The inhomogeneous complex partial differential equations for bi-polyanalytic functions

Yanyan Cui; Chaojun Wang; Yanyan Cui; Chaojun Wang

doi:10.3934/math.2024801

AIMS Mathematics

2024, Volume 9, Issue 6: 16526-16543. doi: 10.3934/math.2024801

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The inhomogeneous complex partial differential equations for bi-polyanalytic functions

Yanyan Cui ^,,
Chaojun Wang

College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, Henan 466001, China

Received: 01 March 2024 Revised: 15 April 2024 Accepted: 18 April 2024 Published: 13 May 2024
MSC : 32A30, 30C45

In this paper, we study a Riemann-Hilbert problem related to inhomogeneous complex partial differential operators of higher order on the unit disk. Applying the Cauchy-Pompeiu formula, we find out the solvable conditions and obtain the representation of the solutions. Then, we investigate the boundary value problems for bi-polyanalytic functions with the Dirichlet and Riemann-Hilbert boundary conditions, obtain the specific solution and the solvable conditions, and extend the conclusion to the corresponding higher-order problems. Therefore, we obtain the solution to the half-Neumann problem of higher order for bi-polyanalytic functions.
- Riemann-Hilbert problems,
- complex partial differential equations,
- Cauchy-Pompeiu formula,
- bi-polyanalytic functions,
- Dirichlet problems
Citation: Yanyan Cui, Chaojun Wang. The inhomogeneous complex partial differential equations for bi-polyanalytic functions[J]. AIMS Mathematics, 2024, 9(6): 16526-16543. doi: 10.3934/math.2024801

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Abstract

In this paper, we study a Riemann-Hilbert problem related to inhomogeneous complex partial differential operators of higher order on the unit disk. Applying the Cauchy-Pompeiu formula, we find out the solvable conditions and obtain the representation of the solutions. Then, we investigate the boundary value problems for bi-polyanalytic functions with the Dirichlet and Riemann-Hilbert boundary conditions, obtain the specific solution and the solvable conditions, and extend the conclusion to the corresponding higher-order problems. Therefore, we obtain the solution to the half-Neumann problem of higher order for bi-polyanalytic functions.

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