Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

On Fredholm Boundary Value Problems for the Cauchy-Riemann Equation When the Carleman Conditions Do Not Hold

Author(s):

Ramin Mubariz Zeynalov

Associate Professor of the Department of Engineering Mathematics and Artificial Intelligence, Azerbaijan Technical University,
PhD in Mathematics, Leading Researcher of the Institute of Control Systems of the Ministry of Science and Education
of the Azerbaijan Republic

raminz.math@gmail.com

Abstract:

This paper is devoted to the study of the solution of boundary value problems for the Cauchy-Riemann equation with nonlocal boundary conditions. In the considered case, the Carleman conditions do not take place, i.e. at least two points that follow each other move simultaneously on the boundary. In this case, using the fundamental solution of the equation under consideration, the basic relation is determined, which consists of two parts. The first part gives an arbitrary solution of the Cauchy-Riemann equation defined in the domain D⊂R^2 and the second part gives the necessary conditions for the solvability of the boundary problems. Unlike the ordinary differential equation, here the necessary conditions contain global terms, i.e., integrals over the boundaries. These conditions include singular integrals, which in general case are regularized by means of the boundary condition, and thus the obtained regular expressions together with the boundary conditions determine the Fredholm character of the problem.

Keywords

References:

  1. Aliyev N, Fatemi M, Jahanshahi M. Analytic solution for the Cauchy-Riemann equation with non-local boundary conditions in the first semi-quarter. Quarterly journal of science of Tarbiat Moallem University. 2010; 9(1):29-40
  2. Zeynalov R. M., Aliev N. A. Determining a spectrum of a boundary value problem. Trans­actions of National Academy of Sciences of Azerbaijan. Series of physical-technical and mathematical sciences. 2019; 39(1):170-177
  3. Zeynalov R. M., Aliev N. A. The Zaremba-Steklov problem for the Cauchy-Riemann equation. Bulletin of Dagestan State University, 2015; 30(6):74-79. (In Russ. )
  4. Zeynalov R. M. Investigation of a boundary value problem for an integro-differential equation with a Cauchy-Riemann principal part. Bulletin of the Buryat State University of Mathematics, Computer Science. 2023: (1):3-10. DOI: 10. 18101/2304-5728-2023-1-3-10 (In Russ. )
  5. Aliyev F. A., Aliyev N. A., Hasanov K. G., Guliyev A. P., Turarov A. K., Isayeva G. V., Numerical-analytical method for solving of the first order partial quasi-linear equations. TWMS Journal of Pure and Applied Mathematics. 2015; 6(2):158-164
  6. Aliev N. A., Mustafayeva Y. Y., Murtuzayeva S. M., The influence of the Carleman con­dition on the fredholm property of the boundary value problem for Cauchy-Riemann equation. Proceedings of the Institute of Applied Mathematics. 2012; 1(2):153-162
  7. Vladimirov V. S. Uravneniya matematicheskoj fiziki [Equations of mathematical physics]. Moscow, Nauka, 1981. 512 p. (In Russ. )
  8. Petrovsky I. G. Lectures on partial differential equations [Lectures on partial differential equations]. Moscow: SPHPML, 1961. 400 p. (In Russ. )

Full text (pdf)