Using examples of boundary value problems for the Helmholtz and Laplace equations, the method is discussed, which allows significantly expanding the class of mathematical physics problems that allow a complete analytical description of fields in non-canonical domains. The basic concept that defines the procedure for constructing analytical solutions is the concept of a general solution of a boundary value problem. Examples are considered that reveal the essence of the method and illustrate the construction of such solutions using partial solutions in different coordinate systems. It is shown that the theorem on the uniqueness of the solution of a boundary value problem does not exclude the possibility of different analytical forms of its representation. Specific problems that arise due to the ambiguity of the form of general solutions and the possibility of different formulations of functional equations that determine the boundary conditions and conjugation conditions of conjugation on the boundaries of partial domains are indicated. The procedures for finding the values of coefficients in general solutions are discussed.
Grinchenko et al. (Wed,) studied this question.