Abstract
We consider linear homogeneous systems of integro-differential and integral equations with Volterra and Fredholm matrix kernels with zero initial conditions. The case is studied where the unknown vector function depends on one (integro-differential systems) or two (systems of integral equations) arguments and the matrix multiplying the principal part is square and identically singular. We point out the fundamental difference between the systems in question and systems solved for the principal part: there exists not only a trivial solution. In terms of matrix pencils and polynomials, we state sufficient conditions under which problems for the systems in question have only the trivial solution. Illustrative examples are given.
Similar content being viewed by others
REFERENCES
Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 1976.
Krasnov, M.L., Integral’nye uravneniya. Vvedenie v teoriyu (Integral Equations. Introduction to the Theory), Moscow: Nauka, 1975.
Bel’tyukov, B.A., Nekotorye voprosy teorii priblizhennykh metodov resheniya integral’nykh uravnenii (Some Questions of the Theory of Approximate Methods for Solving Integral Equations), Irkutsk: IGPI, 1994.
Verlan’, A.F. and Sizikov, V.S., Metody resheniya integral’nykh uravnenii s programmami dlya EVM (Methods for Solving Integral Equations with Computer Programs), Kiev: Nauk. Dumka, 1978.
Apartsin, A.S., Neklassicheskie uravneniya Vol’terry I roda: teoriya i chislennye metody (Nonclassical Volterra Equations of the First Kind: Theory and Numerical Methods), Novosibirsk: Nauka, 1999.
Brunner, H. and van der Houwen, P., The Numerical Solution of Volterra Equations, New York: North-Holland, 1986.
Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge: Cambridge Univ. Press, 2004.
Linz, P., Analytical and Numerical Methods for Volterra Equations, Philadelphia: SIAM, 1985.
Yan Ten Men, Approximate solution of linear Volterra integral equations of the first kind, Cand. Sci. (Phys.-Math.) Dissertation, Irkutsk Melentiev Energy Syst. Inst., 1985.
Bulatov, M.V. and Lima, P.M., Two-dimensional integral-algebraic systems: analysis and computational methods, J. Comput. Appl. Math., 2011, vol. 236, no. 2, pp. 132–140.
Brunner, H. and Hui, L., Collocation methods for integro-differential algebraic equations with index 1, IMA J. Numer. Anal., 2020, vol. 40, no. 2, pp. 850–885.
Chistyakova, E.V. and Chistyakov, V.F., Solution of differential algebraic equations with the Fredholm operator by the least squares method, Appl. Numer. Math., 2020, vol. 149, pp. 43–51.
Bulatov, M.V. and Chistyakova, E.V., Numerical solution of integro-differential systems with a degenerate matrix multiplying the derivative by multistep methods, Differ. Equations, 2006, vol. 42, no. 9, pp. 1317–1325.
Bulatov, M.V. and Do Tien Thanh, Multistep method for solving degenerate integral-differential equations, Vestn. YuUrGU. Ser. Mat. Model. Progr., 2014, vol. 7, no. 3, pp. 93–106.
Bulatov, M.V., On a family of matrix triples, Lyapunovskie chteniya i prezentatsiya informatsionnykh tekhnologii: mater. konf. (Lyapunov Readings Presentation Inf. Technol. Proc. Conf.) (Irkutsk, 2002), p. 10.
Bulatov, M.V. and Ming-Gong Lee, Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order, Differ. Equations, 2008, vol. 44, no. 10, pp. 1353–1360.
Chistyakov, V.F., Algebro-differentsial’nye operatory s konechnomernym yadrom (Algebraic-Differential Operators with a Finite-Dimensional Kernel), Novosibirsk: Nauka, 1996.
Funding
This work was supported by the Russian Science Foundation, project no. 20-51-S52003-a.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
Bulatov, M.V., Solovarova, L.S. On Systems of Integro-Differential and Integral Equations with Identically Singular Matrix Multiplying the Principal Part. Diff Equat 58, 1217–1224 (2022). https://doi.org/10.1134/S0012266122090063
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266122090063