CC BY
48
УДК 517.986.7
Parameter Determination in a Differential Equation of Fractional Order with Riemann-Liouville Fractional Derivative in a Hilbert Space
Dmitry G. Orlovsky*
National Research Nuclear University MEPhI Kashirskoye shosse, 31, Moscow, 115409
Russia
Received 10.10.2014, received in revised form 20.11.2014, accepted 15.12.2014 The Cauchy type problem for a differential equation with fractional derivative and self-adjoint operator in a Hilbert space is considered. The problem of parameter determination in equation by the value of the solution at a fixed point is presented. Theorems of existence and uniqueness of the solution are proved.
Keywords: equation of fractional order, Hilbert space, self-adjoint operator, Cauchy-type problem, Mittag-Leffler function, inverse problem, characteristic function.
Introduction
In recent years, theory of differential equations with fractional derivatives has attracted considerable interest. This is primarily due to the application of this theory to new models of physical processes. However the theory has been developed for some time. The first steps were made by L.Euler, P.Laplace and J.Furier. Further development of this theory was associated with the works of N.H.Abel, J.Liouville and B.Riemann. An important advancement in the development of the theory is presented in [1-4].
One of the modern methods for studying equations of mathematical physics is the theory of differential equations in Banach and Hilbert spaces. Abstract differential equations of fractional order are considered in [5-8].
Let us consider a closed linear operator A in a Banach space H. The operator domain D(A) is dense and 0 < a < 1, u0 G H. Consider the following Cauchy-type problem
Dau(t) = Au(t) + f (t), t > 0, (1)
lim Da-1u(t)= u0, (2)
where Da is the Riemann-Liouville fractional derivative of order a:
, ™ w s 1 d [f' u(s)
(D"u)(t) = W-a)dtJ,, iT-ifds
(D°-1u> <f) = F(Tb)i' ds
© Siberian Federal University. All rights reserved
and r is the Gamma function.
A solution of (1), (2) is a continuous function u(t) in H for t > 0 such that Da-1u(t) is continuously differentiable in H for t > 0. Function u(t) takes its values in D(A) and satisfies
(1), (2) ( [7]).
Along with problem (1), (2), let us consider the homogeneous problem
Dau(t) = Au(t), t > 0, (3)
lim Da-1u(t) = u0. (4)
t^0
We say that the problems (1), (2) and (3), (4) are uniformly well-posed ( [7]) if there is operator function V(t) (solution operator) that commutes with A. There are also two constants M > 0 and u G R such that for every u0 e D(A) function V(t)u0 is the only solution of problem (3), (4) and for any t > 0 the following estimate
||V(t)\\ < Mta-1eut (5)
is valid.
If space H has the Radon-Nikodym property (every absolutely continuous function represented by the integral of its derivative) then problem (3), (4) is uniformly well-posed if and only if when ReX > u the operator A has the resolvent R(Xa) = (XaI — A) 1. The resolvent satisfies the following inequality [9]
dnR(Xa)
dXn
< M r(n + a)
(ReX - u)
n+a '
In this case the solution operator has the following representation [9]
Wo+iro
V(t) = Dl-a — v.p. Í Xa-1extR(Xa)dX (u0 > max(0,u)). 2ni J
Wo -iro
Let us consider the Cauchy type problem (1), (2) assuming that problem is uniformly well-posed. The solvability of this problem can be proved if
(C1) functions f (t) and Af (t) are continuous for t > 0 in H and absolutely integrable at t = 0;
(C2) function Da-1 f (t) is continuous for t > 0 and continuously differentiable for t > 0, function Daf (t) is absolutely integrable at t = 0.
If any of these two conditions is satisfied then the solution of problem (1), (2) exists. The solution is unique and it is given by the formula [9]
u(t) = V(t)u0 + í V(t - s)f (s)ds. (6)
0
The principle of subordination is one of the important properties of equations with fractional derivatives. Along with problem (3), (4) we consider another Cauchy type problem with other differentiation order ¡3 (a < ¡3 < 1)
D13u(t) = Au(t), t > 0, (7)
lim D3-lu(t)= u0. (8)
t—^0
We also consider function fT,v (t) defined by the equality
fTv(t)=< ¿i J eXp(tZ - TzV)dz>f > 0 (9)
a —itt t < °
where a > t> 0, 0 < v < 1 and branch zv is selected such that Re zv > 0 for Re z > 0. This function is non-negative ( [10] and [11])
/ fr,v (t)eXT dT = tv—1Ev,v (A tv), (10)
J 0
where (x) is the generalized Mittag-Leffler function
Ea}!3 (x) = V] ^-■
n=0 r(an + $>
The principle of subordination ( [11]) states that if problem (7), (8) is uniformly well-posed then problem (3), (4) is also uniformly well-posed. Moreover, the solution operator V(t) of problem (3), (4) is expressed in terms of the solution operator U(t) of problem (7), (8) by the following relation
V (t)= fr,v (t)U (t )dT, J 0
where v = a/p.
Throughout this paper it is assumed that the space H is a Hilbert space and operator A is self-adjoint and non-positive. Any reflexive Banach space has the Radon-Nikodym property so in our case this property is satisfied.
Lemma 1. Let us assume that H is a Hilbert space, operator A is self-adjoint and bounded from above, element u0 G D(A) and one of the conditions (C1) or (C2) is satisfied. Then the solution of Cauchy type problem (1), (2) exists, it is unique and has the form
u(t) = ta-1Eaa(taA)uo + / (t - s)a-1 Ea,a((t - s)aA)f (s)ds. (11)
0
Proof. It is known that any self-adjoint bounded from above operator is a generator of a strongly continuous semigroup. This means that problem (7), (8) is uniformly well-posed when p =1. Applying the principle of subordination, we obtain that problem (3), (4) is also uniformly well-posed for any 0 < a < 1. The solution operator of problem (7), (8) is U(t) = exp(At), therefore by virtue of the principle of subordination (taking into account that v = a) we have the equality
/* œ œ ft' b \ Cœ C b
V (t)= fT,a(t)exp(AT )dT = fr,a(t)U eXT dEA dT = / fT,a(t)eXr dExdT, ■J0 JO \J-œ J J0 J-œ
where E\ is a spectral resolution of the operator A and b is its upper bound. As the integrand is non-negative function, we can change the order of integration. Taking into account equation (10), we obtain
V (t)=/ [ fr,a(t)eXT dT\dEx =i ta-1 Ea,a(Xta)dE\ = ta-1Ea^aA).
-œ -œ
0
Relation (11) directly follows from equality (6).
1. Inverse problem
Let us assume that H is a Hilbert space, operator A is self-adjoint and non-positive, 0 < a < 1, p(t) is a scalar non zero function and u0,ui e H. Let us find function u(t) with values in the space H and element p e H that satisfy the following system of equations
Dau(t) = Au(t) + p(t)p, t > 0, (12)
lim Da-iu(t) = u0, (13)
t—► 0
u(T ) = u1. (14)
where T > 0.
According to the adopted terminology problem (12)-(14) belongs to the class of inverse problems. A review of publications on this subject can be found in [12-19]. Inverse problems for equations of fractional order in Banach spaces was considered in papers [20,21]. Similar problem was considered for the first order equation (when a = 1) but with a fractional integral in the additional boundary condition [22]. The behavior of the entire function
F(z) = f (T - s)a-iEa,a(z(T - s)a)p(s)ds. (15)
0
is the key to the solvability of problem (12)-(14).
Function (15) will be denoted the characteristic function of inverse problem (12)-(14).
Theorem 1. Let us assume that H is a Hilbert space, operator A is self-adjoint and non-positive, 0 < a < 1, function p(t) e C[0; T] n Cx(0; T], it is not identically equal to zero for t e (0; T]. The derivative of function tp(t) is absolutely integrable at t = 0 and the elements u0,u\ e H. The solution of problem (12)-(14) is unique if and only if point spectrum of the operator A does not contain zeros of the characteristic function (15). The solution of problem (12)-(14) exists if and only if
i |F(A)|-2 d(Exh,h) < to, (16)
•J
where E\ is a spectral resolution of the operator A, and the element
h = ui - Ta-1Eaia(TaA)uo. (17)
Proof. According to lemma 1 the solution of direct problem (12)-(13) is given by formula (11) where one should set f (t) = <p(t)p. Setting in this formula t=T and taking into account (14), we obtain the following equation for the unknown element p
r T
ui = Ta-iEaa(TaA)uo + (T - s)a-iEa,a((T - s)aA)ip(s)pds.
0
This equation can be written in the following form
Bp = h,
where operator B is defined by the relation
Bp = fT(T - s)a-iEa,a((T - s)aA)p(s)pds,
0
and element h is given by (17).
Let us transform the expression that represents the operator B. For this we use the spectral decomposition of the operator A
A = i XdEx.
J — œ
According to the operator calculus of self-adjoint operators, we have
Ea
Therefore
B
Ea,a((T - s)aA)= í Ea,a(X(T - s)°)dE\. J —tt
J (y (T - s)a—1Ea,a(X(T - s)a)^(s)dE^j ds.
Since the measure dE\® ds is finite on the (-to; 0] x [0; T] then by Fubini's theorem we have B =1 (f (T - s)a-i Ea,a(A(T - s)a)p(s)ds\ dEx = i F(A)dEx = F(A).
■J-x \Jo J J-x
Therefore, the equation for the unknown element p takes the form
F (A)p = h.
(18)
Equation (18) was studied in [23]. It was proved that the solution of this equation is unique if and only if point spectrum of the operator A does not contain zeros of F(z). The existence of a solution of the equation follows from condition (16). The application of this result proves theorem 1. □
Theorem 2. Let us assume that H is a Hilbert space, operator A is self-adjoint and non-positive, 0 < a < 1, function <p(t) G C]0\T] n C 1(0; T] and its derivative is absolutely integrable at t = 0, everywhere either tp(t) ^ 0 or tp(t) ^ 0 and <p(T) = 0, u0,u1 G D(A). Then solution of problem (12)-(14) exists and it is unique.
Proof. Since u0 G D(A) and every function of the self-adjoint operator A commutes with A, then the element Ea¡a(TaA)u0 belongs to D(A). Since u1 G D(A) then h G D(A). It follows directly from the definition of the Mittag-Leffler function that
sa—1Eaa(zsa) = d [saEa,a+i(zsa)] . ds
Integrating (15) by parts, we obtain
F(z) = ( (T - s)a—1 Ea,a(z(T - s)a)cp(s)ds =( sa—1 Eaa(zsa)cp(T - s)ds
Jo Jo
= fT v(T - s)d
o
saEa¡a+1(zsa) + -
ds = ip(T - s)
saEa¡a+1(zsa ) + -
+
saEa¡a+1(zsa) + -
z
rT
+
ip'(T - s)ds = ip(0) saEa¡a+1(zsa) + -
T aEa¡a+1 (zTa ) + -
f'(T - s)ds.
o
<P(T )
+ +
(19)
Let us consider first and last terms in the right hand side of (19).
o
o
It is known ( [24, p. 12]) that when 0 < a < 1
Ea,ß (z) = -
1
r(ß - a)z
+ O(|z| ) (z G R, z ^-œ).
Then for z ^ -œ we have
Ea,a+i(z) = - z + O(|z|-2).
From the last relation we obtain
T aEa,a+i(zTa) = Ta ( - + O(1/(zTa)2)^ = -1 + O(1/z2)
when z G R, z ^ -œ. Thus
¥>(0)
T a Ea,a+i(zT a) + -
= O(1/z2).
It follows from (20) that for sufficiently large negative values of z
Ea,a+i(zsa) + -1—
<
C
(zsa)2
(20)
(21)
where constant C does not depend on s. Multiplying the last inequality by sa we obtain the following estimate
saEa,a+i(zsa) + I
<
C
and we have
saEa,a+i(zsa) +— z
f'(T - s)ds
* C r (T - s)|ds = C
z2
z J 0
|y'(s)|
z2Jo (T - s)
-ds. (22)
According to the theorem condition, the function |y'(s)| is integrable at zero and the integrand in the last integral is equivalent to |<^'(s)|/Ta. Hence the integral converges at zero. For s ^ T the same function is equivalent to |y'(T)|/(T — s)a. Since a < 1 then the integral also converges at s = T.
It follows from (21), (22) that the characteristic function
F (z)--
<p(T )
z G R, z ^ -œ.
(23)
Let us use now the results presented in [24]. It follows from reviews of Chapter 4 of this work (theorem 4.4.1, p.101) that when 0 < a < 1 and ft ^ a the Mittag-Leffler function Ea p(z) > 0 when z ^ 0. Therefore, it follows from conditions of the theorem that characteristic function (15) has no zeros on the real semiaxis z < 0. It follows from the calculus of self-adjoint operators that operator B = F(A) is invertible. We obtain from the asymptotic equality (23) that domain of the operator
f0 dEx
B-i =
F (A)
z2sa
a.
s
0
z
œ
coincides with D(A). The element h is defined by formula (17) and it belongs to D(A) then equation (18) has the unique solution p = B-1h and function u(t) is defined by equality (11) where one should set f (t) = y(t)p. This completes the proof of theorem 2. □
Let us consider the model case y(t) = tp. The conditions of theorem 2 for this case are valid when 3 ^ 0. Moreover, in this case one can write the characteristic function explicitly. Performing the substitution s = in (15), we obtain
F (z) = fT (T - s)a-1Ea,a(z(T - s)a)sp ds = fT (T - sr-wf: ^ - ^ ds =
Jo J0 n=0 r(an +a)
'—' fi '—' ..nn^an+a+p p 1
= v —r (t - s)an+a-lsp ds = y -- (i - e)an+a-1de =
n=0r(an + a) Jo n=o r(an + a) Jo
(24)
~ znTan+a+p ~ znTan+a+p r(3 +1)r(an + a)
= > -;-r- B (3 +1,an + a) = > —;-— •—;-- =
^ r(an + a) ^ ' ' r(an + a) r(an + a + 3 + 1)
n=0 V ' n=0 V ' K ' '
^ ( zT a )n
= r(3 + 1)T_( + + 3 +i) = r(3 + 1)Ta+PEa,a+p+l(zTa). ^T(an + a + 3 + 1)
n=0
It follows from the above-mentioned properties of the Mittag-Leffler function that characteristic function (24) is positive on the negative semiaxis and it has asymptotic bound const/z at -to. Hence it follows that operator B is invertible on D(A).
Corollary 1. Let us assume that H is a Hilbert space, operator A is self-adjoint and non-positive, 0 < a < 1, 3 ^ 0 and elements u0,u1 G D(A). Then the solution of problem
Dau(t) = Au(t) + tpp, t > 0,
lim Da-1u(t) = u0,
t^0
u(T) = ul .
exists, it is unique and has the form
1
r(3 + 1)T a+p
P = r(3 +1)Ta+p E-!a+p+l(T(ui - Ta-lEa,a(TaA)u0) ,
u(t) = ta-1Ea,a(taA)u0 + f (t - s)a-1Ea,a((t - s)aA)sppds.
0
Remark 1. The last formula can be simplified. Let us note that equality (11) can be written in the form
u(t) = ta-1Ea a (t aA)u0 + Ft(A)p,
where Ft(z) is obtained from characteristic function (15) by replacing T with t. Since in our particular case F(z) = r(3 + 1)Ta+pEa,a+p+1(zTa), then function
Ft(z) = r(3 + 1)t a+p Ea ,a+p+1(zta )
and
u(t) = ta-1Ea,a (taA)u0 + r(3 + 1)t a+PEa, a+p+1(taA)p.
We should also mention the case 3 = 0. The study of inverse problems of the type considered here began with this option (see [14-16]).
Corollary 2. Let us assume that H is a Hilbert space, operator A is self-adjoint and non-positive,
0 < a < 1, elements u0,u1 e D(A). Then the solution of problem
Dau(t) = Au(t)+ p, t > 0,
lim Da-1u(t) = u0, t— 0
u(T) = ui
exists, it is unique and has the form
p = TaE-la+i(TaA) (ui - Ta-1Ea,a(TaA)u0),
u(t)= ta-i Ea,a(ta A)u0 + taEa,a+i(taA)p.
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Определение параметра дифференциального уравнения дробного порядка с производной Римана-Лиувилля в гильбертовом пространстве
Дмитрий Г. Орловский
В гильбертовом пространстве рассмотрена задача типа Коши для дифференциального уравнения с дробной производной и самосопряженным оператором. Поставлена задача определения параметра в неоднородном члене уравнения по значению решения в фиксированной точке. Доказаны теоремы существования и единственности решения.
Ключевые слова: уравнение дробного порядка, гильбертово пространство, самосопряженный оператор, задача типа Коши, функция Миттаг -Леффлера, обратная задача, характеристическая функция.
