ДИСКРЕТНО-АНАЛІТИЧНИЙ МЕТОД РОЗВ’ЯЗАННЯ ЗАДАЧ МАТЕМАТИЧНОЇ ФІЗИКИ

This paper discusses a specific approach to solving problems in mathematical physics as described by linear differential equations. For this class of equations, a substantial body of solutions has been developed for particular problems within so-called canonical domains. A common feature of these domains is that their boundaries coincide with coordinate surfaces in Cartesian, cylindrical, or spherical coordinate systems. Significant experience has been accumulated in constructing partial solutions for linear equations in these coordinate systems. The focus of the article is on exploring the potential of using these known partial solutions to solve problems involving the determination of quantitative characteristics of physical fields in domains of arbitrary shape. An analysis of existing analytical solutions for canonical domains enables the formulation of a new concept: a general solution for boundary value problems defined on a class of non-canonical domains. A distinctive feature of these domains is that their boundary surfaces are made up of segments of coordinate surfaces from different coordinate systems. A key aspect of this approach is the superposition principle, which holds for linear equations: any linear combination of partial solutions is also a valid solution to the boundary value problem. The study of general solutions in non-canonical domains reveals the possibility of constructing multiple solution forms tailored to specific problems. Employing such general solutions presents opportunities to reduce the computational effort required for problems in general non-canonical domains. This is achieved through the concept of domain decomposition: one part of the domain employs an analytical general solution to represent the field, while the remainder is treated using traditional discretization methods such as the finite element method. The article also addresses the use of various forms of the general solution and presents specific calculation examples. Additionally, it discusses different algorithms for handling boundary and interface conditions on auxiliary surfaces.