Method of Riesz potentials applied to solution to nonhomogeneous singular wave equations

  • Shishkina Elina L., ilina_dico@mail.com Voronezh State University, Faculty of Applied Mathematics, Informatics and Mechanics, Universitetskaya square, 1, Voronezh, 394018, Russia
  • Abbas Syed, sabbas.iitk@gmail.com School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, H.P., 175005, India
Keywords: Riesz potential, Bessel operator, Euler-Poisson-Darboux equation, singular Cauchy problem

Abstract

Riesz potentials are convolution operators with fractional powers of some distance (Euclidean, Lorentz or other) to a point. From application point of view, such potentials are tools for solving differential equations of mathematical physics and inverse problems. For example, Marsel Riesz used these operators for writing the solution to the Cauchy problem for the wave equation and theory of the Radon transform is based on
Riesz potentials. In this article, we use the Riesz potentials constructed with the help of generalized convolution for solution to the wave equations with Bessel operators. First, we describe general method of Riesz potentials, give basic definitions, introduce solvable equations and write suitable potentials (Riesz hyperbolic $B$-potentials). Then, we show that these potentials are absolutely convergent integrals for some functions and for some values of the parameter representing fractional powers of the Lorentz distance. Next we show the connection of the Riesz hyperbolic B-potentials with d’Alembert operators in which the Bessel operators are used in place of the second derivatives. Next we continue analytically considered potentials to the required parameter values that includes zero and show that when value of the parameter is zero these operators are identity operators. Finally, we solve singular initial value hyperbolic problems and give examples.

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How to Cite
Shishkina, E. and Abbas, S. ( ) “Method of Riesz potentials applied to solution to nonhomogeneous singular wave equations”, Mathematical notes of NEFU, 25(3), pp. 68-91. doi: https://doi.org/10.25587/SVFU.2018.99.16952.
Section
Mathematics