Monogenic functions in commutative algebras associated with classical equations of mathematical physics

Abstract

The methods involving the functions analytic in a complex plane for plane potential fields inspire the search for the analogous efficient methods for solving the spatial and multidimensional problems of mathematical physics. Many such methods are based on the mappings of hypercomplex algebras. The essence of the algebraic-analytic approach to elliptic equations of mathematical physics consists in the finding of a commutative Banach algebra such that the differentiable functions with values in this algebra have components satisfying the given equation with partial derivatives. The use of differentiable functions given in commutative Banach algebras combines the preservation of basic properties of analytic functions of a complex variable for the mentioned differentiable functions and the convenience and the simplicity of construction of solutions of PDEs. The paper contains the review of results reflecting the formation and the development of the mentioned approach.