Abstract
In this overview we have pointed out some boundary value problems in a subset of complex plane. We start with Cauchy-Riemann operator and their conjugates. Then we introduce the Cauchy- Riemann operator and its conjugate. Firstly, we introduce the polyanalytic Pompeiu integral representation 𝑇 𝑓 and its conjugate which vanish at infinity. The basic polyanalytic Schwarz and polyanalytic Dirichlet problems are introduced. The later part is devoted to polyanalytic problems and discussions on polyharmonic problems. We have also summarized polyanalytic Neumann problem in the unit disk for 𝜕𝑧𝑤 = 𝑓. In this case, we may have three types of boundaryvalueproblems.ThosearepolyharmonicDirichletproblem,polyharmonicNeumannproblemandpolyharmonic Riquier (Navier) problem. In this later part we have given the iterated polyharmonic Green function.
References
- Aksoy, Ü., Begehr, H., Çelebi, A., Shupeyeva, B., 2025, Complex partial differential equations, Journal of Mathematical Sciences, Vol. 287, No. 6. google scholar
- Begehr, H., 2025, Polyanalytic Neumann-n problem in the unit disk, Complex Variables and Elliptic Equations, 70:2, 278-286. google scholar
- Vekua, I.N., 1962, Generalized Analytic Functions, Pergamon Press, Oxford. google scholar
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | June 24, 2025 |
| Submission Date | May 16, 2025 |
| Acceptance Date | June 19, 2025 |
| Published in Issue | Year 2025 Volume: 3 Issue: 1 |
