Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University2050110.14498/vsgtu1511Research ArticleThe evaluation of the order of approximation of the matrix method for numerical integration of the boundary value problems for systems of linear non-homogeneous ordinary differential equations of the second order with variable coefficients. Message 1. Boundary value problems with boundary conditions of the first kindMaklakovVladimir N(Cand. Phys. & Math. Sci.; makvo63@yandex.ru), Associate Professor, Dept. of Higher Mathematics and Applied Informaticsmakvo63@yandex.ruSamara State Technical University1509201620338940914022020Copyright © 2016, Samara State Technical University2016We present the first message of the cycle from two articles where the rearrangement of the order of approximation of the matrix method of numerical integration depending on the degree in the Taylor’s polynomial expansion of solutions of boundary value problems for systems of ordinary differential equations of the second order with variable coefficients with boundary conditions of the first kind were investigated. The Taylor polynomial of the second degree use at the approximation of derivatives by finite differences leads to the second order of approximation of the traditional method of nets. In the study of boundary value problems for systems of ordinary differential equations of the second order we offer the previously proposed method of numerical integration with the use of matrix calculus where the approximation of derivatives by finite differences was not performed. According to this method a certain degree of Taylor polynomial can be selected for the construction of the difference equations system. The disparity is calculated and the order of the method of approximation is assessed depending on the chosen degree of Taylor polynomial. It is theoretically shown that for the boundary value problem with boundary conditions of the first kind the order of approximation method increases with the degree of the Taylor polynomial and is equal to this degree only for its even values. For odd values of the degree the order of approximation is less by one. The theoretical conclusions are confirmed by a numerical experiment for boundary value problems with boundary conditions of the first kind.ordinary differential equationsordinary differential equation systemsboundary value problemsboundary conditions of the firstsecond and third kindorder of approximationnumerical methodsTaylor polynomialsобыкновенные дифференциальные уравнениясистемы обыкновенных дифференциальных уравненийкраевые задачиграничные условия первоговторого и третьего родапорядок аппроксимациичисленные методымногочлены Тейлора[Keller H. B. Accurate Difference Methods for Nonlinear Two-point Boundary Value Problems // SIAM J. Numer. Anal., 1974. vol. 11, no. 2. pp. 305-320. doi: 10.1137/0711028.][Lentini M., Pereyra V. A Variable Order Finite Difference Method for Nonlinear Multipoint Boundary Value Problems // Mathematics of Computation, 1974. vol. 28, no. 128. pp. 981-1003. doi: 10.2307/2005360.][Keller H. B. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations: Survey and Some Resent Results on Difference Methods / Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. New York: Academic Press, 1975. pp. 27-88. doi: 10.1016/b978-0-12-068660-5.50007-7.][Годунов С. К., Рябенький В. С. Разностные схемы. М.: Наука, 1977. 439 с.][Формалеев В. Ф., Ревизников Д. Л. Численные методы. М.: Физматлит, 2004. 400 с.][Самарский А. А. Теория разностных схем. М.: Наука, 1977. 656 с.][Самарский А. А., Гулин А. В. Численные методы. М.: Наука, 1973. 432 с.][Самарский А. А., Гулин А. В. Устойчивость разностных схем. М.: Наука, 1973. 416 с.][Boutayeb A., Chetouani A. Global extrapolations of numerical methods for solving a parabolic problem with non local boundary conditions // International Journal of Computer Mathematics, 2003. vol. 80, no. 6. pp. 789-797. doi: 10.1080/0020716021000039209.][Boutayeb A., Chetouani A. A Numerical Comparison of Different Methods Applied to the Solution of Problems with Non Local Boundary Conditions // Applied Mathematical Sciences, 2007. vol. 1, no. 44. pp. 2173-2185, http://www.m-hikari.com/ams/ams-password-2007/ams-password41-44-2007/boutayebAMS41-44-2007.pdf.][Радченко В. П., Усов А. А. Модификация сеточных методов решения линейных дифференциальных уравнений с переменными коэффициентами на основе тейлоровских разложений // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. Науки, 2008. № 2(17). С. 60-65. doi: 10.14498/vsgtu646.][Маклаков В. Н. Оценка порядка аппроксимации матричного метода численного интегрирования краевых задач для линейных неоднородных обыкновенных дифференциальных уравнений второго порядка // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 2014. № 36. С. 143-160. doi: 10.14498/vsgtu1364.][Самарский А. А., Николаев Е. С. Методы решения сеточных уравнений. М.: Наука, 1978. 592 с.][Рябенький В. С. Необходимые и достаточные условия хорошей обусловленности краевых задач для систем обыкновенных разностных уравнений // Ж. вычисл. матем. и матем. физ., 1964. Т. 4, № 2. С. 242-255.][Фихтенгольц Г. М. Курс дифференциального и интегрального исчисления. Т. 1. М.: Наука, 1970. 608 с.][Курош А. Г. Курс высшей алгебры. М.: Наука, 1975. 431 с.][Турчак Л. И. Основы численных методов. М.: Наука, 1987. 320 с.][Закс Л. Статистическое оценивание. М.: Статистика, 1976. 598 с.]