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 CMFD, 2019, Volume 65, Issue 2, Pages 157–338 (Mi cmfd382)

This article is cited in 1 scientific paper (total in 1 paper)

General Euler–Poisson–Darboux equation and hyperbolic $B$-potentials

E. L. Shishkina

Voronezh State University, Voronezh, Russia

Abstract: In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz $B$-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler–Poisson–Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

DOI: https://doi.org/10.22363/2413-3639-2019-65-2-157-338

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UDC: 517.956.45, 517.968.74

Citation: E. L. Shishkina, “General Euler–Poisson–Darboux equation and hyperbolic $B$-potentials”, Partial differential equations, CMFD, 65, no. 2, PFUR, M., 2019, 157–338

Citation in format AMSBIB
\Bibitem{Shi19} \by E.~L.~Shishkina \paper General Euler--Poisson--Darboux equation and hyperbolic $B$-potentials \inbook Partial differential equations \serial CMFD \yr 2019 \vol 65 \issue 2 \pages 157--338 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd382} \crossref{https://doi.org/10.22363/2413-3639-2019-65-2-157-338} 

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This publication is cited in the following articles:
1. N. V. Zaitseva, “Uniqueness of the Solution of a Nonlocal Problem for an Elliptic-Hyperbolic Equation with Singular Coefficients”, Math. Notes, 109:4 (2021), 563–569
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