Polyanalytic Nonlinear Boundary Value Problems for Upper Half Plane

Year 2024, Volume: 16 Issue: 1, 90 - 95, 30.06.2024

Pelin Ayşe Gökgöz

https://doi.org/10.47000/tjmcs.1277234

Abstract

Our aim is to improve the techniques of partial differential equations in complex variable. In this article we consider the explicit representations for polyanalytic Dirichlet boundary value problems in the upper half plane. We also point out the solutions of some simple nonlinear equations.

Keywords

Dirichlet problem, Cauchy-Pompeiu representations, nonlinear complex equations

References

• Akal, M., Begehr, H., On nonlinear Riemann-Hilbert boundary value problems for second order elliptic systems in the plane, Applicable Analysis, 63(3-4)(1996), 331–351.
• Akel, M., Begehr, H., Schwarz problem for first order elliptic systems in unbounded sectors, Eurasian Mathematical Journal, 5(4)(2014), 6–24.
• Aksoy, Ü., Begehr, H., Çelebi, A.O., Schwarz problem for higher-order complex partial differential equations in the upper half plane, Math. Nachr., 292(6)(2019), 1183–1193.
• Aksoy, Ü ., Çelebi, A.O., Dirichlet problem for a generalized inhomogeneous polyharmonic equation in an annular domain, Complex Var. Elliptic Equ., 57(2-4)(2012), 229–241.
• Aksoy, Ü, Çelebi, A.O., Polyharmonic Robin problem for complex linear partial differential equations, Complex Var. Elliptic Equ., 59(12)(2014), 1679–1695.
• Aksoy, Ü, Çelebi, A.O., Norm estimates of a class of Caldero´n-Zygmund type strongly singular integral operators, Integral Transforms Spec. Funct., 18(1-2)(2007), 87–93.
• Begehr, H., Kumar, A., Boundary value problems for the inhomogeneous polyanalytic equation I, Analysis (Munich), 25(1)(2005), 55–71.
• Begehr, H., Kumar, A., Boundary value problems for the inhomogeneous polyanalytic equation II, Analysis (Munich), 27(4)(2007), 359–373.
• Begehr, H., Kumar, A., Boundary value problems for bi-polyanalytic functions, Appl. Anal., 85(9)(2006), 1045–1077.
• Begehr, H., Du, J., Wang, Y., A Dirichlet problem for polyharmonic functions, Ann. Mat. Pura Appl., 187(2008), 435–457.
• Begehr, H., Hile, G.N., A hierarchy of integral operators, Rocky Mountain J. Math., 27(3)(1997), 669–706.
• Begehr, H., Gaertner, E., A Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane, Georgian Math. J., 14(1)(2007), 33–52.
• Begehr, H., Complex Analytic Methods for Partial Differential Equations: An introductory text, World Scientific, Singapore, 1994.
• Chaudhary, A., Kumar, A., Boundary value problems in upper half plane, Complex Var. Elliptic Equ., 54(5)(2009), 441–448.
• Çelebi, A.O., Gökgöz, P.A., Schwarz problem in a ring domain, Appl. Anal., 101(11)(2022), 3912–3924.
• Çelebi, A.O., Gökgöz, P.A., A note on the Schwarz problem in a ring domain. In: Current Trends in Analysis and its Applications and Computation: Trends Math., Switzerland, (2022), 31–39.
• Gaertner, E., Basic complex boundary value problems in the upper half plane, Ph.D. Thesis, FU Berlin. Available at http://www.diss.fuberlin.de/2006/320/.
• Kumar, A., Prakash, R., Mixed boundary value problems for the inhomogeneous polyanalytic equation, Complex Var. Elliptic Equ., 51(3)(2006), 209–223.
• Kumar, A., Prakash, R., Iterated boundary value problems for the inhomogeneous polyanalytic equation, Complex Var. Elliptic Equ., 52(10-11)(2007), 921–932.
• Vekua, I.N., Generalized Analytic Functions, Pergamon Press, Oxford, 1962.