Int. J. Open Problems Compt. Math., Vol. 4, No. 1, March 2011 ISSN 1998-6262; Copyright © ICSRS Publication, 2011 www.i-csrs.org Cauchy Problem and Modified Lacunary Interpolations for Solving Initial Value Problems *Faraidun K. Hama-Salh and** Karwan H. F. Jwamer *University of Sulaimani-College of Science Education-Department of Mathematics-IRAQ e-mail: faraidun78@yahoo.com **University of Sulaimani-College of Science-Department of Mathematics, IRAQ e-mail: jwameri1973@gmail.com Abstract The purpose of this paper is to obtain approximate solution of Cauchy problem and solving initial values problems by modified spline functions of degree six which interpolate the lacunary data third and fifth derivatives. Other purpose of this construction is to study the convergence analysis and the upper bounds of errors of the approximate solution is investigated, also we compared with the exact solution to demonstrate the prescribed lacunary spline function interpolation. Keywords: Cauchy Problem, Lacunary Interpolation, Convergence Analysis. AMS subject classifications: 65D05, 65D07 and 65D32. 1 Introduction Consider the Cauchy's initial value problem: 2 0 1 0 0 ) ( , ) ( ], 1 , 0 [ , )) ( ), ( , ( ) ( y x y y x y x x y x y x f x y          (1)