Representation of solutions of partial differential equations by contour integrals in two-dimensional complex space

Abstract

The study aimed was to develop an efficient method for obtaining solutions to partial differential equations (PDE), in particular, equations with polynomial coefficients and Helmholtz-type equations. The methodology included the analysis of the properties of differential forms, which made it possible to represent solutions of the PDEs in the form of series, terms being contour integrals in a two-dimensional complex space. For some cases of equations, contour integrals were reduced to special functions of two variables. This approach significantly simplified the analysis and application of such solutions. The main results of the study show that the proposed scheme for constructing solutions was effective and can be applied to a wide class of PDEs, including equations with polynomial coefficients and Helmholtz-type equations. Specifically, the study determined that solutions of Helmholtz-type equations can be represented in the form of special functions, particularly Bessel functions, which ensures the accuracy and universality of the results obtained. It was also proved that the use of contour integrals to construct solutions in the form of series provided stable and convergent results, which was critical for practical applications. The obtained results also indicate the possibility of applying this methodology to the analysis of more complex systems described by the PDE. The conclusions of the study emphasise the importance of the obtained results for the development of the PDE theory. The proposed approach is based on the use of the method of contour integrals, which allows us to construct explicit solutions of boundary value problems in the half-plane, strip and circular domains. The obtained formulas, which use variable substitution and contour integrals, ensure the accuracy and efficiency of solutions, and also allow the use of methods of the theory of analytic functions for the construction of boundary conditions.