Geodesic
On two-dimensional systems of Volterra integral equations of the first kind
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 1, Tome 208 (2022), pp. 3-10.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider two-dimensional systems of Volterra integral equations of the first kind. The case where a system of integral equations of the second kind is obtained by differentiating the equations is well studied. We examine the case where this approach leads to a system of integral equations with an degenerate matrix of the principal part. We formulate sufficient conditions for the existence of a unique smooth solution in terms of matrix pencils.
Keywords: two-dimensional integral equation of Volterra type, integro-algebraic equation, matrix pencil. 
@article{INTO_2022_208_a0,
     author = {M. V. Bulatov and L. S. Solovarova},
     title = {On two-dimensional systems of {Volterra} integral equations of the first kind},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--10},
     publisher = {mathdoc},
     volume = {208},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_208_a0/}
}
TY  - JOUR
AU  - M. V. Bulatov
AU  - L. S. Solovarova
TI  - On two-dimensional systems of Volterra integral equations of the first kind
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 3
EP  - 10
VL  - 208
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_208_a0/
LA  - ru
ID  - INTO_2022_208_a0
ER  - 
%0 Journal Article
%A M. V. Bulatov
%A L. S. Solovarova
%T On two-dimensional systems of Volterra integral equations of the first kind
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 3-10
%V 208
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_208_a0/
%G ru
%F INTO_2022_208_a0
M. V. Bulatov; L. S. Solovarova. On two-dimensional systems of Volterra integral equations of the first kind. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 1, Tome 208 (2022), pp. 3-10. http://geodesic.mathdoc.fr/item/INTO_2022_208_a0/

[1] Botoroeva M. N., Modelirovanie razvivayuschikhsya sistem na osnove integralnykh uravnenii Volterra, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Buryat. gos. un-t, 2019

[2] Boyarintsev Yu. E., Regulyarnye i singulyarnye sistemy lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, Novosibirsk, 1980

[3] Budnikova O. S., Chislennoe reshenie integro-algebraicheskikh uravnenii mnogoshagovymi metodami, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, MGU, 2015

[4] Bulatov M. V., “Chislennoe reshenie sistem integralnykh uravnenii Volterra pervogo roda I roda”, Zh. vychisl. mat. mat. fiz., 38:4 (1998), 607–611 | MR | Zbl

[5] Bulatov M. V., “Primenenie matrichnykh polinomov k issledovaniyu lineinykh differentsialno-algebraicheskikh uravnenii vysokogo poryadka”, Differ. uravn., 44:10 (2008), 1299–1306 | MR

[6] Gantmakher F. R., Teoriya matrits, Nauka, M., 1966 | MR

[7] Kolmogorov A. N., Elementy teorii funktsii i funktsionalnogo analiza, Fizmatlit, M., 2004

[8] Krasnov M. L., Integralnye uravneniya. Vvedenie v teoriyu, Nauka, M., 1975

[9] Chistyakov V. F., Algebro-differentsialnye operatory s konechnomernym yadrom, Nauka, Novosibirsk, 1996

[10] Balakumar V., “Numerical solution of Volterra integral-algebraic equations using block pulse functions”, Appl. Math. Comp., 263 (2015), 165–170 | DOI | MR | Zbl

[11] Brunner H., Collocation Methods for Volterra Integral and Related Functioal Equations, Cambridge Univ. Press, Cambridge, 2004 | MR

[12] Brunner H., Volterra Integral Equations: An Introduction to Theory and Applications, Cambridge Univ. Press, Cambridge, 2017 | MR | Zbl

[13] Bulatov M. V., “Two-dimensional integral-algebraic systems: Analysis and computational methods”, J. Comput. Appl. Math., 236:2 (2011), 132–140 | DOI | MR | Zbl

[14] Bulatov M. V., “Existence and uniqueness of solutions to weakly singular integral, algebraic and integro-differential equations”, Centr. Eur. J. Math., 12:2 (2014), 308–321 | MR | Zbl

[15] Bulatov M. V., “Construction of implicit multistep methods for solving integral algebraic equations”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. mat. Inform. Protsessy upravl., 1:3 (2019), 310–322 | MR

[16] Hadizadeh M., “Jacobi spectral solution for integral-algebraic equations of index $2$”, Appl. Numer. Math., 61 (2011), 131–148 | DOI | MR | Zbl

[17] Lamour R., März R., Tischendorf C., Differential-Algebraic Equations: A Projector- Based Analysis, Springer-Verlag, Berlin–Heidelberg, 2013 | MR | Zbl

[18] Pishbin S., “The semi-explicit Volterra integral algebraic equations with weakly singular kernels: the numerical treatments”, J. Comput. Appl. Math., 245 (2013), 121–132 | DOI | MR | Zbl

[19] Pishbin S., “Optimal convergence results of piecewise polynomial collocation solutions for integral-algebraic equations of index $3$”, J. Comput. Appl. Math., 279 (2015), 209–224 | DOI | MR | Zbl

[20] Pishbin S., “Numerical solution and structural analysis of two-dimensional integral-algebraic equations”, Numer. Algorithms., 73:2 (2016), 305–322 | DOI | MR | Zbl