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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 3, Pages 389–409 (Mi vsgtu1511)  

This article is cited in 2 scientific papers (total in 2 papers)

Differential Equations and Mathematical Physics

The 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 kind

V. N. Maklakov

Samara State Technical University, Samara, 443100, Russian Federation

Abstract: We 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.

Keywords: ordinary differential equations, ordinary differential equation systems, boundary value problems, boundary conditions of the first, second and third kind, order of approximation, numerical methods, Taylor polynomials

DOI: https://doi.org/10.14498/vsgtu1511

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Bibliographic databases:

Document Type: Article
UDC: 517.927:519.624
MSC: 34B99
Original article submitted 15/VII/2016
revision submitted – 27/VIII/2016

Citation: V. N. Maklakov, “The 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 kind”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:3 (2016), 389–409

Citation in format AMSBIB
\Bibitem{Mak16}
\by V.~N.~Maklakov
\paper The 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 kind
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2016
\vol 20
\issue 3
\pages 389--409
\mathnet{http://mi.mathnet.ru/vsgtu1511}
\crossref{https://doi.org/10.14498/vsgtu1511}
\zmath{https://zbmath.org/?q=an:06964517}
\elib{http://elibrary.ru/item.asp?id=28282240}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
    Cycle of papers

    This publication is cited in the following articles:
    1. V. N. Maklakov, “Otsenka poryadka approksimatsii matrichnogo metoda chislennogo integrirovaniya kraevykh zadach dlya sistem lineinykh neodnorodnykh obyknovennykh differentsialnykh uravnenii vtorogo poryadka s peremennymi koeffitsientami. Soobschenie 2. Kraevye zadachi s granichnymi usloviyami vtorogo i tretego roda”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 21:1 (2017), 55–79  mathnet  crossref  elib
    2. V. N. Maklakov, Ya. G. Stelmakh, “Chislennoe integrirovanie matrichnym metodom kraevykh zadach dlya lineinykh neodnorodnykh obyknovennykh differentsialnykh uravnenii tretego poryadka s peremennymi koeffitsientami”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 22:1 (2018), 153–183  mathnet  crossref  zmath  elib
  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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