Uniqueness of solutions of boundary value problems for Cauchy- Riemann equation with local and non- local boundary conditions

Authors
Javad Ebadpur Golanbar
Mohammad Jahanshahi
Nihan Aliyev
Abstract
In This paper, we consider two boundary value problems for non homogenuous Cauchy- Riemann equation with local and non-local boundary conditions separatly.In the first problem boundary condition is local and for proving uniqueness of solution, we show that homogenuous boundary value problem has only non-trivial solution. In the second problem, the boundary condition is non-local. At first, the given problem changed to a second kind system of Fredholm integral equation. Then the singularities in the Kernels of integral equations are removed. Finaly, for the uniqueness of solution, we show that the regularized kernels satify in contraction mapping theorem.



./files/site1/files/%D8%B9%D8%A8%D8%A7%D8%AF%D9%BE%D9%88%D8%B1.pdf
Keywords
Cauchy- Riemann equation
uniqueness of solution
fundamental solution
Regularization
system of Fredholm integral equations
[1] V.S.Vladimirov, “ Equations of Mathematical Physics” Mir Publishers Moscow 1984.

[2] L. Evance, partial differential equations, American Mathematical society, second edition, 2010.

[3] J‎. ‎Kondo‎, Integral equations‎, Oxford University Press ‎(1991)‎.

[4] H. Ted Davis and Kendall T. Thomson, Linear algebra and linear operators in engineering: with applications in Mathematica , Academic press, 2000.

[5] ‎M‎. ‎Jahanshahi and M‎. ‎Sajjadmanesh‎, Analytical solutions for the Stepjen's inverse problem with local boundary conditions including a first order hyperbolic equation‎, Bulletin of the Iranian Mathematical Society Vol 39, No5, 2013.

[6] ‎M‎. ‎Sajjadmanesh‎, ‎M‎. ‎Jahanshahi and N‎. ‎Aleiv‎, Inverse problem of the kind of Tikhonov-Lavrentev including the Cauchy-Riemann equation‎,Azerbaijan Journal of Mathematics(2013)‎, ‎no‎. ‎1‎.

[7] M. Jahanshahi, M. Fatehi, Analytic solution for the Cauchy – Riemann equation with nonlocal boundary conditions in the first quarter, International Journal of Pure and Applied Mathematics Volume 46, No.2, page 245-249, 2008.

[8] M. Jahanshahi, N. Aliev, Determining of an Analytic Function on Analytic Domain by Cauchy – Riemann Equation with Special Kind of Boundary Conditions, Southeast Asian Bulletin of Mathematics; Vol. 28 Issue 1,p33, 2004

[9] M. Jahanshahi. J. Ebadpour Golanbar , N.Aliev,Spectral problem for an initial- boundary value problems involving first order two dimensional generalized nongomogenous Cauchy – Rieman equation with general non-local boundary conditions, Proceedings of the 6th international conference on control and optimization with industrial applications, Vol 1, page 212-215,11-13 July, 2018, Baku, Azerbaijan.

[10] A. B. Rasulov, A. P. Soldatov, Boundary value problem for a generalized Cauchy–Riemann equation with singular coefficients, Differential equations, May 2016, Volume 52, Issue 5, pp 616–629

11. محمد جهانشاهی، مجتبی سجادمنش، روش جدید برای بررسی و تشخیص خود الحاق بودن مسائل مقدار مرزی شامل معادلات دیفرانسیل عادی، نشریه علوم دانشگاه خوارزمی، سال دوازدهم، شماره 1، بهار 1391.

12. نهان علی اف، محمد حسین فاتحی، محمد جهانشاهی، ارایه جواب تحلیلی برای معادله کوشی – ریمان با شرایط مرزی غیر موضعی در ناحیه نصف ربع اول، نشریه علوم دانشگاه خوارزمی، سال دهم، شماره 1، بهار 1388.
Mathematical Research
Volume 7, Issue 4 - Serial Number 19
Spring 2021
Pages 843-857