Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of the sub-department of higher and applied mathematics, Penza State University (40, Krasnaya street, Penza, Russia), E-mail: email@example.com
Kudryashova Natal'ya Yur'evna, Candidate of physical and mathematical sciences, associate professor, sub-department of higher and applied mathematics, Penza State University (40, Krasnaya street, Penza, Russia), E-mail: firstname.lastname@example.org
Shaldaeva Anastasiya Aleksandrovna, Masters’s degree student, Penza State University (40, Krasnaya street, Penza, Russia), E-mail: email@example.com
Background. The work is devoted to the study of sets of functions in which the condition for the unique solvability of degenerate singular integral equations is satisfied. At present, the study of many sections of singular integral equations can be considered completed. An exception is singular integral equations that vanish on manifolds with a measure greater than zero. The theory of singular integral equations in degenerate cases is constructed, from which it follows that, firstly, degenerate singular integral equations have an infinite number of solutions; secondly, the first and second Noether theorems are not valid for these equations. But specific algorithms and approximate methods for solving singular integral equations in degenerate cases are absent. Due to the fact that many processes in physics and technology are modeled by degenerate singular integral equations, it becomes necessary to develop approximate methods for solving them. In addition, since in the Holder space and in the space L2 of functions summable in a square, degenerate singular integral equations have an infinite number of solutions, the actual problem of distinguishing the uniqueness sets of the solutions of these equations arises, as well as the equally urgent problem of constructing approximate methods for solving them.
Materials and methods. To distinguish classes of functions in which degenerate singular integral equations have a unique solution, methods of the theory of functions of a complex variable, Riemann boundary value problems, and the theory of singular integral equations are used. When constructing approximate methods, iterative-projection methods are used.
Results. Classes of functions are constructed on which solutions, if they exist, are uniquely determined. In this regard, a new formulation of the solution of degenerate singular integral equations is proposed. Collocation and mechanical quadrature methods for solving degenerate singular integral equations on the constructed classes of functions are proposed and substantiated.
Conclusions. The proposed results can be directly used in solving many problems of physics and technology, in particular, in the problems of integral geometry, aerodynamics, and hydrodynamics. It is of interest to extend these results to degenerate polysingular integral equations.
1. Gakhov F. D. Kraevye zadachi [Boundary value problems]. Moscow: Nauka, 1963, 640 p. [In Russian]
2. Muskhelishvili N. I. Singulyarnye integral'nye uravneniya [Singular integral equations]. Moscow: Nauka, 1966, 707 p. [In Russian]
3. Ivanov V. V. Teoriya priblizhennykh metodov i ee primenenie k chislennomu resheniyu singulyarnykh integral'nykh uravneniy [The theory of approximate methods and its application to the numerical solution of singular integral equations]. Kiev: Naukova Dumka, 1968, 288 p. [In Russian]
4. Gokhberg I. Ts., Fel'dman I. A. Uravneniya v svertkakh i proektsionnye metody ikh resheniya [Convolution equations and projection methods for solving them]. Moscow: Nauka, 1971, 352 p. [In Russian]
5. Belotserkovskiy S. M., Lifanov I. K. Chislennye metody v singulyarnykh integral'nykh uravneniyah [Numerical methods in singular integral equations]. Moscow: Nauka, 1985, 256 p. [In Russian]
6. Lifanov I. K. Metod singulyarnykh integral'nykh uravneniy i chislennyy eksperiment [The method of singular integral equations and numerical experiment]. Moscow: TOO «Yanus», 1995, 520 p. [In Russian]
7. Mikhlin S. G., Prossdorf S. Singulare Integraloperatoren. Berlin, Acad. Verl., 1980, 514 p.
8. Prossdorf S., Silbermann B. Numerical Analysis for Integral and Related Operator Equations. Berlin: Acad. Verl., 1991, 544 p.
9. Boykov I. V. Priblizhennoe reshenie singulyarnykh integral'nykh uravneniy [An approximate solution of singular integral equations]. Penza: Izd-vo PGU, 2004, 316 p. [In Russian]
10. Chikin L. A. Uchenye zapiski Kazanskogo gosudarstvennogo universiteta [Proceedings of Kazan State University]. 1953, vol. 113, bk. 10, pp. 57 – 105. [In Russian]
11. Lavrent'ev M. M. Uspekhi matematicheskikh nauk [Advances in mathematical sciences]. 1979, vol. 34, no. 4, p. 143. [In Russian]
12. Lavrent'ev M. M. Sibirskiy matematicheskiy zhurnal [Siberian mathematical journal]. 1980, vol. 21, no. 3, pp. 225–228. [In Russian]
13. Boykov I. V. Primenenie vychislitel'nykh metodov v nauchno-tekhnicheskikh issledovaniyakh: mezhvuz. sb. nauchn. tr. [The use of computational methods in scientific and technical research: interuniversity collected papers]. Issue 6. Penza: Izdvo Penz. politekh. in-ta, 1984, pp. 3–11. [In Russian]
14. Boykov I. V., Kudryashova N. Yu. Differentsial'nye uravneniya [Differential equations]. 2000, vol. 36, no. 9, pp. 1230–1237. [In Russian]
15. Boykov I. V., Zelina Ya. V. Voprosy radioelektroniki [Radioelectronics’ issues]. 2017, no. 12, pp. 89–92. [In Russian]
16. Boykov I. V., Zelina Ya. V. Voprosy radioelektroniki [Radioelectronics’ issues]. 2018, no. 12, pp. 64–68. [In Russian]
17. Gakhov F. D., Cherskiy Yu. I. Uravneniya tipa svertki [Convolution type equations]. Moscow: Nauka, 1978, 296 p. [In Russian]