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The Cauchy problem for the multi-time fractional diffusion equation.

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118 НАУЧНЫЕ ВЕДОМОСТИ
Серия: Математика. Физика. 2015. ќ17(214). Вып. 40
MSC 26A33
THE CAUCHY PROBLEM
FOR THE MULTI-TIME FRACTIONAL DIFFUSION EQUATION
A.V. Pskhu
Scientic Research Institute of Applied Mathematics and Automation, KBSC, RAS,
89A, Shortanov street, Nalchik, Russia, e-mail: pskhu@mail333.com
Key words: Cauchy;s problem, multi-time equations, diusion equation.
Consider the equation
m
?
?k
k=1
? ?k
u(x, y) ? ?x u(x, y) = f (x, y) .
?yk?k
(1)
Here x = (x1 , · · · , xn ) ? Rn , y = ?
(y1 , · · · , ym ) ? Rm and ? = (?1 , · · · , ?m ) ? Rm , ?k > 0;
?x is the Laplace operator, ?x = ni=1 ? 2 /?x2i ; ? ?k /?yk?k is an operator of fractional partial
dierentiation of order ?k , ?k ? (0, 1), with respect to yk and with origin at yk = 0. The
fractional dierentiation is given by the Dzhrbashyan-Nersesyan operator associated with
{? ,? }
?k ?1 ?k
the sequence {?k , ?k }, ?k , ?k ? (0, 1], ?k = ?k + ?k ? 1, ? ?k /?yk?k = D0ykk k = D0y
D0yk
k
?k ?1
?k
(see [1]), where D0yk and D0yk are the Riemann-Liouville fractional integral and derivative.
For a survey on results relating the initial and boundary value problems for a fractional
diusion equation and its generalizations, we refer to papers [2] and [3].
For any element z ? Rm , we denote by zk the k -th coordinate of z . Let z and ? be
elements of Rm . The expressions z? , z ? , z? and z?,k denote the vectors (z1 ?1 , ..., zm ?m ) and
?
?m
?m
(z1?1 , ..., zm
), and the quantities m
i=1 zi and
i=1, i?=k zi respectively.
Consider the function
? ?
m
?
?? ? 1t
t e
? (??k , µk ; ?zk t) dt ,
(2)
fm,? (z; ?; µ) =
0
k=1
where m ? N, ? ? R, and z , ? , µ ? Rm , zk > 0, k = 1, m. Here, ? (?, ?; t) =
?
?
i=0
ti
i!?(?i + ?)
is the Wright function (see [4]). In terms of function (2), we dene the function
( 2
)
|x|
1
2?n ?1
?
??
?m,n (x, y) = Cn |x| y? fm,n/2
?y ; ?; 0 , where Cn = ? ?n/2 .
4
4
We put T = {y : yk ? (0, Tk ), k = 1, m} and ? = {(x, y) : x ? Rn , y ? T }. By T(k) and
y(k) we denote the projections of T and y ? Rm onto Rm?1 along yk . Also we write
Iy = (0, y1 ) Ч · · · Ч (0, ym ),
Iy(k) = (0, y1 ) Ч · · · Ч (0, yk?1 ) Ч (0, yk+1 ) Ч · · · Ч (0, ym ).
Серия: Математика. Физика. 2015. ќ17(214). Вып. 40 119
НАУЧНЫЕ ВЕДОМОСТИ
By ?k we denote the interior points of the set ?k = ?? ? {yk = 0}, k = 1, m.
( )
A function u(x, y) is called a regular solution of equation (1) if y?1?? u(x, y) ? C ? for
{? ,? }
?k ?1
some ? ? Rm with positive ?k , D0y
u ? C(? ? ?k ), D0ykk k u and uxj xj belong to C(?),
k
k = 1, m, j = 1, n. This function satises equation (1) at all points (x, y) ? ?.
In this work, we study the following problem: nd a regular solution u = u(x, y) of
equation (1) in ? such that
?k ?1
u(x, y) = ? (x, y(k) ),
lim D0y
k
yk ?0
x ? Rm ,
y(k) ? T(k) ,
k = 1, m.
(3)
Formulate the main results of the work.
(
)
1?µ
Theorem 1. Suppose that y?,k
?k (x, y(k) ) ? C Rn Ч T (k) and y?1?µ f (x, y) ? C(?) for
some µ ? Rm with positive µk , and
(
)
)
(
2
2
1?µ
lim y?1?µ f (x, y) exp ??k |x| 2??k = 0,
lim y?,k
?k (x, y(k) ) exp ??k |x| 2??k = 0,
|x|??
|x|??
(
where ?k < 1 ?
?k
2
?k
) ( ? ) 2??
k
and k = 1, m. Then a regular solution u(x, y) of problem (1),
k
2Tk
(3) that satises the condition
)
(
2
lim y?1?? u(x, y) exp ??k |x| 2??k = 0,
|x|??
has the form
k = 1, m,
? ?
u(x, y) =
+
m
?
k=1
?
?k
?
(k)
Iy
Rn
Iy
Rn
f (?, ?)??m,n (x ? ?, y ? ?) d?d? +
]
[ ? ?1 ?
Dykk?k ?m,n (x ? ?, y ? ?) ?
k =0
?k (?, ?(k) ) d?d?(k) .
Theorem 2. There is at most one regular solution of problem (1), (3) in the class of
functions that satisfy the following condition for some positive constant ?:
(
)
2
lim y?1?? u(x, y) exp ??|x| 2??0 = 0,
|x|??
where ?0 = min{?1 , ?2 , ..., ?m }.
References
1. Dzhrbashyan M.M., Nersesyan A.B. Fractional derivatives and the Cauchy problem for
dierential equations of fractional order // Izv. Akad. Nauk Armenian SSR Matem. 1968. 3, No.1. P.3-29. (Russian)
2. Kilbas A.A., Srivastava H.M. and Trujillo J.J. Theory and applications of fractional dierential
equations / North-Holland Math. Stud. vol.204, Amsterdam: Elsevier, 2006.
3. Pskhu A.V. The fundamental solution of a diusion-wave equation of fractional order //
Izvestiya: Mathematics. 2009. 73, No.2 P.351-392.
4. Wright E.M. On the coecients of power series having exponential singularities // J.London
Math. Soc. 1933. 8, No.29. P.71-79.
120 НАУЧНЫЕ ВЕДОМОСТИ
Серия: Математика. Физика. 2015. ќ17(214). Вып. 40
ЗАДАЧА КОШИ ДЛЯ МНОГОВРЕМЕННОГО ДРОБНОГО
УРАВНЕНИЯ ДИФФУЗИИ
А.В. Псху
Научно-исследовательский институт прикладной математики и автоматики, КБАР,
ул. Шортанова, 89A, Нальчик, Россия, e-mail: pskhu@mail333.com
Ключевые слова: задача Коши, многовременные уравнения, уравнение диффузии.
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