Geodesic
On Stable Algorithms for Numerical Solution of Integral-Algebraic Equations
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 5-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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There is the necessity to study integral-algebraic equations if a prototype process has an aftereffect at the analysis of various areas of science. Particularly, a system of interrelated Volterra equations of the first and second kind and algebraic equations can be written as integral-algebraic equation. In this paper linear integral-algebraic equations are considered. We have constructed multistep methods for numerical solutions of IAEs. These methods are based on Adams quadrature formulas and on extrapolation formulas as well. We have proven suggested algorithms convergence. In this paper we show that our multistep methods have a property of self-regularizing; and regularization parameter is the step of a grid, which is connected with the level of accuracy of right-part error of the system under consideration. The results of numerical experiments illustrate theoretical computations.
Keywords: integral-algebraic equations; multistep methods; self-regularization. 
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M. V. Bulatov; O. S. Budnikova. On Stable Algorithms for Numerical Solution of Integral-Algebraic Equations. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 5-14. http://geodesic.mathdoc.fr/item/VYURU_2013_6_4_a0/

[1] Apartsyn A. S., Nonclassical Linear Volterra Equations of the First Kind, VSP, Utrecht, 2003 | MR | Zbl

[2] Apartsyn A. S., Bakushinskii A. B., “Approximate Solution of Volterra Integral Equations of the First Kind by the Quadrature Method”, Differential and Integral Equations, 1, Irkutsk. Gos. Univ., Irkutsk, 1973, 107–116

[3] Boyarintsev Yu. E., Regular and Singular Systems of Linear Ordinary Differential Equations, Nauka, Novosibirsk, 1980, 222 pp. | MR | Zbl

[4] Boyarintsev Yu. E., “Application of Generalized Inverse Matrices to Solve and to Research Systems of Partial Differential Equations of First Order”, Methods of Optimization and Operations Research, SEI SO AN USSR, Irkutsk, 1984, 123–141

[5] Boyarintsev Yu. E., Methods of Solution of Singular Systems of Ordinary Differential Equations, Nauka, Novosibirsk, 1988, 158 pp. | MR | Zbl

[6] Boyarintsev Yu. E., Methods of Solution of Continuous and Discontinuous Problems for Singular Systems of Equations, Nauka, Novosibirsk, 1996, 261 pp. | MR | Zbl

[7] Boyarintsev Yu. E., Korsukov V. M., “Application of Difference Methods to Solve Regular Systems of Ordinary Differential Equations”, Questions of Applied Mathematics, SEI SO AN USSR, Irkutsk, 1975, 140–152 | Zbl

[8] Boyarintsev Yu. E., Orlova I. V., Pencil Matrix and Algebraic-Differential Systems, Nauka, Novosibirsk, 2006, 124 pp.

[9] Budnikova O. S., Bulatov M. V., “Numerical Solution of Integral-Algebraic Equations of Multistep Methods”, Computational Mathematics and Mathematical Physics, 52:5 (2012), 691–701 | DOI | MR | Zbl

[10] Bulatov M. V., Chistjakov V. F., “Solution of Algebro-Differential Systems by Least Square Method”, Optimization Methods and Applications, Prod. XI Baikal Int. workshop, v. 4, ESI SB RAS, Irkutsk, 1998, 72–75

[11] Bulatov M. V., “Regularization of Singular Systems of Volterra Integral Equations”, Computational Mathematics and Mathematical Physics, 42:3 (2002), 315–320 | MR | Zbl

[12] Verlan' A. F., Sizikov V. S., Integral Equations: Methods, Algoritms, Solutions, Naukova dumka, Kiev, 1986 | MR

[13] Ten Men Yan, Approximate Solution of Linear Volterra Integral Equations of the First Kind, Candidate's Dissertation in Mathematics and Physics, Irkutsk, 1985

[14] Chistyakov V. F., “On Singular Systems of Ordinary Differential Equations and Their Integrals Analogues”, Lyapunov Functions and Applications, Nauka, Novosibirsk, 1987, 231–239

[15] Harrier E., Wanner G., Solving Ordinary Differential Equations, v. II, Springer-Verlag, 1991

[16] K. F. Brenan, S. L. Campbell, L. R. Petzold, “Numercal Solution of Initial-Value Problems in Differental-Algebraic Equations”, Appl. Math., Philadelphia, 1996 | MR

[17] H. Brunner, P. J. van der Houwen, The Numercal Solution of Volterra Equations, CWI Monographs, 3, North-Holland, Amsterdam, 1986 | MR | Zbl

[18] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, University Press, Cambridge, 2004 | MR | Zbl

[19] J. P. Kauthen, “The Numerical Solution of Integral-Algebraic Equations of Index-1 by Pollinomial Spline Collocation Methods”, Math. Comp., 236 (2000), 1503–1514 | DOI | MR

[20] P. Linz, “A Survey of Methods for the Solution of Volterra Integral Equations of the First Kind”, Collocation Methods for Volterra Integral and Related Functional Equations, University Press, Cambridge, 2004

[21] M. Hadizadeh, F. Ghoreishi, S. Pishbin, “Jacobi Spectral Solution for Integral Algebraic Equations of Index-2”, Appl. Numer. Math., 61:1 (2011), 131–148 | DOI | MR | Zbl