ISSN 1064-5624, Doklady Mathematics, 2014, Vol. 90, No. 3, pp. 1–6. © Pleiades Publishing, Ltd., 2014.
Original Russian Text © L.N. Lyakhov, I.P. Polovinkin, E.L. Shishkina, 2014, published in Doklady Akademii Nauk, 2014, Vol. 459, No. 5, pp. 533–538.
1
1. INTRODUCTION
The integral representation of the solution of the
Cauchy problem for the classical wave equation uses
Asgeirsson’s relations for the spherical means of solu-
tions of ultrahyperbolic equations (see monographs
[1, pp. 81–90 of the Russian translation; 2, p. 679 of
the Russian translation; 3, p. 465 of the Russian trans-
lation]). Two types of solutions corresponding to even
and odd dimensions of the space are known. In our
study, the role of the dimension of the space is played
by the number n + |γ|, which may be noninteger (the
multi-index γ is not required to consist of integers).
This has led to a general formula, which implies the
classical formula for the solution as the special case
where the multi-index γ consists of positive integers.
We had to introduce a general ultrahyperbolic equa-
tion, which includes both the classical equation and
the singular equation considered in [4], and derive the
corresponding relation (similar to Asgeirsson’s rela-
tion) to solve it.
The classical ultrahyperbolic equation has the form
(1)
In [4], we considered the equation obtained from
Eq. (1) by replacing the second derivatives by the
Bessel singular differential operators = +
(here, γ
i
> 0, i = 1, 2, …, m' + m'', γ = (γ', γ'') =
(γ
1
, …, γ
m'+ m''
), γ' = ( , …, ), γ'' = ( , …, )):
Δ
x
uxy , ( ) Δ
y
uxy , ( ) , xy ,
n
. ∈ =
B
γ
i
∂
2
∂ x
i
2
------
γ
i
x
i
--
∂
∂ x
i
-----
γ
1
'
γ
m'
'
γ
1
''
γ
m''
''
(2)
Equation (2) with γ
i
= 0 for i = 1, 2, …, m' + m''
coincides with the classical ultrahyperbolic equation
(1). However, it has turned out to be impossible to pass
to the limit as γ
i
0 in the main result of [4] (the
Asgeirsson relation for weighted spherical means gen-
erated by the generalized shift), because generalized
shifts are not defined at γ
i
= 0. It is usually assumed
(see [5, 6]) that the parameters γ
i
in Eq. (2) are fixed
and strictly positive.
In this paper, we study the general case, which
unites Eqs. (1) and (2) and is more fundamental in
many respects.
1
For the solutions of the corresponding
general singular ultrahyperbolic equation, we obtain a
relation of the type of Asgeirsson’s relations, by using
which we find two fractional integro-differential rep-
resentations for the solutions of the Cauchy problem
for a singular wave equation with Bessel time operator.
Notation. Suppose given positive integers n = n' +
n'' ≥ 2 and m = m' + m'' ≥ 2, where each of the integers
n', n'', m', and m'' is nonnegative and fixed. We set N =
n + m. We use the notation
1
I.A. Kipriyanov assumed in 1991 that the Bessel operators on the
left- and right-hand sides of a singular hypergeometric equation
act on only one variable and the Bessel operators in the equation
may have different indices γ. But, in this case, the sums of the
indices and the dimensions of Euclidean spaces on the left- and
right-hand sides of the equation must be equal.
B
γ
i
'
( )
x'
ux ()
i 1 =
m'
∑
B
γ
i
''
( )
x''
ux () ,
j 1 =
m''
∑
=
m' γ' + m'' γ'' , + =
x x' x'' , ( )
m'
+
m''
+
. × ∈ =
z x ; y ( )
N
+
∈
n
+
m
+
; × = =
x x' x'' , ( )
n
+
∈
n'
+
n''
, × = =
n'
+
x
n'
: x
1
0 … x
n'
, , 0 > > ∈ { } , =
y y' y'' , ( )
m
+
∈
m'
+
m''
, × = =
Formulas for the Solution of the Cauchy Problem
for a Singular Wave Equation with Bessel Time Operator
L. N. Lyakhov, I. P. Polovinkin, and E. L. Shishkina
Presented by Academician V.A. Il’in May 5, 2014
Received June 25, 2014
DOI: 10.1134/S106456241407028X
Voronezh State University, Universitetskaya pl. 1,
Voronezh, 394006 Russia
e-mail: levnlya@mail.ru, polovinkin@yandex.ru,
ilina_dico@mail.ru
MATHEMATICS