The Poisson Formula for Solutions to Initialboundary Value Problems for B-Hyperbolic Equations with Bessel Operators wit
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
THE POISSON FORMULA FOR SOLUTIONS TO INITIALBOUNDARY VALUE PROBLEMS FOR B-HYPERBOLIC EQUATIONS WITH BESSEL OPERATORS WITH NEGATIVE INDEX L. N. Lyakhov∗ Voronezh State University 1, Universitetskaya pl., Voronezh 394006, Russia [email protected]
K. S. Yeletskikh Bunin Yelets State University 28, Kommunarov St., Yelets 399770, Russia [email protected]
E. L. Sanina Voronezh State University 1, Universitetskaya pl., Voronezh 394006, Russia [email protected]
UDC 517.951
For the Euler–Poisson–Darboux type equations with the Bessel operators with negative index we consider the Cauchy problem with the boundary conditions. We obtain a representation of the solution in the form of the Poisson formula. Bibliography: 8 titles.
1
Bessel Operators with Negative Index
This paper continues the study started by the authors in [1]. The Bessel operator B−β with negative index −β is defined by d2 β d d d = tβ t−β , β > 0. B−β = 2 − dt t dt dt dt The B-hyperbolic equation B−β,t u(x, t) = Bγ,x u(x, t), (x, t) ∈ D = {(0, 1) × (0, T )},
(1.1)
where γ > 0, β > 0, T > 0, is an Euler–Poisson–Darboux type equation. Here, the Bessel operator acts on the time variable (i.e., the variable relative to which the initial conditions are ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 47-56. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0051
51
imposed). We consider the Sobolev–Kipriyanov space of odd B-smoothness m d 2 2m+1 2 k 2 m Hγ (0, 1) = f : f H 2m+1 = Bγ f Lγ (0,1) + Bγ f γ 0.
k=0
(Similar weighted classes of functions are studied in [2] and [3]). We look for a solution to Equation (1.1) in the class of functions u(x, t) ∈ C 1 (D) ∩ C 2 (D) ∩ Hγ3 (0, 1; C(0, T ))
(1.2)
satisfying the initial condition u(x, 0) = ϕ(x),
0 x l,
(1.3)
and one of the following (first, second, third kind) boundary conditions u(0, t) = 0, 0 t T ; ux (0, t) = 0, 0 t T,
u(l, t) = 0, 0 t T, ux (l, t) = 0, 0 t T,
ux (0, t) − Hu(0, t) = 0, 0 t T,
(1.4)
ux (l, t) + hu(0, t) = 0, 0 t T.
Functions satisfying the singular Bessel equation Bs u(x) =
d2 u(x) s du(x) + = −λ2 u(t) dx2 x dx
(1.5)
are called B-cylindrical (Bessel) functions. Two classes of B-cylindrical functions are considered. The first class corresponds to eigenfunctions of the operator Bγ with s = γ > 0, and the second one consists of eigenfunctions of the operator B−β with s = −β, 0 < β < 1. The functions of the first class are called j-Bessel functions and are connected with the Bessel functions Jν of the first kind by the equality jν = Γ(ν + 1)(x/2)−ν Jν (x) (cf. details in [4]). The number ν (the order of the j-Bessel functions is not less than −1/2) is connected with the index γ of the Bessel operator by the equality ν = (γ − 1)/2. The uniform convergence of the series with respect to these functions was studied in [1] and the results of [1] are used in this paper. 1.1. B−β -cyli
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