Systems of two-dimensional complex partial differential equations for bi-polyanalytic functions

Yanyan Cui; Chaojun Wang; Yanyan Cui; Chaojun Wang

doi:10.3934/math.20241265

AIMS Mathematics

2024, Volume 9, Issue 9: 25908-25933. doi: 10.3934/math.20241265

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Research article

Systems of two-dimensional complex partial differential equations for bi-polyanalytic functions

Yanyan Cui ^,,
Chaojun Wang

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

Received: 27 June 2024 Revised: 21 August 2024 Accepted: 27 August 2024 Published: 06 September 2024
MSC : 30C45, 32A30

A class of Schwarz problems with the conditions concerning the real and imaginary parts of high-order partial differentiations for polyanalytic functions was discussed first on the bicylinder. Then, with the particular solution to the Schwarz problem for polyanalytic functions, a Dirichlet problem for bi-polyanalytic functions was investigated on the bicylinder. From the perspective of series, the specific representation of the solution was obtained. In this article, a novel and effective method for solving boundary value problems, with the help of series expansion, was provided. This method can also be used to solve other types of boundary value problems or complex partial differential equation problems of other functions in high-dimensional complex spaces.
- bi-polyanalytic functions,
- polyanalytic functions,
- Schwarz problems,
- Dirichlet problems,
- complex partial differential equations
Citation: Yanyan Cui, Chaojun Wang. Systems of two-dimensional complex partial differential equations for bi-polyanalytic functions[J]. AIMS Mathematics, 2024, 9(9): 25908-25933. doi: 10.3934/math.20241265

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Abstract

A class of Schwarz problems with the conditions concerning the real and imaginary parts of high-order partial differentiations for polyanalytic functions was discussed first on the bicylinder. Then, with the particular solution to the Schwarz problem for polyanalytic functions, a Dirichlet problem for bi-polyanalytic functions was investigated on the bicylinder. From the perspective of series, the specific representation of the solution was obtained. In this article, a novel and effective method for solving boundary value problems, with the help of series expansion, was provided. This method can also be used to solve other types of boundary value problems or complex partial differential equation problems of other functions in high-dimensional complex spaces.

References

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