Abstract
Problem statement: The lacunary problem, which we had investigated in this study, consider in finding the spline function of degree six S(x) of deficiency four, interpolating data given on the function value and third, forth order in the interval [0,1]. Also, on the extra initial condition was prescribed on the first derivative. Other purpose of this construction was to solve the second order initial value problem by one example showed that the spline function being interpolation very well compare to [1]. The convergence analysis and the stability of approximation solution were investigated and compared with the exact solution to demonstrate the prescribed lacunary spline (0,3,4) function interpolation. Approach: An approximation with spline functions of degree six and deficiency four is developed for solving initial value problems, with prescribed nonlinear endpoint conditions. Under suitable assumptions with applications showed this spline of the type (0,3,4) are existences, uniqueness and error bounds of the deficient of the solution. Result: Numerical example showed that the presented spline function their effectiveness in solving the second order initial value problem and also showed that our result more well to result in [1]. Also, we note that, the better error bounds were obtained for small step size h. Conclusion: In this study we showed that the lacunary data (0,3,4) are more well approximate to the given second order initial value problem compare with the lacunary data (0,3,5) used in [1].
Highlights
The lacunary interpolation problem we investigate in this paper consists in finding the sixtic spline S(x) of deficiency four, interpolating data given on the function and its Cauchy’s problem for second derivative at a given set of nodes of the interval [0, 1]
We approximate the problem (1) by the six degree spline functions as the first boundary condition respect to known third and fourth order derivatives and found the best error bound with convergence analysis
We present numerical results to demonstrate the convergence of the spline (0,3,4) function of degree six which constructed before to the second order initial value problem
Summary
The lacunary interpolation problem we investigate in this paper consists in finding the sixtic spline S(x) of deficiency four, interpolating data given on the function and its Cauchy’s problem for second derivative at a given set of nodes of the interval [0, 1]. Siddiqi and Akram[7] used the quintic spline to find approximation solution of fourth order boundary-value problems. In several years various authors have been used spline functions for finding an approximate solution of initial value problem including as[3,4] studies the for third and fourth order boundary value problem.
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