Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)13396Research ArticleLocal-Cubic Spline for Approximate Solution of Boundary Value ProblemsZhanlavT-MijiddorjR-National University of MongoliaMongolian State University of Education150220162132317092016Copyright © 2016,2016We have constructed an explicit local-cubic spline for the approximation of the smooth functions and have studied the behavior of the approximation. To solve numerically boundary value problems, a spline-scheme based on the properties of the local-cubic spline and the standard cubic spline collocation is proposed. The scheme is implemented by sequentially solving two tridiagonal systems, which allow to use the three-point sweep method and diﬀer from each other only by matrix of the right-hand side of the equation. It indicates that this algorithm is eﬃcient. The number of operations depends linearly on the number of grid nodes. It is proved that the constructed spline possesses the same approximation properties as the local-cubic spline. Thus, in this paper we actually considered the approximation of the solutions of the boundary value problems. The proposed scheme also allows to ﬁnd the ﬁrst and second derivatives of the solution of the boundary value problem on the uniform grid nodes of the fourth-order accuracy with respect to the step-size of the grid. The numerical experiments conﬁrm the theoretical order of convergence. Due to good approximation properties and the simplicity of the algorithm implementation, the proposed method can be applied to solve numerically the boundary value problems for the second order ordinary diﬀerential equations, which often occur in mathematics, physics, and in the ﬁeld of natural and engineering sciences.boundary value problemscubic splinehigh accuracyкраевые задачикубический сплайнповышенная точностьЗавьялов Ю.С., Квасов Б.И., Мирошниченко В.Л. Методы сплайн-функций. - М.: Наука, 1980.Жанлав T. О методе сплайн-аппроксимации решения обыкновенных дифференциальных уравнений второго порядка. - 1992.Жанлав T. О трехточечной сплайн-схеме повышенной точности // ЖВМ и МФ. - 1991. - Т. 31, № 1. - С. 40-51.Дронов С.Г., Лигун А.А. Об одном сплайн-метод решения краевой задачи // Укр. матем. журнал. - 1989. - Т. 41, № 5. - С. 703-707.Жанлав T. О представлении интерполяционных кубических сплайнов через B-сплайны // Методы сплайн-функций (Новосибирск). - 1981. - № 87. - С. 3-10.Sablonnier P. Univariate spline quasi-interpolants and applications to numerical analysis // Rend. Sem. Mat. Univ. Pol. Torino.|2005.|Vol. 63, No 3.|Pp. 211-222.Zhanlav T., Mijiddorj R. The local integro cubic splines and their approximation properties // Appl. Math. Comput. - 2010. - Т. 216, № 7.Zhu C.G., Wang R.-H. Numerical solution of Burgers equation by cubic B-spline quasi-interpolation // Appl. Math. Comput. - 2009. - Т. 208, № 1.Zhu C., Kang W.-S. Numerical solution of Burgers-Fisher equation by cubic B-spline quasi-interpolation // Appl. Math. Comput. 2010. Vol. 216, No 9. Pp. 2679-2686.