COMPLEX BOUNDARY VALUE PROBLEMS FOR THE CAUCHY–RIEMANN OPERATOR ON A TRIANGLE

Mohamed Akel, Muajebah Hidan and Mohamed Abdalla
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Mohamed Akel: Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
Muajebah Hidan: ��Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia
Mohamed Abdalla: Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt†Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia

FRACTALS (fractals), 2022, vol. 30, issue 10, 1-15

Abstract: There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

Keywords: Cauchy–Pompeiu Integral Formula; Cauchy–Riemann Equations; Dirichlet Problem; Neumann Problem; Robin Problem (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22402526

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