
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 nonhomogeneous 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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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 nonhomogeneous 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 nonhomogeneous 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 389409
\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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http://mi.mathnet.ru/eng/vsgtu1511 http://mi.mathnet.ru/eng/vsgtu/v220/i3/p389
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Cycle of papers
 The evaluation of the order of approximation of the matrix method for numerical integration
of the boundary value problems for systems of linear nonhomogeneous 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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, 20:3, 389–409
 The evaluation of the order of approximation of the matrix method for numerical integration
of the boundary value problems for systems of linear nonhomogeneous ordinary differential equations
of the second order with variable coefficients.
Message 2. Boundary value problems with boundary conditions of the second and third kind
V. N. Maklakov Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, 21:1, 55–79
This publication is cited in the following articles:

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. unta. Ser. Fiz.mat. nauki, 21:1 (2017), 55–79

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. unta. Ser. Fiz.mat. nauki, 22:1 (2018), 153–183

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