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DOI: 10.46698/f7969-2225-7035-j
Boundary Value Problems for Inhomogeneous Polyanalytic Equations in a Triangle
Karaca, B.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 4.
Abstract: In this paper, we investigate Dirichlet and Schwarz type boundary value problems for both the inhomogeneous Cauchy-Riemann equation and higher-order polyanalytic equations in a nonstandard domain, specifically a triangular region formed by the intersection of three circular disks in the complex plane. Such domains introduce additional geometric complexity, which requires careful analytical treatment. By constructing appropriate kernel functions tailored to the geometry of the domain, we develop integral operator techniques that allow us to derive explicit solution formulas for the given boundary conditions. In addition, we establish necessary and sufficient conditions for the solvability of these problems, depending on the compatibility of boundary data and the properties of the inhomogeneous terms. Our approach generalizes classical methods used for standard domains, extending their applicability to more intricate geometric settings. The results presented in this work contribute to the broader theory of boundary value problems for complex partial differential equations and offer new tools for addressing similar problems in applied mathematical physics and complex analysis
For citation: Karaca, B. Boundary Value Problems for Inhomogeneous Polyanalytic Equations in a Triangle, Vladikavkaz Math. J., 2025, vol. 27, no. 4, pp. 46-60. DOI 10.46698/f7969-2225-7035-j
1. Begehr, H. Complex Analytic Methods for Partial Differential Equations. An introductory
Text, Singapore, World Scientific, 1994.
2. Begehr, H. and Hile, G. N. A Hierarchy of Integral Operators, Rocky Mountain Journal of Mathematics, 1997, vol. 27, no. 3, pp. 669-706. DOI: 10.1216/rmjm/1181071888.
3. Celebi, A. O. and Aksoy U. Schwarz Problem for Higher-Order Complex Elliptic Partial Differential Equations, Integral Transforms and Special Functions, 2008, vol. 19, no. 6, pp. 413-428. DOI: 10.1080/10652460801933645.
4. Chaudhary, A. Neumann and Mixed Boundary Value Problems on the Upper Half Plane, Advances in the Theory of Nonlinear Analysis and its Application, 2022, vol. 6, no. 1, pp. 135-142. DOI: 10.31197/atnaa.950920.
5. Karaca, B. Schwarz Problem for Model Partial Differential Equations with One Complex
Variable, Sakarya University Journal of Science, 2024, vol. 28, no. 2, pp. 410-417. DOI: 10.16984/saufenbilder.1390617.
6. Yongzhi, X. Riemann Problem and Inverse Riemann Problem of (\( \lambda \),1) bi-Analytic Functions, Complex Variables and Elliptic Equations, 2007, vol. 52, no. 10-11, pp. 853-864. DOI: 10.1080/17476930701483809.
7. Wang, Y. and Wang, Y. Schwarz-Type Problem Of Nonhomogeneous Cauchy-Riemann Equation on a Triangle, Journal of Mathematical Analysis and Applications, 2011, vol. 377, no. 2, pp. 557-570. DOI: 10.1016/j.jmaa.2010.11.023.
8. Hao, Y. and Hua, L. Schwarz Boundary Value Problem on Reuleaux Triangle, Complex Variables and Elliptic Equations, 2021, vol. 67, no. 10, pp. 2444-2457. DOI: 10.1080/17476933.2021.1931150.
9. Wang, Y. and Wang, Y. Two Boundary-Value Problems for the Cauchy-Riemann Equation in a Sector, Complex Analysis and Operator Theory, 2012, vol. 6, pp. 1121-1138. DOI: 10.1007/s11785-010-0107-0.
10. Wang, Y. and Zhao, X. Schwarz Boundary Value Problem for the Cauchy-Riemann Equation in a Rectangle, Boundary Value Problems, 2016, vol. 2016, Article no. 7. DOI: 10.1186/s13661-016-0520-z.
11. Gencturk, I. Robin Boundary Value Problem Depending on Parameters in a Ring Domain, Fundamental Journal of Mathematics and Applications, 2020, vol. 3, no. 2, pp. 161-167. DOI: 10.33401/fujma.795538.
12. Akel, M., Hidan, M. and Abdalla, M. Complex Boundary Value Problems for the Cauchy-Riemann Operator on a Triangle, Fractals, 2022, vol. 30, no. 10, Article no. 2240252. DOI: 10.1142/S0218348X22402526.
13. Akel, M. S. and Alabbad, F. A Riemann-Hilbert Boundary Value Problem in a Bounded Sector, Complex Variables and Elliptic Equations, 2014, vol. 60, no. 4, pp. 493-509. DOI: 10.1080/17476933.2014.944866.
14. Darya, A. and Taghizadeh, N. Schwarz and Dirichlet Problems for \(\bar{\partial}\)-Equation in a Triangular Domain, Russian Mathematics, 2024, vol. 68, pp. 9-17. DOI: 10.3103/S1066369X24700853.
15. Soldatov, A. P. On Boundary Properties of Conformal Mappings, Computational Mathematics and Mathematical Physics, 2024, vol. 64, no. 9, pp. 2011-2025. DOI: 10.1134/S0965542524701070.
16. Koshanov, B. D. and Soldatov, A. P. On the Solvability of the Generalized Neumann Problem for a Higher-Order Elliptic Equation in an Infinite Domain, Contemporary Mathematics Fundamental Directions, 2021, vol. 67, no. 3, pp. 564-575. DOI: 10.22363/2413-3639-2021-67-3-564-575.
17. Soldatov, A. P. On the First and Second Boundary Value Problems for Elliptic Systems on the Plane, Differential Equations, 2003, vol. 39, no. 5, pp. 712-725. DOI: 10.1023/A:1026102322259.
18. Kozhanov, A. I. Boundary Value Problems for Fourth-Order Sobolev Type Equations, Journal of Siberian Federal University Mathematics & Physics, 2021, vol. 14, no. 4, pp. 425-432. DOI: 10.17516/1997-1397-2021-14-4-425-432.
19. Kozhanov, A. I. and Kenzhebay, Kh. Boundary Value Problems with an Integro-Differential Non-Local Condition for Composite Type Differential Equations of the Fourth Order, Chelyabinskiy Fiziko-Matematicheskiy Zhurnal, 2023, vol. 8, no. 4, pp. 516-527. DOI: 10.47475/2500-0101-2023-8-4-516-527.
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