 0, 1, …, n1.
Where: and W_{6}(h) denotes the modules of continuity of y^{(6)}.
Proof of lemma 1: If then
using Taylor’s expansion formula, we have:
where, and
similar expressions for the derivatives of y(x) can be used.
Now from Eq. 16 and using (17) we obtain:
where for
i = 1, 2, …, n1; s = 1, 2, 3, 4 and:
where, .
We see that the system of Eq. 18 and 19
is the unknowns e_{i,1}, i = 1, 2, …,n1 has the unique solution:
Where:
It is clear that:
Hence:
Which completes the proof of the Lemma 1.
Proof of Theorem 2: Let where
i = 1, 2, …, n1.
We have from Eq. 3 by applying Taylor’s expansion formula we have:
Using (20) and (15), we have:
From Eq. 3 we have:
from which we obtain: From Eq. 21 we get:
by using Eq. 15 and using Taylor series expansion on y^{(5)}(x)
and about
x = x_{i}.
From (5), we have ,
from which we obtain:
To find ,
we need the following equation:
where, .
From (50) and using Taylor series expansion, we get:
where, .
From (22) and using (23) we get that:
By (5), ,
from which we obtain:
To find we
need the following equations:
and from Eq. 3 that:
Then:
Using Eq. 13, 14 and apply Taylor’s expansion
formula, we can show that:
and From (25), using (26), (27) and Lemma 1 we can get:
Carrying on similar arguments we easily find that:
This proves Theorem 2 for .
For ,
we have from (2):
and
Carrying on similar steps as for the case ,
i=1,2,…,n1, we find the following inequalities:
and .
But for the first derivative of ,
we have the following inequality:
Also for S(x), we get:
This proves Theorem 2 for x∈[x_{0}, x_{1}]. Hence, the proof of Theorem 2 is completed. RESULTS AND DISCUSSION We present numerical results to demonstrate the convergence of the spline (0, 3, 5) function of degree six which constructed before to the second order initial value problem.
Problem 1: we consider that the second order initial value problem where
x∈[0,1] and y(0) = y`(0) = 1 with the exact solution y(x) = e^{x [6]}.
Problem 2: we consider that the second order initial value problem y˝y = x where x∈[0,1] and y(0) = y`(0) = 0. From Eq. 2 it’s easy to verify that:
Also it is easy from Eq. 2 and 3 to verify
that:
S_{i}(x_{i+1}) = y_{i+1} for i = 0,1,…, n1:
and
From (5) we have:
From 1 and 2, with using the values of a_{i,j}, i = 0, 1, …, n1 and
j = 2,4,6 given in the Eq. 916, we get:
It turns out that the six degree spline which presented in this study, yield approximate solution that is O(h^{6}) as stated in Theorem 2. The results are shown in the Table 1 and 2 for different step sizes h.
Table 1: 
An absolute maximum error for S(x) and it’s derivative’s
for problem 1 

Table 2: 
An absolute maximum error for S(x) and it’s derivative’s for
problem 2 

CONCLUSION In this study we treat for a first time a lacunary data (0,3,5) by constructing spline function of degree six which interpolates the lacunary data (0,3,5) and the constructed spline function applied to solve the second order initial value problems. Numerical examples, showed that the presented spline function proved their effectiveness in solving the second order initial value problems. Also, we note that, the better error bounds are obtained for a small step size h. " target="_blank">View Fulltext
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