We establish conditions under which the Bessel operator of “infinite order” is defined and continuous in certain spaces of type S: The correct solvability of the nonlocal multipoint (in time) problem is proved for evolutionary equations of parabolic type with the indicated operators and the initial condition in the form of a generalized function of the ultradistribution type. The properties of the fundamental solution of the posed problem and the properties of the Bessel transform of generalized functions from the spaces of type S; convolutions, convolvers, and multipliers are investigated.
Similar content being viewed by others
References
I. M. Gelfand and G. E. Shilov, Spaces of Test and Generalized Functions [in Russian], Fizmatgiz, Moscow (1958).
V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).
M. L. Gorbachuk and P. I. Dudnikov, “On the initial data of the Cauchy problem for parabolic equations for which solutions are infinitely differentiable,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 4, 9–11 (1981).
V. V. Horodets’kyi, Limit Properties of the Solutions of Parabolic-Type Equations Smooth in a Layer [in Ukrainian], Ruta, Chernivtsi (1998).
V. V. Horodets’kyi, Sets of Initial Values of Smooth Solutions of Differential-Operator Equations of the Parabolic Type [in Ukrainian], Ruta, Chernivtsi (1998).
Ya. I. Zhitomirskii, “Cauchy problem for systems of linear partial derivative equations with differential Bessel operator,” Mat. Sb., 36, No. 2, 299–310 (1955).
V. V. Horodets’kyi and O. V. Martynyuk, Parabolic Pseudodifferential Equations with Analytic Symbols in Spaces of Type S [in Ukrainian], Tekhnodruk, Chernivtsi (2019).
A. M. Nakhushev, Equations of Mathematical Biology [in Russian], Vysshaya Shkola, Moscow (1995).
I. A. Belavin, S. P. Kapitsa, and S. P. Kurdyumov, “Mathematical model of global demographic processes with regard for space distribution,” Zh. Vychisl. Mat. Mat. Fiz., 38, No. 6, 885–902 (1998).
A. A. Dezin, General Problems of the Theory of Boundary-Value Problems [in Russian], Nauka, Moscow (1980).
V. K. Romanko, “Boundary-value problems for one class of differential operators,” Differents. Uravn., 10, No. 11, 117–131 (1974).
V. K. Romanko, “Nonlocal boundary-value problems for some systems of equations,” Mat. Zametki, 37, No. 7, 727–733 (1985).
A. A. Makarov, “Existence of a correct two-point boundary-value problem for systems of pseudodifferential equations,”Differents. Uravn., 30, No. 1, 144–150 (1994).
V. I. Chesalin, “Problem with nonlocal boundary conditions for abstract hyperbolic equations,” Differents. Uravn., 15, No. 11, 2104–2106 (1979).
V. V. Horodets’kyi and Ya. M. Drin, “Multipoint (in time) problem for one class of evolutionary pseudodifferential equations,” Ukr. Mat. Zh., 66, No. 5, 619–633 (2014); English translation: Ukr. Math. J., 66, No. 5, 690–706 (2014).
V. V. Horodets’kyi and O. V. Martynyuk, Cauchy Problem and Nonlocal Problems for Singular Evolutionary Equations of Parabolic Type [in Ukrainian], Knyhy XXI, Chernivtsi (2010).
J. Chabrowski, “On the non-local problems with a functional for parabolic equation,” Funkcial. Ekvac., 27, 101–123 (1984).
N. L. Lazetic, “On classical solutions of mixed boundary problems for one-dimensional parabolic equation of second order,” Publ. Inst. Math. (Beograd) (N.S.), 67, 53–75 (2010).
Ya. M. Drin’, “Multipoint problem for evolutionary pseudodifferential equations,” Dop. Nats. Akad. Nauk Ukr., No. 7, 7–11 (2010).
I. M. Gelfand and G. E. Shilov, Some Problems of the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).
B. L. Gurevich, “Some spaces of test and generalized functions and the Cauchy problem for finite-difference schemes,” Dokl. Akad. Nauk SSSR, 99, No. 6, 893–896 (1954).
T. I. Hotynchan and R. M. Atamanyuk, “Different forms of the definition of spaces of type W;” Nauk. Visn. Chernivets. Univ.: Zb. Nauk. Prats’. Mat. [in Ukrainian], Ruta, Chernivtsi, Issue 111, 21–26 (2001).
B. I. Levitan, “Expansion in Bessel functions in Fourier series and integrals,” Usp. Mat. Zh., 6, Issue 2, 102–143 (1951).
V. V. Horodets’kyi and T. I. Hotynchan, “Bessel transformation in spaces of type\(\dot{S}\);” Bukovyn. Mat. Zh., 5, No. 3-4, 50–55 (2017).
M. I. Matiichuk, Parabolic Singular Boundary-Value Problems [in Ukrainian], Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (1999).
G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, and Formulas for Reference and Review, McGraw-Hill, New York (1968).
I. M. Gelfand and G. E. Shilov, “Fourier transformations of rapidly increasing functions and problems of uniqueness of the Cauchy problem,” Usp. Mat. Nauk, 8, Issue 6, 3–54 (1953).
V. V. Horodets’kyi, Ya. M. Drin’, and M. I. Nahnybida, Generalized Functions. Methods for the Solution of Problems [in Ukrainian], Knyhy XXI, Chernivtsi (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 25, No. 4, pp. 291–324, October–December, 2022.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Horodets’kyi, V.V., Kolisnyk, R.S. & Shevchuk, N.M. Multipoint (In Time) Problem for Singular Parabolic Equations in Spaces of Type S. J Math Sci 277, 201–239 (2023). https://doi.org/10.1007/s10958-023-06828-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06828-w