On the solution of one integro-differential equation with singular and hypersingular integrals
https://doi.org/10.29235/1561-2430-2020-56-3-298-309
Abstract
A linear integro-differential equation of the first order given on a closed curve located on the complex plane is studied. The coefficients of the equation have a special structure. The equation contains a singular integral, which can be understood as the main value by Cauchy, and a hypersingular integral which can be understood as the end part by Hadamard. The analytical continuation method is applied. The equation is reduced to a sequential solution of the Riemann boundary value problem and two linear differential equations. The Riemann problem is solved in the class of analytic functions with special points. Differential equations are solved in the class of analytical functions on the complex plane. The conditions for the solvability of the original equation are explicitly given. The solution of the equation when these conditions are fulfilled is also given explicitly. Examples are considered. A non-obvious special case is analyzed.
About the Author
A. P. Shilin
Belarusian State University
Belarus
Belarus
Andrey P. Shilin – Ph. D. (Physics and Mathematics), Assistant Professor, Assistant Professor of the Department of Higher Mathematics and Mathematical Physics
4, Nezavisimosti Ave., 220030, Minsk
References
1. Muskhelishvili N. I. Singular Integral Equations. Moscow, Nauka Publ., 1968. 513 p. (in Russian).
2. Boykov I. V., Boykova A. I. Analytical methods of solving hypersingular integral equations. Izvestiyavuzow. Povolzshiy region. Fiziko-matematicheskiye nauki = University proceedings. Volga region. Physical and mathematical sciences, 2017, no. 2 (42), pp. 63–78. https://doi.org/10.21685/2072-3040-2017-2-6
3. Zverovich, E. I. Solution of the hypersingular integro-differential equation with constant coefficients. Doklady Nacionalnoi Akademii Nauk Belarusi = Proceedings of the National Academy of Sciences of Belarus, 2010, vol. 54, no. 6, pp. 5–8 (in Russian).
4. Zverovich E. I., Shilin A. P. Integro-differential equations with singular and hypersingular integrals. Vestsi Natsyianal’nai akademii navuk Belarusi. Seryiafizika-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2018, vol. 54, no. 4, pp. 404–407 (in Russian). https://doi.org/10.29235/1561-2430-2018-54-4-404-407
5. Shilin A. P. Riemann’s differential boundary-value problem and its application to integro-differential equations. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2019, vol. 63, no. 4, pp. 391–397 (in Russian). https://doi.org/10.29235/1561-8323-2019-63-4-391-397
6. Shilin A. P. A hypersingular integro-differential equations of the Euler type. Vestsi Natsyianal’nai akademii navuk Belarusi. Seryia fizika-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2020, vol. 56, no. 1, pp. 17–29 (in Russian). https://doi.org/10.29235/1561-2430-2020-56-1-17-29
7. Zverovich E. I. Generalization of Sohotsky formulas. Vestsi Natsyianal’nai akademii navuk Belarusi. Seryia fizikamatematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series, 2012, no. 2, pp. 2–28 (in Russian).
8. Zverovich E. I. Boundary value problems in the theory of analytic functions in Holder classes on riemann surfaces. Uspekhi matematicheskikh nauk = Advances in mathematical Sciences, 1971, vol. 26, no. 1 (157), pp. 117–192 (in Russian).
9. Gakhov F. D. Boundary Value Problems. Moscow, Nauka Publ., 1977. 640 p. (in Russian).
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