Exact Three-Point Scheme and Schemes of High Order of Accuracy for a Forth-Order Ordinary Differential Equation

We propose an exact three-point scheme and schemes of high order of accuracy, which are two systems of linear algebraic equations. Each equation of the system contains five unknown values of the exact solution and its first derivative at three grid points on the interval. In constructing the scheme,...

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Veröffentlicht in:	Cybernetics and systems analysis Jg. 56; H. 4; S. 566 - 576
1. Verfasser:	Prikazchikov, V.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.07.2020 Springer Springer Nature B.V
Schlagworte:	Accuracy Algorithms Artificial Intelligence Control Differential equations Exact solutions Linear algebra Linear equations Mathematical analysis Mathematics Mathematics and Statistics Matrix methods Ordinary differential equations Processor Architectures Software Engineering/Programming and Operating Systems Superposition (mathematics) Systems Theory system of linear algebraic equations exact scheme superposition of solutions forth-order differential equation Green function boundary-value problem spectral problem Cauchy problem Wronskian scheme of high order of accuracy linearly independent solutions grid method tridiagonal matrix algorithm functional series
ISSN:	1060-0396, 1573-8337
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Abstract	We propose an exact three-point scheme and schemes of high order of accuracy, which are two systems of linear algebraic equations. Each equation of the system contains five unknown values of the exact solution and its first derivative at three grid points on the interval. In constructing the scheme, the principle of superposition of solutions was used. Partial sums of the functional series representing independent solutions provide schemes of arbitrary order of accuracy for the boundary-value poblem and for the spectral one. To solve systems of linear equations, the modified tridiagonal matrix algorithm is proposed.
AbstractList	We propose an exact three-point scheme and schemes of high order of accuracy, which are two systems of linear algebraic equations. Each equation of the system contains five unknown values of the exact solution and its first derivative at three grid points on the interval. In constructing the scheme, the principle of superposition of solutions was used. Partial sums of the functional series representing independent solutions provide schemes of arbitrary order of accuracy for the boundary-value poblem and for the spectral one. To solve systems of linear equations, the modified tridiagonal matrix algorithm is proposed. We propose an exact three-point scheme and schemes of high order of accuracy, which are two systems of linear algebraic equations. Each equation of the system contains five unknown values of the exact solution and its first derivative at three grid points on the interval. In constructing the scheme, the principle of superposition of solutions was used. Partial sums of the functional series representing independent solutions provide schemes of arbitrary order of accuracy for the boundary-value poblem and for the spectral one. To solve systems of linear equations, the modified tridiagonal matrix algorithm is proposed. Keywords: forth-order differential equation, boundary-value problem, spectral problem, Cauchy problem, linearly independent solutions, Wronskian, superposition of solutions, Green function, grid method, exact scheme, scheme of high order of accuracy, functional series, system of linear algebraic equations, tridiagonal matrix algorithm.
Audience	Academic
Author	Prikazchikov, V.
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Keywords	system of linear algebraic equations exact scheme superposition of solutions forth-order differential equation Green function boundary-value problem spectral problem Cauchy problem Wronskian scheme of high order of accuracy linearly independent solutions grid method tridiagonal matrix algorithm functional series
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References	CR10 Samarskii (CR12) 1983 CR3 Show (CR5) 1963; 3 CR6 Zav’yalov, Kvasov, Miroshnichenko (CR9) 1980 Prikazchikov, Klunnik (CR11) 1994; 30 Tikhonov, Samarskii (CR1) 1963; 3 Show (CR4) 1963; 3 Prikazchikov, Klunnik, Lyubomyrska (CR8) 1992; 76 Prikazchikov (CR2) 1969; 9 Prikazchikov (CR7) 2017; 53 VG Prikazchikov (273_CR11) 1994; 30 AA Samarskii (273_CR12) 1983 X Show (273_CR4) 1963; 3 VG Prikazchikov (273_CR8) 1992; 76 YS Zav’yalov (273_CR9) 1980 VG Prikazchikov (273_CR7) 2017; 53 VG Prikazchikov (273_CR2) 1969; 9 273_CR6 AN Tikhonov (273_CR1) 1963; 3 273_CR10 X Show (273_CR5) 1963; 3 273_CR3
References_xml	– volume: 3 start-page: 841 issue: 5 year: 1963 end-page: 860 ident: CR4 article-title: Homogeneous difference schemes for the fourth-order equation with discontinuous coefficients publication-title: Zh. Vych. Mat. Mat. Fiz. – volume: 3 start-page: 1014 issue: 6 year: 1963 end-page: 1031 ident: CR5 article-title: The difference Sturm–Liouville problem for the fourth-order equation with discontinuous coefficients publication-title: Zh. Vych. Mat. Mat. Fiz. – ident: CR6 – year: 1980 ident: CR9 publication-title: Methods of Spline Functions [in Russian] – ident: CR3 – volume: 53 start-page: 186 issue: 2 year: 2017 end-page: 192 ident: CR7 article-title: Methods to construct the accurate difference scheme for a differential equation of order 4 publication-title: Cybern. Syst. Analysis doi: 10.1007/s10559-017-9918-6 – volume: 9 start-page: 315 issue: 2 year: 1969 end-page: 335 ident: CR2 article-title: Homogeneous difference schemes of high order of accuracy for the Sturm–Liouville problem publication-title: Zh. Vych. Mat. Mat. Fiz. – volume: 30 start-page: 1800 issue: 10 year: 1994 end-page: 1805 ident: CR11 article-title: A priori estimate of solution of a biharmonic equation publication-title: Diff. Uravneniya – volume: 3 start-page: 99 issue: 3 year: 1963 end-page: 108 ident: CR1 article-title: On homogeneous difference schemes of high order of accuracy on non-uniform grids publication-title: Zh. Vych. Mat. Mat. Fiz. – volume: 76 start-page: 49 year: 1992 end-page: 59 ident: CR8 article-title: Spline projection schemes for equations of fourth order publication-title: Obchysl. ta Prykladna Matematyka – ident: CR10 – year: 1983 ident: CR12 publication-title: Theory of Difference Schemes [in Russian] – ident: 273_CR6 – volume: 9 start-page: 315 issue: 2 year: 1969 ident: 273_CR2 publication-title: Zh. Vych. Mat. Mat. Fiz. – volume: 3 start-page: 1014 issue: 6 year: 1963 ident: 273_CR5 publication-title: Zh. Vych. Mat. Mat. Fiz. – ident: 273_CR10 – volume: 30 start-page: 1800 issue: 10 year: 1994 ident: 273_CR11 publication-title: Diff. Uravneniya – volume: 3 start-page: 841 issue: 5 year: 1963 ident: 273_CR4 publication-title: Zh. Vych. Mat. Mat. Fiz. – volume: 53 start-page: 186 issue: 2 year: 2017 ident: 273_CR7 publication-title: Cybern. Syst. Analysis doi: 10.1007/s10559-017-9918-6 – volume-title: Theory of Difference Schemes [in Russian] year: 1983 ident: 273_CR12 – volume: 76 start-page: 49 year: 1992 ident: 273_CR8 publication-title: Obchysl. ta Prykladna Matematyka – volume: 3 start-page: 99 issue: 3 year: 1963 ident: 273_CR1 publication-title: Zh. Vych. Mat. Mat. Fiz. – ident: 273_CR3 – volume-title: Methods of Spline Functions [in Russian] year: 1980 ident: 273_CR9
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SubjectTerms	Accuracy Algorithms Artificial Intelligence Control Differential equations Exact solutions Linear algebra Linear equations Mathematical analysis Mathematics Mathematics and Statistics Matrix methods Ordinary differential equations Processor Architectures Software Engineering/Programming and Operating Systems Superposition (mathematics) Systems Theory
Title	Exact Three-Point Scheme and Schemes of High Order of Accuracy for a Forth-Order Ordinary Differential Equation
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