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Commutative Algebras Associated with Classic Equations of Mathematical Physics

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Advances in Applied Analysis

Part of the book series: Trends in Mathematics ((TM))

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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.

Mathematics Subject Classification (2010). Primary 30G35; Secondary 35J05, 31A30.

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Authors and Affiliations

  1. Institute of Mathematics of the National Academy of Sciences of Ukraine, 3, Tereshchenkivska Street, Kiev, Ukraine

    S. A. Plaksa

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  1. S. A. Plaksa
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Correspondence to S. A. Plaksa .

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  1. , Department of Economics, Belarusian State University, Nezavisimosti ave 4, Minsk, 220030, Belarus

    Sergei V. Rogosin

  2. , Department of Economics, Belarus State University, Nezavisimosti ave 4, Minsk, 220030, Belarus

    Anna A. Koroleva

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Plaksa, S.A. (2012). Commutative Algebras Associated with Classic Equations of Mathematical Physics. In: Rogosin, S., Koroleva, A. (eds) Advances in Applied Analysis. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0417-2_5

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