Abstract
Integro-differential equations with kernels including hypergeometric Gaussian function that depends on the arguments ratio are studied over a closed curve in the complex plane. Special cases of the equations considered are the special integro-differential equation with Cauchy kernel, equations with power and logarithmic kernels. By means of the curvilinear convolution operator with the kernel of special kind, the equations with derivatives are reduced to the equations without derivatives. We find out the connection between special cases of the above-mentioned convolution operator and the known integral representations of piecewise analytical functions applied in the study of boundary value problems of the Riemann type. The correct statement of Noetherian property for the investigated class of equations is given. In this case, the operators corresponding to the equations are considered acting from the space of summable functions into the space of fractional integrals of the curvilinear convolution type. Examples of integro-differential equations solvable in a closed form are given.
Similar content being viewed by others
References
Gakhov, F.D. Boundary value problems (Nauka, Moscow, 1977) fin Russian].
Muskhelishvili, N.I. Singular integral equations (Nauka, Moscow, 1968) fin Russian].
Vekua, N.R Systems of singular integral equations (Nauka, Moscow, 1970) fin Russian].
Peschanskii, A.I. “Integral Equations of Curvilinear Convolution Type with Hypergeometric Function in a Kernel”, Russian Math. 63 (9), 43–54 (2019).
Krikunov, Yu.M. “On solution of the inverse boundary value Riemann problem and singular integro- differential equation”, Uchen. zap. Kaz. univ. 112 (10), 191–199 (1952).
Isakhanov, R.S. “Differential boundary value problem of linear conjugation and its application to the theory of integro-differential equations”, Soobsch. AN GruzSSR 20 (6), 659–666 (1958).
Vekua, N.P. “On one system of singular integro-differential equations and its applications to boundary value problems of linear conjugation”, Tr. Tbilissk. matem. inst. AN GruzSSR 24, 135–147 (1957).
Zhegalov, V.I. “On problems with derivatives in boundary conditions”, Tr. semin. po kraevim zadacham 10, 38–52 (1973).
Rogozhin, V.S. “New integral representation for piecewise holomorphic functions and its applications”, DAN SSSR 135 (4), 791–793 (1960).
Saks, R.S. Boundary value problems for elliptic systems of differential equations (Novosibirsk, Izd. NGU, 1975) fin Russian].
Tovmasyan, N.E. “On a theory of singular integral equations”, Differential Equations 3 (1), 69–81 (1967).
Peschanskii, A.I., Cherskii, Yu.I. “An integral equation with curvilinear convolutions on a closed contour”, Ukrainian Mathematical Journal 36 (3), 301–305 (1984).
Peschanskii, A.I. “Description of a space of fractional integrals of curvilinear convolution type”, Soviet Math. (Iz. VUZ) 33 (7), 37–50 (1989).
Peschanskii, A.I. “On description of a space Lpη (Г) of fractional integrals of curvilinear convolution type” (in: Proc. XXY Int. sci.-tech. conf. “Applied problems in mathematics”, Sevastopol, 18–22 September 2017 (Izd. SevGU, Sevastopol, 2017)) fin Russian].
Gokhberg, I.Ts., Krupnik, N.Ya. Introduction to the theory of one-dimensional singular integral operators (Shtiintsa, Kishinev, 1973) fin Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 1, pp. 84–93.
About this article
Cite this article
Peschansky, A.I. Integro-Differential Equations over a Closed Circuit with Gaussian Function in the Kernel. Russ Math. 64, 78–87 (2020). https://doi.org/10.3103/S1066369X20010077
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X20010077