On Sixtic Lacunary Spline Solutions of Second Order Initial Value Problem

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].