INTRODUCTION
Initial value problems occur in a number of areas of applied mathematics among which are fluid mechanics, elasticity and quantum mechanics as well as science and engineering. Only small class of differential equations can be solved.
Several researchers have investigated some numerical methods for solving initial
value problems, among which include cubic spline method, finite difference method,
multiderivative method and finite element method (Burden
and Faires, 2001; Kanth and Vishnu, 2006).
The literature on the numerical solutions of initial value and boundary value
problems by using lacunary spline functions is not too much. Saxena
and Venturino (1994) used twopoint boundary value problem by using lacunary
spline function which interpolates the lacunary data (0, 2) and also Saki
and Usmani (1983), Saxena and Venturino (1994),
Siddiqi and Akram (2003, 2007),
Siddiqi et al. (2007, 2008)
are used qunitic spline functions for different boundary value problems. Saeed
and Jwamer (2004) showed the existences, uniqueness and error bounds for
lacunary interpolation by six degree spline, that spline has degree six only
on the first and the final subintervals in given partitions and Jwamer
(2005) showed theoretically only this spline which is found by Saeed
and Jwamer (2004) approximate to the second order Cauchy problem but in
the present work we showed the existence, uniqueness and error bounds for the
same lacunary which is (0, 1, 4) but the degree of spline six at the all subintervals
of given partition. Also, we showed theoretically and practically, that this
spline approximate to the solution of fourth order initial value problem which
defined in Eq. 1. AlBayati et al.
(2009) and Sallam and Hussien (1984) are used different
lacunary type of approximation solutions for different initial value problems.
The general fourth order initial value problem is considered of the form:
With the initial conditions:
By using that fεC^{n1} ([0, 1]xR^{4}) and that f is Lipschitz continuous in y, y', y'' and y'''.
The lacunary interpolation problem, which we have investigated in this study, consists in finding the six degree spline S (x) of deficiency four, interpolating data given on the function value and first and fourth order in the interval [0, 1]. Also, an extra initial condition is prescribed on the second derivative.
DESCRIPTIONS OF THE METHOD
Here, we present six degree spline interpolation for one dimensional and given sufficiently smooth function f (x) defined on I = [0, 1] and Δn: 0 = x_{0}<x_{1}<x_{2}<...<x_{n} = 1 denote the uniform partition of I with knots x_{i} = ih,where, I = 0, 1, 2,..., n. We denote by S^{4}_{n,6} the class of six degree splines S (x) such that:
on the interval [x_{0}, x_{1}] where, a_{0,j}, j = 3, 5 and 6 unknowns are to be determined.
Let us examine now intervals [x_{i}, x_{i+t}], 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, a_{i,j}, t = 1 (1) (n1), j = 2, 3, 5 and 6 unknowns we need to be determine it.
On the last interval [x_{n1}, x_{n}] we define S_{n1} (x) as follows:
where, a_{n1,j}, j = 2, 3 5 and 6 unknowns are to be determined.
THEORETICAL RESULTS
Here, the existence and uniqueness theorem for spline function of degree six which interpolate the lacunary data (0, 1, 4 ) is presented and examined.
Theorem 1: Existence and Uniqueness Spline Function
Given the real numbers y^{(r)}, i = 0, 1, 2,…, n and r = 0,
1, 4 and r = 0, 1, 4 and y'' (x_{0}) and y'' (x_{n}) then, there
exist a unique spline of degree six as given in the Eq. 35
such that:
Proof
We defined the 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. 3 by using the condition 7 where, i = 0, we obtain the following:
And
From the boundary condition 8 we have:
Solving Eq. 911 to obtain the following:
Substituting these values of a_{0,3}, a_{0,5} and a_{0,6} we get:
We shall find the coefficients of S_{i} (x) on the interval [x_{i},x_{i+1}] for I = 1, 2, 3,..., n2.
Here we have:
And
From Eq. 9 we have:
substitute it in each of Eq. 17, 18 and
20 and we obtain the following linear system:
Where:
Clearly, the above linear system Eq. 22 has unique solutions and after solving it we obtain the following relations:
Note that the values a_{i,2}, a_{i,3}, a_{i,5}, and a_{i,6} for i = 1, 2,…, n2 has unique and to find each of them only the first step means for i = 1 use Eq. 16 and after that it is easy that we show the above linear has unique solutions.
Finally, for finding the coefficients of S_{n1} (x), we have:
And
Solving Eq. 2729, we see that the coefficients
a_{n1,i}; i = 2, 3, 5 and 6 are uniquely determined. Hence the proof
of Theorem 1 is completed.
CONVERGENCE AND ERROR BOUNDS
The error bound of the spline function S (x) which is a solution of the problem 6 is obtained for the uniform partition I by the following theorem:
Theorem 2
Let y∈C^{6} [0, 1] and S (x) be a unique spline function of
degree six which is a solution of the problem 6. Then for x∈ [x_{i},
x_{i+1}]; i = 0, 1, 2,…, n1:
And for i = 1, 2, 3,…, n2:
And for i = n1:
where, W_{6} (h) denotes the modules of continuity of y^{(6)},
defined by W_{6} (h) = max {y^{(6)} (x)y^{(6)}
(y); where xy<h and x, y∈[0, 1]}
To prove this theorem we need the following lemma:
Lemma 1
Let y∈C^{6} [0, 1]. Then,
for i = 0, 1, …, n1
Where:
And W_{6} (h) denotes the modules of continuity of y^{(6)}.
Proof of Lemma 1
If y∈C^{6} [0, 1] then using Taylor’s expansion formula,
we have:
where, x_{i}<θ_{i}<x_{i+1} and similar expressions
for the derivatives of y (x) can be used.
Now from Eq. 23 and using Eq. 31 we obtain:
where, x_{i}<θ_{s,i}<x_{i+1} for i = 1, 2,..., n; s = 1, 2, 3, 4 and:
where, x_{i}<θ_{1,0}, θ_{2,0}, θ_{3,0}, θ_{4,0}<x_{1}
We see that the system of Eq. 32 and 33
is unknown e_{i,2}, i = 1, 2, …,n1 has the unique solution:
Where:
It is clear that:
Hence:
This completes the proof of the lemma 1.
Proof of Theorem 2
Let xε [x_{i}, x_{i+1}] where, i = 1, 2, …, n2.
We have from Eq. 4 with apply Taylor’s expansion formula we have
Using Eq. 26 and 34, lemma 1 and Taylor’s
series, we have
From Eq. 4 we have:
From Eq. 36 we obtain:
From Eq. 25, 35 and 37
and lemma 1 we get:
Since:
From Eq. 6, we have S_{i}^{(4)} (x_{i})y^{(4)}
(x_{i}) = 0, from which and using Eq. 38 and 39
we obtain:
To find S_{i}^{(3)} (x)y^{(3)} (x), from Eq. 4 we have:
Using Eq. 24, 35, 38
and Taylor’s formula in Eq. 41 we obtain
To find S_{i}^{(2)} (x)y^{(2)} (x), from Eq. 4 we have:
Using Eq. 25, 35, 38
and 42, lemma 1 and Taylor’s formula in Eq.
43 we obtain
Since:
From Eq. 6, we have S_{i}' (x_{i})y' (x_{i})
= 0, then using Eq. 44 and 45 we obtain:
Since:
And since from Eq. 6, we have S_{i} (x_{i})y
(x_{i}) = 0, then put it in above Eq. 47 and using
Eq. 46 we obtain:
This proves Theorem 2 for x∈ [x_{i}, x_{i+1}], I = 1,
2,..., n1.
For x∈ [x_{0}, x_{1}], we have from Eq. 3:
Using Eq. 15 and Taylor’s series in Eq.
48 we obtain:
Carrying on similar steps as for the case x∈ [x_{i}, x_{i+1}],
i = 1, 2, …, n1, we find the following inequalities:
This proves Theorem 2 for x∈ [x_{0}, x_{1}]. And for x∈
[x_{n1}, x_{n}] the same manner in above is used to get a_{n1,2},
a_{n1,3}, a_{n1,5} and a_{n1,6}, the proof of Theorem
2 is complete.
NUMERICAL RESULTS
This study presents numerical result to demonstrate the convergence of the spline function of degree six which is constructed in the section three to the fourth order initial value problem.
Problem
We consider that the fourth order initial value problem y^{(4)}y
= 0 where, x∈ [0, 1], y (0) = y'(0) = y''(0) = y'''(0) = 1.
From Eq. 3 it’s easy to verify that:
Table 1: 
Absolute maximum error for S (x) and it’s derivative 

Also it is easy from Eq. 3 and 4 to verify
that:
From Eq. 6 we have:
From Eq. 3 and 4, with using the values
of a_{i,j}, i = 0, 1,…,n1 and j = 2, 3, 5, 6 given in the Eq.
1316 and 2326,
we get:
It turns out that the six degree spline which is presented in this study, yield
approximate solution that is O(h^{6}) as stated in Theorem 1. The results
are shown in the Table 1 for different step sizes h in the
algorithm similar to AlBayati et al. (2009).
CONCLUSION
In this study, we get to the conclusion that, this spline which is defined
in Eq. 35 approximate to the solution of
Eq. 1 and also if we change the step size h by taking small
value we obtain the best approximate solution to the exact solution of Eq.
1.