In this paper, with the properties of $\beta$-Cauchy type integrals and the Plemelj formula for the corresponding singular integrals, the Hilbert boundary value problems for $\beta$-analytic functions were first discussed on the unit disc in $\mathbb{C}$, and the concrete representations of the solutions were obtained. Thereafter, on the basis of the results for the Riemann boundary value problems for analytic functions, several Riemann problems for higher-order complex partial differential systems and $(\lambda ,1)$ bi-analytic functions were investigated for the bicylinder in ${\mathbb{C}}^2$. The solutions to the problems and the corresponding solvable conditions were obtained.

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.

Applying the Cauchy-Pompeiu formula and the properties of the singular integral operators on the unit disc, the specific representation of the solutions to the boundary value problems with the Dirichlet boundary conditions for bi-polyanalytic functions are obtained on the bicylinder. Also, the mixed-type boundary value problems of higher order for bi-polyanalytic functions were investigated. In addition, a system of complex partial differential equations with respect to polyanalytic functions with Neumann boundary conditions was discussed. On this foundation, the solutions to Neumann boundary value problems for bi-polyanalytic functions on the bicylinder were obtained. These results provide a favorable method for discussing other boundary value problems of bi-polyanalytic functions and the related systems of inhomogeneous complex partial differential equations of higher order in spaces of several complex variables.

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.

In this paper, we mainly discussed an inhomogeneous Riemann-Hilbert boundary value problem for complex partial differential operators of higher order on the bicylinder $D^2$ in ${\mathbb{C}}^2$. Applying the classical Cauchy-Pompeiu formula and the Gauss theorem, we found out the solvable condition and obtained the specific solution of the inhomogeneous boundary value problem on the bicylinder $D^2$. The conclusion lays a solid foundation for further research on other types of boundary value problems concerning complex partial differential equations, such as mixed boundary value problems.

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.