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Commutative Algebras Associated with Classic Equations of Mathematical Physics

  • S. A. PlaksaEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

The idea of an algebraic-analytic approach to equations of mathematical physics means to find a commutative Banach algebra such that monogenic functions with values in this algebra have components satisfying to given equations with partial derivatives.

We obtain here a constructive description of monogenic functions taking values in a commutative algebra associated with a two-dimensional biharmonic equation by means of analytic functions of complex variables. For the mentioned monogenic functions we establish basic properties analogous to properties of analytic functions of complex variables: the Cauchy integral theorem and integral formula, the Morera theorem, the uniqueness theorem, and the Taylor and Laurent expansions. Similar results are obtained for monogenic functions which take values in a three-dimensional commutative algebra and satisfy the three-dimensional Laplace equation.

In infinite-dimensional commutative Banach algebras we construct explicitly monogenic functions which have components satisfying the threedimensional Laplace equation. We establish that all spherical functions are components of the mentioned monogenic functions. A relation between these monogenic functions and harmonic vectors is described.

We establish that solutions of elliptic equations degenerating on an axis are constructed by means of components of analytic functions taking values in an infinite-dimensional commutative Banach algebra. In such a way we obtain integral expressions for axial-symmetric potentials and Stokes flow functions in an arbitrary simply connected domain symmetric with respect to an axis.

Keywords

Laplace equation biharmonic equation harmonic vector axialsymmetric potential Stokes flow function harmonic commutative Banach algebra biharmonic algebra monogenic function Cauchy–Riemann conditions Cauchy integral theorem Cauchy integral formula Morera theorem Taylor expansion Laurent expansion. 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institute of Mathematics of the National Academy of Sciences of UkraineKievUkraine

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