Abstract
Systems of Volterra linear integral equations with identically singular matrices in the principal part (called integral-algebraic equations) are examined. Multistep methods for the numerical solution of a selected class of such systems are proposed and justified.
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References
A. S. Apartsyn, Nonclassical Linear Volterra Equations of the First Kind (Nauka, Novosibirsk, 1999; VSP, Utrecht, 2003).
H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations (North-Holland, Amsterdam, 1986).
P. Linz, Analytical and Numerical Methods for Volterra Equations (SIAM, Philadelphia, 1985).
Ten Men Yan, Candidate’s Dissertation in Mathematics and Physics (Irkutsk, 1985).
M. V. Bulatov, “Regularization of Singular Systems of Volterra Integral Equations,” Comput. Math. Math. Phys. 42, 315–320 (2002).
M. V. Bulatov and V. F. Chistyakov, Preprint, September (Memorial Univ. of Newfoundland, 1997).
V. F. Chistyakov, On Singular Systems of Ordinary Differential Equations and Their Integrals Analogues: Lyapunov Functions and Applications (Nauka, Novosibirsk, 1987) [in Russian].
M. V. Bulatov, “On Nonlinear Systems of Integral Equations of the Fourth Kind,” in Proceedings of 11th Baikal International School-Seminar on Optimization Methods and Their Applications July 5–12, 1998 (Baikal, Irkutsk, 1998), Vol. 4, pp. 68–71.
H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations (Cambridge Univ. Press, Cambridge, 2004).
J.-P. Kauthen, “The Numerical Solution of Integral-Algebraic Equations of Index-1 by Polynomial Spline Collocation Methods,” Math. Comput. 236, 1503–1514 (2000).
M. Hadizadeh, F. Ghoreishi, and S. Pishbin, “Jacobi Spectral Solution for Integral Algebraic Equations of Index-2,” Appl. Numer. Math. 1, 131–148 (2011).
M. V. Bulatov, “Transformations of Differential-Algebraic Systems of Equations,” Comput. Math. Math. Phys. 34, 301–311 (1994).
M. V. Bulatov, “Integro-Differential Systems with a Degenerate Matrix Multiplying the Derivative,” Differ. Equations 38, 731–737 (2002).
M. V. Bulatov and E. V. Chistyakova, “Numerical Solution of Integro-Differential Systems with a Degenerate Matrix Multiplying the Derivative by Multistep Methods,” Differ. Equations 42, 1317–1325 (2006).
E. V. Chistyakova, “Properties of Finite-Difference Schemes for Singular Integrodifferential Equations of Index 1,” Comput. Math. Math. Phys. 49, 1507–1515 (2009).
N. S. Bakhvalov, Numerical Methods: Analysis, Algebra, Ordinary Differential Equations (Nauka, Moscow, 1975; Mir, Moscow, 1977).
A. S. Apartsin and A. B. Bakushinskii, “Approximate Solution of Volterra Integral Equations of the First Kind by the Quadrature Method,” in Differential and Integral Equations (Irkutsk. Gos. Univ., Irkutsk, 1973), pp. 107–116 [in Russian].
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Original Russian Text © O.S. Budnikova, M.V. Bulatov, 2012, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2012, Vol. 52, No. 5, pp. 829–839.
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Budnikova, O.S., Bulatov, M.V. Numerical solution of integral-algebraic equations for multistep methods. Comput. Math. and Math. Phys. 52, 691–701 (2012). https://doi.org/10.1134/S0965542512050041
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DOI: https://doi.org/10.1134/S0965542512050041