Monogenic Functions in Commutative Algebras Associated with Classical Equations of Mathematical Physics

Abstract

Analytic function methods in the complex plane for plane potential fields inspire searching analogous effective methods for solving spatial and multidimensional problems of mathematical physics. Many such methods are based on mappings of hypercomplex algebras.An idea of an algebraic-analytic approach to elliptic equations of mathematical physics means a finding of commutative Banach algebra such that differentiable functions with values in this algebra have components satisfying the given equation with partial derivatives. An 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.

On solutions of PDEs by using algebras

“…Several authors cite this work; they put conditions of the type that there exists an algebra A for which Ae 2 1 + Be 1 e 2 + Ce 2 2 = 0 is satisfied in order to construct solutions of PDEs of the type (1); see Pogorui et al 8 and Plaksa. 9 In this paper, we require that A𝜑(e 1 ) 2 + B𝜑(e 1 )𝜑(e 2 ) + C𝜑(e 2 ) 2 + D𝜑(e 1 ) + E𝜑(e 2 ) + Fe 1 = 0, for constructing solutions for PDEs of the type (2). This allows us to build solutions for a wider class of EDPs.…”

“…The method applied in this paper is a more explicit way of the method proposed in Ketchum 1 for solving PDEs of mathematical physics. Several authors cite this work; they put conditions of the type that there exists an algebra for which is satisfied in order to construct solutions of PDEs of the type (); see Pogorui et al 8 and Plaksa 9 . In this paper, we require that for constructing solutions for PDEs of the type ().…”