Several boundary value problems for bi-analytic complex partial differential systems of second order

Yanyan Cui; Chaojun Wang; Yanyan Cui; Chaojun Wang

doi:10.3934/math.2025767

AIMS Mathematics

2025, Volume 10, Issue 7: 17117-17143. doi: 10.3934/math.2025767

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Research article

Several boundary value problems for bi-analytic complex partial differential systems of second order

Yanyan Cui ^,,
Chaojun Wang

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

Received: 17 April 2025 Revised: 13 July 2025 Accepted: 22 July 2025 Published: 31 July 2025
MSC : 30C45, 32A30

In this paper, several boundary value problems for second order complex partial differential systems of bi-analytic functions were investigated on the bicylinder. Homogeneous and nonhomogeneous Dirichlet problems for $ (\lambda, 1) $ bi-analytic functions were first discussed on the bicylinder. Applying the Cauchy-Pompeiu formula and the properties of the Poisson kernel, the expressions of the solutions to the Dirichlet problems were obtained. Thereafter, the Riemann problems and the inverse problems for $ (\lambda, 1) $ bi-analytic functions were explored on the generalized bicylinder in $ \mathbb{C}^2 $. Applying the Plemelj formula, the solutions to the corresponding problems were obtained.
- complex partial differential equations,
- Dirichlet problems,
- Riemann problems,
- bi-analytic functions,
- Cauchy-Pompeiu formula
Citation: Yanyan Cui, Chaojun Wang. Several boundary value problems for bi-analytic complex partial differential systems of second order[J]. AIMS Mathematics, 2025, 10(7): 17117-17143. doi: 10.3934/math.2025767

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Abstract

In this paper, several boundary value problems for second order complex partial differential systems of bi-analytic functions were investigated on the bicylinder. Homogeneous and nonhomogeneous Dirichlet problems for $ (\lambda, 1) $ bi-analytic functions were first discussed on the bicylinder. Applying the Cauchy-Pompeiu formula and the properties of the Poisson kernel, the expressions of the solutions to the Dirichlet problems were obtained. Thereafter, the Riemann problems and the inverse problems for $ (\lambda, 1) $ bi-analytic functions were explored on the generalized bicylinder in $ \mathbb{C}^2 $. Applying the Plemelj formula, the solutions to the corresponding problems were obtained.

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