Abstract
This paper investigates the Schwarz problem. Initially, the focus lies on analyzing the problem for the first, second orders. Subsequently, attention shifts towards studying the same problem for equations of higher order. In the realm of second-order equations, the Schwarz problem is specifically examined for some operators; Laplace, Bitsadze and its complex conjugate. The findings demonstrate that the Schwarz problem for an n-order equation, when equipped with solely one boundary condition, exhibits an infinite number of solutions. However, by incorporating additional boundary conditions, it becomes feasible to obtain a unique solution for problem concerning n-order equations, effectively rendering it a well-posed problem.
Keywords
Schwarz Problem Complex model homogeneous partial differential equation Complex model inhomogeneous partial differential equations
References
- [1] H. Begehr, “Boundary Value Problems in Complex Analysis Ⅰ, Ⅱ,” Boletin de la Asosiacion, vol. Ⅻ, no. 2, pp. 65-85, 217-250, 2005.
- [2] B. Karaca, “Dirichlet Problem for Complex Model Partial Differential Equations,” Complex Variables and Elliptic Equations, vol. 65, no. 10, pp. 1748-1762, 2020.
- [3] H. Begehr, S. Burgumbayeva, B. Shupeyeva, “Harmonic Green Functions for a Plane Domain With Two Touching Circles As Boundary,” Advanced Mathematical Models & Applications, vol. 3, no. 1, pp. 18-29, 2018.
- [4] Ü. Aksoy, AO. Çelebi, “Schwarz Problem for Higher Order Linear Equations in a Polydisc,” Complex Variables and Elliptic Equations, vol. 62, no. 10, pp. 1558-1569, 2017.
- [5] M. Akel, M. Hidan, M. Abdalla, “Complex Boundary Value Problems for the Cauchy–Riemann Operator on a Triangle,” Fractals, vol. 30, no. 10, pp. 1-15, 2022.
- [6] M. Akel, H. Begehr, A. Mohammed, “A Neumann Problem for the Polyanalytic Operator in Planar Domains with Harmonic Green Function,” Applicable Analysis, vol. 101, no. 11, pp. 3816-3824, 2022.
- [7] H. Begehr, S. Burgumbayeva, A. Dauletkulova, H. Lin, “Harmonic Green Functions for the Almaty Apple,” Complex Variables and Elliptic Equations, vol. 65, no. 11, pp. 1814-1825, 2020.
- [8] Ü. Aksoy, H. Begehr, AO. Çelebi, “Schwarz Problem for Higher‐Order Complex Partial Differential Equations in the Upper Half Plane,” Mathematische Nachrichten, vol. 292, no. 6, pp. 1183-1193, 2019.
Abstract
References
- [1] H. Begehr, “Boundary Value Problems in Complex Analysis Ⅰ, Ⅱ,” Boletin de la Asosiacion, vol. Ⅻ, no. 2, pp. 65-85, 217-250, 2005.
- [2] B. Karaca, “Dirichlet Problem for Complex Model Partial Differential Equations,” Complex Variables and Elliptic Equations, vol. 65, no. 10, pp. 1748-1762, 2020.
- [3] H. Begehr, S. Burgumbayeva, B. Shupeyeva, “Harmonic Green Functions for a Plane Domain With Two Touching Circles As Boundary,” Advanced Mathematical Models & Applications, vol. 3, no. 1, pp. 18-29, 2018.
- [4] Ü. Aksoy, AO. Çelebi, “Schwarz Problem for Higher Order Linear Equations in a Polydisc,” Complex Variables and Elliptic Equations, vol. 62, no. 10, pp. 1558-1569, 2017.
- [5] M. Akel, M. Hidan, M. Abdalla, “Complex Boundary Value Problems for the Cauchy–Riemann Operator on a Triangle,” Fractals, vol. 30, no. 10, pp. 1-15, 2022.
- [6] M. Akel, H. Begehr, A. Mohammed, “A Neumann Problem for the Polyanalytic Operator in Planar Domains with Harmonic Green Function,” Applicable Analysis, vol. 101, no. 11, pp. 3816-3824, 2022.
- [7] H. Begehr, S. Burgumbayeva, A. Dauletkulova, H. Lin, “Harmonic Green Functions for the Almaty Apple,” Complex Variables and Elliptic Equations, vol. 65, no. 11, pp. 1814-1825, 2020.
- [8] Ü. Aksoy, H. Begehr, AO. Çelebi, “Schwarz Problem for Higher‐Order Complex Partial Differential Equations in the Upper Half Plane,” Mathematische Nachrichten, vol. 292, no. 6, pp. 1183-1193, 2019.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Methods and Special Functions |
| Journal Section | Research Articles |
| Authors | |
| Early Pub Date | April 26, 2024 |
| Publication Date | April 30, 2024 |
| Submission Date | November 14, 2023 |
| Acceptance Date | January 19, 2024 |
| Published in Issue | Year 2024 Volume: 28 Issue: 2 |
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