A hypersingular integro-differential equation with linear functions in coefficients
https://doi.org/10.29235/1561-2430-2022-58-4-358-369
Abstract
In this paper, we investigated a new linear integro-differential equation of arbitrary order given on the closed curve located on a complex plane. The coefficients of the equation are variables and have a special form. The characteristic feature is the presence of linear functions in the coefficients. The equation is reduced to the consecutive solution of a Riemann boundary value problem on an initial curve and two linear differential equations. Differential equations are solved for analytic functions in areas into which the initial curve separates a complex plane. The corresponding fundamental systems of solutions are found, after that the arbitrary-constant variation method is applied. To achieve the analyticity of the obtained solutions the restrictions are imposed. All the arising conditions of resolvability of the input equation are written down explicitly, and if they are carried out then the solution is written in an explicit form. We represent the example demonstrating the existence of the cases when all conditions of resolvability are satisfied.
About the Author
A. P. Shilin
Belarusian State University
Russian Federation
Russian Federation
Andrey P. Shilin – Ph. D. (Physics and Mathematics), Associate Professor, Associate Professor of the Department of Higher Mathematics and Mathematical Physics, Belarusian State University.
4, Nezavisimosti Ave., 220030, Minsk
References
1. Hadamard J. Le Problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Paris, Hermann et Cie Éditeurs, 1932. 560 p. (in French).
2. Boykov I. V. Analytical and numerical methods for solving hypersingular integral equations. Dinamicheskie sistemy [Dynamic Systems], 2019, vol. 9, no. 3, pp. 244–272 (in Russian).
3. Zverovich E. I. Generalization of Sohotsky formulas. Vestsi Natsyianal’nai akademii navuk Belarusi. Seryia fizika-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2012, no. 2, pp. 24–28 (in Russian).
4. 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).
5. Zverovich E. I., Shilin A. P. Integro-differential equations with singular and hypersingular integrals. Vestsi Natsyianal’nai akademii navuk Belarusi. Seryia fizika-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
6. Shilin A. P. On the solution of one integro-differential equation with singular and hypersingular integrals. 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. 3, pp. 298–309 (in Russian). https://doi. org/10.29235/1561-2430-2020-56-3-298-309
7. 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
8. Shilin A. P. Hypersingular integro-differential equation with power factors in coefficients. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika = Journal of the Belarusian State University. Mathematics and Informatics, 2019, no. 3, pp. 48–56 (in Russian). https://doi.org/10.33581/2520-6508-2019-3-48-56.
9. Gakhov F. D. Boundary Value Problems. Moscow, Nauka Publ., 1977. 640 p. (in Russian).
10. Zaytsev V. F., Polyanin A. D. Handbook of Ordinary Differential Equations. Moscow, Fizmatlit Publ., 2001. 576 p. (in Russian).
11. Kamke E. Handbook of Ordinary Differential Equations. Saint Peterburg, Lan’ Publ., 2003. 576 p. (in Russian).
Review
Views:
294
Email this article 