Geodesic
Solution of one hypersingular integro-differential equation defined by determinants
Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2021), pp. 17-28 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper provides an exact analytical solution to a hypersingular integro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with variable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.
Keywords: integro-differential equations; hypersingular integrals; generalised Sokhotsky formulas; differential equations; Riemann boundary problem. 
@article{BGUMI_2021_2_a1,
     author = {A. P. Shilin},
     title = {Solution of one hypersingular integro-differential equation defined by determinants},
     journal = {Journal of the Belarusian State University. Mathematics and Informatics},
     pages = {17--28},
     year = {2021},
     volume = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/BGUMI_2021_2_a1/}
}
TY  - JOUR
AU  - A. P. Shilin
TI  - Solution of one hypersingular integro-differential equation defined by determinants
JO  - Journal of the Belarusian State University. Mathematics and Informatics
PY  - 2021
SP  - 17
EP  - 28
VL  - 2
UR  - http://geodesic.mathdoc.fr/item/BGUMI_2021_2_a1/
LA  - ru
ID  - BGUMI_2021_2_a1
ER  - 
%0 Journal Article
%A A. P. Shilin
%T Solution of one hypersingular integro-differential equation defined by determinants
%J Journal of the Belarusian State University. Mathematics and Informatics
%D 2021
%P 17-28
%V 2
%U http://geodesic.mathdoc.fr/item/BGUMI_2021_2_a1/
%G ru
%F BGUMI_2021_2_a1
A. P. Shilin. Solution of one hypersingular integro-differential equation defined by determinants. Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2021), pp. 17-28. http://geodesic.mathdoc.fr/item/BGUMI_2021_2_a1/

[1] I. V. Boykov, E. S. Ventsel, A. I. Boykova, “An approximate solution of hypersingular integral equations”, Applied Numerical Mathematics, 60(6) (2010), 607–628 | DOI | MR | Zbl

[2] I. V. Boikov, “Analiticheskie i chislennye metody resheniya gipersingulyarnykh integralnykh uravnenii”, Dinamicheskie sistemy, 9(3) (2019), 244–272

[3] E. I. Zverovich, “Reshenie gipersingulyarnogo integro-differentsialnogo uravneniya s postoyannymi koeffitsientami”, Doklady Natsionalnoi akademii nauk Belarusi, 54(6) (2010), 5–8 | Zbl

[4] E. I. Zverovich, A. P. Shilin, “Reshenie integro-differentsialnykh uravnenii s singulyarnymi i gipersingulyarnymi integralami spetsialnogo vida”, Izvestiya Natsionalnoi akademii nauk Belarusi. Seriya fiziko-matematicheskikh nauk, 54(4) (2018), 404–407 | DOI

[5] A. P. Shilin, “Gipersingulyarnoe integro-differentsialnoe uravnenie eilerova tipa”, Izvestiya Natsionalnoi akademii nauk Belarusi. Seriya fiziko-matematicheskikh nauk, 56(1) (2020), 17–29 | DOI | MR

[6] A. P. Shilin, “O reshenii odnogo integro-differentsialnogo uravneniya s singulyarnym i gipersingulyarnym integralami”, Izvestiya Natsionalnoi akademii nauk Belarusi. Seriya fiziko-matematicheskikh nauk, 56(3) (2020), 298–309 | DOI

[7] A. P. Shilin, “Differentsialnaya kraevaya zadacha Rimana i ee prilozhenie k integro-differentsialnym uravneniyam”, Doklady Natsionalnoi akademii nauk Belarusi, 63(4) (2019), 391–397 | DOI

[8] E. I. Zverovich, “Kraevye zadachi teorii analiticheskikh funktsii v gelderovskikh klassakh na rimanovykh poverkhnostyakh”, Uspekhi matematicheskikh nauk, 26(1) (1971), 113–179 | MR | Zbl

[9] E. I. Zverovich, “Obobschenie formul Sokhotskogo”, Izvestiya Natsionalnoi akademii nauk Belarusi. Seriya fiziko-matematicheskikh nauk, 2 (2012), 24–28

[10] F. D. Gakhov, “Kraevye zadachi”, 3, Nauka, Moskva, 1977, 640 | MR

[11] M. A. Evgrafov, K. A. Bezhanov, Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin, “Sbornik zadach po teorii analiticheskikh funktsii”, 2, Nauka, Moskva, 1972, 416 | MR