Abstract
We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.
[1] Boyarintsev Yu., Methods of Solving Singular Systems of Ordinary Differential Equations, Pure Appl. Math. (N.Y.), John Wiley & Sons, Chichester, 1992 Search in Google Scholar
[2] Brenan K.E., Campbell S.L., Petzold L.R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Classics Appl. Math., 14, SIAM, Philadelphia, 1996 10.1137/1.9781611971224Search in Google Scholar
[3] Brunner H., Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monogr. Appl. Comput. Math., 15, Cambridge University Press, Cambridge, 2004 http://dx.doi.org/10.1017/CBO978051154323410.1017/CBO9780511543234Search in Google Scholar
[4] Brunner H., Bulatov M.V., On singular systems of integral equations with weakly singular kernels, In: Optimization Methods and their Applications, Irkutsk, July 5–12, 1998, Melentiev Energy Systems Institute, Irkutsk, 1998, 64–67 Search in Google Scholar
[5] Brunner H., van der Houwen P.J., The Numerical Solution of Volterra Equations, CWI Monogr., 3, North-Holland, Amsterdam, 1986 Search in Google Scholar
[6] Bulatov M.V., Transformation of algebro-differential systems of equations, Comput. Math. Math. Phys., 1994, 34(3), 301–311 Search in Google Scholar
[7] Bulatov M.V., Integro-differential systems with a degenerate matrix multiplying the derivative, Differ. Equ., 2002, 38(5), 731–737 http://dx.doi.org/10.1023/A:102022712815810.1023/A:1020227128158Search in Google Scholar
[8] Bulatov M.V., Chistyakova E.V., On a family of singular integro-differential equations, Comput. Math. Math. Phys., 2011, 51(9), 1558–1566 http://dx.doi.org/10.1134/S096554251109006510.1134/S0965542511090065Search in Google Scholar
[9] Bulatov M.V., Lee M.-G., Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order, Differ. Equ., 2008, 44(10), 1353–1360 http://dx.doi.org/10.1134/S001226610810001710.1134/S0012266108100017Search in Google Scholar
[10] Bulatov M.V., Lima P.M., Two-dimensional integral-algebraic systems: Analysis and computational methods, J. Comput. Appl. Math., 2011, 236(2), 132–140 http://dx.doi.org/10.1016/j.cam.2011.06.00110.1016/j.cam.2011.06.001Search in Google Scholar
[11] Chistyakov V.F., Algebro-Differential Operators with a Finite-Dimensional Kernel, Nauka, Novosibirsk, 1996 (in Russian) Search in Google Scholar
[12] Chistyakov V.F., On singular systems of ordinary differential equations and their integral analogs, In: Lyapunov Functions and their Applications, Nauka, Novosibirsk, 1987, 231–239 (in Russian) Search in Google Scholar
[13] Gantmacher F.R., The Theory of Matrices, 1&2, Chelsea Publishing, New York, 1977 Search in Google Scholar
[14] Griepentrog E., März R., Differential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte Math., 88, Teubner, Leipzig, 1986 Search in Google Scholar
[15] Hadizadeh M., Ghoreishi F., Pishbin S., Jacobi spectral solution for integral-algebraic equations of index-2, Appl. Numer. Math., 2011, 61(1), 131–148 http://dx.doi.org/10.1016/j.apnum.2010.08.00910.1016/j.apnum.2010.08.009Search in Google Scholar
[16] Hairer E., Wanner G., Solving Ordinary Differential Equations, II, Springer Ser. Comput. Math., 14, Springer, Berlin, 1991 http://dx.doi.org/10.1007/978-3-662-09947-610.1007/978-3-662-09947-6Search in Google Scholar
[17] Kunkel P., Mehrmann V., Differential-Algebraic Equations, EMS Textbk. Math., European Mathematical Society, Zürich, 2006 10.4171/017Search in Google Scholar
[18] Lamour R., März R., Tischendorf C., Differential-Algebraic Equations: A Projector Based Analysis, Differ.-Algebr. Equ. Forum, Springer, Heidelberg, 2013 10.1007/978-3-642-27555-5Search in Google Scholar
[19] Lancaster P., Theory of Matrices, Academic Press, New York, London, 1969 Search in Google Scholar
[20] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993 Search in Google Scholar
© 2014 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
