APPROXIMATE SOLUTION OF HYPERSINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND WITH SECOND ORDER FEATURES ON THE CLASS OF FUNCTIONS WITH WEIGHT ((1+ x) / (1− x)^±1/2 

Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of the subdepartment of higher and applied mathematics, Penza State University (40, Krasnaya street, Penza, Russia), E-mail: boikov@pnzgu.ru
Boykova Alla Il'inichna, 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: math@pnzgu.ru 

517.392 

10.21685/2072-3040-2019-3-6 

Background. Approximate methods for solving hypersingular integral equations are an actively developing area of computational mathematics. This is due to the numerous applications of hypersingular integral equations in aerodynamics, electrodynamics, physics, and the fact that analytical solutions of hypersingular integral equations are possible only in exceptional cases. In addition to direct applications in physics and technology, hypersingular integral equations of the first kind arise in the approximate solution of boundary-value problems of mathematical physics. Recently, interest in the study of analytical and numerical methods for solving hypersingular integral equations has significantly increased in connection with their active use in modeling various problems in radio engineering and radar. It turned out that one of the main methods of mathematical modeling of antennas is hypersingular integral equations. In this paper, projection methods for solving hypersingular integral equations of the first kind with second-order singularities are proposed and justified. The case when the solution has the form of x(t) = (1− t2 )^±1/2ϕ(t) .
Materials and methods. Methods of functional analysis and approximation theory are used. Function spaces in which hypersingular operators act are introduced. To prove the solvability of the proposed computational scheme and assess the accuracy of the approximate method, the general theory of Kantorovich approximate methods is used.
Results. A computational scheme is constructed for the approximate solution of hypersingular integral equations with second-order singularities on a class of solutions of the form of x(t) = (1− t2 )^±1/2ϕ(t) . Estimates of the speed of convergence and the error of the computational scheme are obtained.
Conclusions. A computational scheme for the approximate solution of first-type hypersingular integral equations defined on a segment [–1,1]. The results can be used to solve problems of aerodynamics (finite-wing equation), electrodynamics (diffraction on different screens), hydrodynamics (hydrofoil theory), and to solve equations of mathematical physics by the method of boundary integral equations. 

hypersingular integral equations, collocation method, mechanical quadrature method 

1. Gakhov F. D. Kraevye zadachi [Boundary value problems]. Moscow: Nauka, 1977, 640 p. [In Russian]
2. Muskhelishvili N. I. Singulyarnye integral'nye uravneniya [Singular integral equations]. Moscow: Nauka, 1968, 612 p. [In Russian]
3. Boykov I. V., Boykova A. I. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2017, no. 2, pp. 63–78. [In Russian]
4. Boykov I. V., Boykova A. I. Analytical methods for solution of hypersingular and polyhypersingular integral equations. arXiv:1901.04880v1 [math.NA] 15 Janury 2019, 22 p.
5. 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, 287 p.
6. 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]
7. Belotserkovskiy S. M., Lifanov I. K. Chislennye metody v singulyarnykh integral'nykh uravneniyakh [Numerical methods in singular integral equations]. Moscow: Nauka, 1985, 256 p. [In Russian]
8. 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]
9. Prossdorf S., Silberman B. Numerical Analysis for Integral and Related Operator Equations. Berlin, Acad. Verl., 1991.
10. Mikhlin S. G., Prossdorf S. Singular Integral Operatoren. Berlin, Acad. Verl., 1980.
11. 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]
12. Boykov I. V. Numerical methods for solution of singular integral equations. arXiv: 1610.09611[math.NA]. 182 p.
13. Boykov I. V. Funktsional'nyy analiz i teoriya funktsiy: sb. [Functional analysis and theory of functions: selected articles]. Issue. 7. Kazan, 1970, pp. 3–23. [In Russian]
14. Boykov I. V., Roudnev V. A., Boykova A. I., Baulina O. A. Applied Numerical Mathematics. 2018, vol. 127, pp. 280–305.
15. Boykov I. V., Ventsel E. S., Boykova A. I. Applied Numerical Mathematics. 2010, vol. 60, no. 6, pp. 607–628.
16. Boykov I. V., Ventsel E. S., Roudnev V. A., Boykova A. I. Applied Numerical Mathematics. 2014, vol. 86, pp. 1–21.
17. Boykov I. V., Zakharova Yu. F., Semov M. A., Esaf'ev A. A. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2014, no. 3, pp. 101–113. [In Russian]
18. Boykov I. V. Uchenye zapiski Penzenskogo politekhnicheskogo instituta [Proceedings of Penza Polytechnic Institute]. Issue 4. Penza, 1973, pp. 42–61. [In Russian]
19. Boykov I. V., Zakharova Yu. F. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2010, no. 1, pp. 80–90. [In Russian]
20. Boykov I. V., Zakharova Yu. F. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2012, no. 3 (23), pp. 99–114. [In Russian]
21. Golberg M. A. Journal of Integral Equations. 1983, vol. 5, pp. 329–340.
22. Golberg M. A. Journal of Integral Equations. 1985, vol. 9, pp. 267–275.
23. Eshkuvatov Z. K., Narzullaev A. Indian Journal of Industrial and Applied Mathematics. 2019, vol. 10, no. 1 (Special Issue), pp. 11–37.
24. Eshkuvatov Z. K., Zulkarnain F. S., N. M. A. Nik Long, Muminov Z. SpringerPlus. 2016, vol. 5, – p. 1473.
25. Eminov S. I., Petrova S. Yu. Vestnik YuUrGU. Ser. Mat. Model. Progr. [Proceedings of South Ural State University. Series of mathematical modeling and programming]. 2018, vol. 11, iss. 2, pp. 139–146. [In Russian]
26. Lifanov I. K. Uspekhi sovremennoy radioelektroniki [Advances in modern electronics]. 2006, no. 8, pp. 62–67. [In Russian]
27. Vaynikko G. M., Lifanov I. K., Poltavskiy L. N. Chislennye metody v gipersingulyarnykh integral'nykh uravneniyakh i ikh prilozheniya [Numerical methods in hypersingular integral equations and their applications]. Moscow: Yanus-K., 2001, 508 p. [In Russian]
28. Oseledets I. V., Tyrtyshnikov E. E. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki [Journal of calculus mathematics and mathematical physics]. 2005, vol. 45, no. 2, pp. 315–326. [In Russian]
29. Boykov I. V., Boykova A. I. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2017, no. 2, pp. 79–90. [In Russian]
30. Boykov I. V., Boykova A. I., Semov M. A. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2015, no. 3, pp. 11–27. [In Russian]
31. Hadamard J. Lecons sur la Propagation des Ondes et les Equations de l'Hydrodynamique [Lessons on wave propagation and hydrodynamic equations]. Herman. Paris, 1903, 320 p. (reprinted by Chelsea. New York, 1949).
32. Adamar Zh. Zadacha Koshi dlya lineynykh uravneniy s chastnymi proizvodnymi giperbolicheskogo tipa [Cauchy problem for linear partial differential equations of hyperbolic type]. Moscow: Nauka, 1978, 351 p. [In Russian]
33. Adamar Zh. Issledovanie psikhologii protsessa izobreteniya v oblasti matematiki [The study of the psychology of the invention process in the field of mathematics]. Moscow: Sovetskoe radio, 1970, 152 p. [In Russian]
34. Chikin L. A. Uchenye zapiski Kazanskogo gosudarstvennogo universiteta [Proceedings of Kazan State University]. 1953, vol. 113, bk. 10, pp. 57–105. [In Russian]
35. Kaya A. C., Erdogan E. Quatery of applied mathematics. 1987, vol. 45, no. 1, pp. 105–122.
36. Gel'fand I. M., Shilov G. E. Obobshchennye funktsii i deystviya nad nimi [Generalized functions and actions on them]. Issue 1. Moscow: GIFML, 1958, 439 p. [In Russian]
37. Kantorovich L. V., Akilov G. P. Funktsional'nyy analiz [Functional analysis]. Moscow: Nauka, 1977, 750 p. [In Russian]