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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Vydané v:	Cybernetics and systems analysis Ročník 56; číslo 4; s. 566 - 576
Hlavný autor:	Prikazchikov, V.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.07.2020 Springer Springer Nature B.V
Predmet:	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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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
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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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