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ISSN 2074-1871 Уфимский математический журнал. Том 16. № 1 (2024). С. 111-125.
INVERSE PROBLEM FOR SUBDIFFUSION EQUATION WITH FRACTIONAL CAPUTO DERIVATIVE
R.R. ASHUROV, M.D. SHAKAROVA
Abstract. We consider an inverse problem on determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative. The right-hand side of the equation has the form f (x)g(t) and the unknown is the function f (x). The condition u(x, t0) = ^(x) is taken as the over-determination condition, where t0 is some interior point of the considered domain and ^(x) is a given function. By the Fourier method we show that under certain conditions on the functions g(t) and ^(x) the solution of the inverse problem exists and is unique. We provide an example showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions g(t). For such functions g(t) we find necessary and sufficient conditions on the initial function and on the function from the over-determination condition, which ensure the existence of a solution to the inverse problem.
Keywords: subdiffusion equation, forward and inverse problems, the Caputo derivatives, Fourier method.
Mathematics Subject Classifications: 35R11, 34A12
1. Introduction
Given a fixed number p e (0,1], we consider the following initial-boundary value problem ' D%u(x, t) - Au(x,t) = F(x,t) = f (x)g(t), x e H, t e (0,T], u(x,t)\an = 0, (1.1)
u(x, 0) = tp(x), x e H.
Here f (x), g(t) and p(x) are continuous functions in the domain H c RN and h(t) stands for the Caputo fractional derivative (see, for instance, [1])
t
f d tp-1 D^h(t) = J ui-p(t - s) — h(s)ds, up(t) = ,
0
where r(p) is the gamma function. If we first integrate and then differentiate, then we get the Riemann-Liouville derivative.
It should be noted that if p = 1, then both the Caputo derivative and the Riemann-Liouville derivative coincide with the classical first order derivative. Therefore, if p =1, then problem (1.1) coincides with the usual initial-boundary value problem for the diffusion equation.
Problem (1.1) is also called the forward problem. The main purpose of this study is the inverse problem on determining the right-hand side of the equation, namely, the function f (x).
R.R. Ashurov, M.D. Shakarova, Inverse problem for the subdiffusion equation with fractional Caputo derivative.
© Ashurov R.R., Shakarova M.D. 2024.
The authors acknowledge financial support from the Ministry of Innovative Development of the Republic of Uzbekistan, Grant No F-FA-2021-424. Submitted November 02, 2022.
Ill
To solve the inverse problem, one needs an extra condition. Following A.I. Prilepko and A.B. Kostin [2] and K.B. Sabitov [3] (see also [4]), we consider the additional condition in the form:
u(x,t0) = ^>(x), x E Q, (1.2)
where t0 is a given fixed point of the segment (0, T\.
We call the initial-boundary value problem (1.1) together with the additional condition (1.2) the inverse problem on finding the part f (x) of the right-hand side of the equation.
The authors usually impose an additional condition (1.2) at the final time t0 = T (see, for instance, [5], [6] for classical diffusion equations and [7], [8] for subdiffusion equations). The meaning of taking condition (1.2) at t0 is that in some cases the uniqueness of the solution of the inverse problem is violated if t0 = T and by choosing t0 it is possible to achieve uniqueness in these cases as well.
We are interested in the classical solution (we simply call it a solution) of the problems under consideration, i.e. such solutions that themselves and all the derivatives involved in the equation are continuous, moreover, all the given functions are continuous and the equation is obeyed at each point. As an example, let us give the definition of the solution to the inverse problem.
Definition 1.1. A pair of functions {u(x,t),f (x)} with the properties
1. D%u(x, t), Au(x,t) E C(Q x (0.T]),
2. u{x,t) E Cjp x [0.T]),
3. f (x) E C(Q),
and satisfying conditions (1.1), (1.2) is called a solution of the inverse problem.
We note that in this definition the requirement of continuity in a closed domain of all derivatives of the solution appearing in (1.1) was proposed by O.A. Ladyzhenskaya [9]. The advantage of this choice is that the uniqueness of such a solution is proved quite simply, moreover, the solution found by the Fourier method satisfies the above conditions.
Inverse problems on determining the right hand side of various subdiffusion equations were studied by a number of authors due to the importance of such problems for applications. However, it should be immediately noted that for the abstract case of the source function F(x,t) there is no general theory vet, see survey paper [10] and the references therein. In all known works, the split source function F(x,t) = f (x)g(t) is considered and the methods of investigation depend on whether f (x) or g(t) is unknown. It is somewhat more difficult to study the case when function g(t) is unknown. For example, in papers [11] and [12] the questions of finding the non-stationarv source function g(t) were studied. It should be noted that in these papers the over-determination condition is taken in a fairly general form: B[u(-,t)] = ^(t), where B is a linear continuous functional. In particular, one can take u(x0,t) or fQ u(x,t)dx as B[u(-, t)]. The determination of the unknown function g(t) for subdiffusion equations was studied in the articles [10] and [13].
For subdiffusion and diffusion equations, the case g(t) = 1 and the unknown is f (x) was studied by many authors, see, for example, [14]—[20]. We mention only some of these articles.
Subdiffusion equations with an elliptic part as an ordinary differential expression were considered in papers [14],[15], [16]. The authors of papers [17], [18] studied subdiffusion equations, the elliptic part of which is the Laplace operator or a second-order differential operator. Paper [19] is devoted to study the inverse problem for a subdiffusion equation with the Caputo fractional derivative and an arbitrary elliptic self-adjoint differential operator. The authors of this paper proved the uniqueness and existance of a generalized solution. The case of the Riemann-Liouville derivative was considered in [20]. Here the uniqueness and existence of a classical solution were proved. In papers [14] and [18], the fractional derivative in the subdiffusion equation is a two-parameter generalized Hilfer fractional derivative.
In [21], the authors considered the inverse problem of simultaneous determination of the order of the Riemann-Liouville fractional derivative and the source function in subdiffusion equations. Using the classical Fourier method, the authors proved that the solution to this inverse problem exists and is unique.
In monograph by K.B, Sabitov [22] the solvability of forward and inverse problems for equations of mixed parabolic-hyperbolic type was studied.
We note some results obtained for the case g(t) ^ 1. For classical diffusion equations, such an inverse problem was studied in detail, see the well-known monograph by S, Kabanikhin [23, Ch, 8] as well as [2], [3], [4], [5], [6], Since the equation considered by us also covers the diffusion equation, we will dwell on these works in more detail at the end of Section 4,
In paper [24] the problem on finding function f (x) for an abstract subdiffusion equation with the Caputo derivative was studied. To find the function f (x), the authors used the following additional condition J^ u(t)dp(t) = uT.
M, Slodichka et al, [7] and [8] studied the uniqueness of a solution of the inverse problem for a subdiffusion equation, the elliptic part of which depends on time. It was proved that if function g(t) is sign-definite, then the solution of the inverse problem is unique. It should be especially noted that in [8] the authors constructed an example of a function g(t) that changes sign in the domain under consideration and this resulted in the loss of the uniqueness of the solution to the inverse problem.
It is well known that the considered inverse problem is ill-posed, i.e., the solution does not depend continuously on the given data. Therefore, in the works of some authors, various regularization methods were proposed for constructing an approximate solution of the inverse problem, see, for instnace, [25], [26], In paper [25] the inverse problem for the fractional diffusion equation with the Riemann-Liouville derivative was considered. Assuming that solutions to the equation can be represented by a Fourier series, the authors applied the Tikhonov regularization method to find an approximate solution. Convergence estimates for exact and regularized solutions were presented for a priori and a posteriori rules for choosing parameters. In [26], similar questions were investigated for the stochastic fractional diffusion equation.
This work is devoted to the study of forward problem (1,1) and inverse problem (1.1), (1.2) on determining the right-hand side of the equation. Let us list the main results of this paper.
1) First, in Section 3, we prove the existence and uniqueness theorem for the forward problem (1.1) by using the Fourier method. We present conditions on the initial function p(x) and on the right-hand side of the equation that ensure the validity of the application of the Fourier method. Due to the fact that the elliptic part of the equation is the Laplace operator, the conditions on the functions f (x) and g(t) turned out to be easier to check than in the case of a general elliptic operator, see [27];
2) Then in Section 4, under a certain condition on function g(t) (for example, the constant sign is sufficient), we prove the existence and uniqueness of a solution to the inverse problem. Further, we show that if this condition is violated, then for the existence of a solution to the inverse problem it is sufficient the functions in the initial condition and the over-determination condition to be orthogonal to some eigenfunctions of the Laplace operator with the Dirichlet condition;
3) An example of function g(t) is constructed in Section 4, for which the condition noted above is not satisfied and, as a result, the inverse problem has more than one solution.
The following Section 2 is auxiliary and contains definitions and well-known assertions necessary for further presentation. The section Conclusions completes this work.
2. Preliminaries
In this auxiliary section we define fractional powers of a self-adjoint extension of the Laplace operator, formulate a lemma from book by Krasnoselskii et al. [28], a fundamental result
bv V.A. Il'in [29] about the convergence of the Fourier coefficients and indicate some needed properties of the Mittag-Leffler function.
We denote by {\k} and {vk(x)} a set of positive eigenvalues and an associated complete system of orthonormal eigenfunetions in L2(Q) of the following spectral problem
— Av(x) = Xv(x), x E Q,
= 0.
Let a be an arbitrary real number. Consider an operator Aa acting in L2(Q) as
Aag(x) = ^ Xk9kVkgk = (g,vk),
k=l
on the domain
D(A°) = jg E L2(Q): £ X? I9k|2 < On the elements of D(Aa) we introduce the norm
<x
Ml = E A2CT19k I2 = № g\\2. k=l
Let A be the operator acting in L2(Q) as Ag(x) = —Ag(x) on the domain D(A) = {g E C2(Q) : g(x) = 0,x E dQ^, then bv A = A1 we denote the self-adjoint extension of A in L2(Q).
In our reasoning the following lemma from the book Krasnoselskii et al, [28] plays an important role.
Lemma 2.1. Let a > 4. Then operator A-a continuously maps the space L2(Q) into C(Q), and moreover, the following estimate holds
P-(J g\\ C (Q) ^ c\\g\\
In order to prove the existence of solutions of forward and inverse problems, it is necessary
to study the convergence of the following series:
^ N
Y,Xk Ihk I2, t>- , (2.1)
k=l
where hk are the Fourier coefficients of function h(x). In the case of integers r, in fundamental paper [29] by V.A. Il'in, conditions were obtained for the convergence of such series in terms of the membership of the function h(x) in the classical Sobolev spaces WJk(Q). To formulate these conditions, we introduce the class ^(Q) as the closure in the W2 (Q)-norm of the set of
QQ The theorem of V.A. Il'in states that if the function h(x) satisfies the conditions
h(x) E W^ +l(Q) and h(x), Ah(x),..., A^h(x) E W^(Q), (2.2)
then scalar series (2.1) converges. Here [a] denotes the integer part of a number a. Similarly, if in (2.1) we replace r by r + 2, then the convergence conditions becomes
h(x) e wl^ +3(Q) and h(x), Ah(x),..., A^hlh(x) e Wl(Q). (2.3)
For 0 < p < 1 and an arbitral complex number let Ep^(z) denote the Mittag-Leffler function with two parameters of the complex argument z:
ro k
^ > = £ rw^ ■ ™
For p = 1 we have the classical Mittag-Leffler function Ep(z) = EPi1(z).
We recall some properties of the Mittag-Leffler functions, see, for instance, [30],
Lemma 2.2. For any t ^ 0 one has
\EPA-t)\ < Y+~t, (2-5)
where constant C is independent of t and p.
Lemma 2.3. (see [31]J. The classical Mittag-Leffler function of the negative argument Ep(-t) is monotonically decreasing function for all 0 < p < 1 and
0 < Ep(-t) < 1, Ep(0) = 1.
Lemma 2.4. (see [30, Eq, (2,30)] and [32, Lm. 4]J. Let p he an arbitrary complex number. Then the following asymptotic estimate holds
A-1
t-
EPA-t) -
T(p - p) where C is an absolute constant.
i £ t> i,
Lemma 2.5. (see [31, Eq, (4,4,5)]J. Let p > 0, V > 0 and X e C. Then for all positive t one has
t
J(t - v)^-1vp-1Ep,p(XVp)dV = t^-1Ep,p+,(\tp). (2.6)
0
3. Well-posedness of forward problem (1.1) First we consider the following problem for a homogeneous equation
'Dpty(x, t) - Ay(x,t) = 0, (x,t) e H x (0,T], y(x,t)\dn = 0, (3.1)
y(x, 0) = >-p(x), x e H,
where p(x) is a given function.
Theorem 3.1. Let function p(x) satisfy conditions (2.2). Then problem (3.1) has a unique solution:
<x
y(x,t) = ^ PkEp(-\ktp)vk(x), (3.2)
k=1
where are the Fourier coefficients of function <p(x).
Proof. This theorem for a more general subdiffusion equation was proved in [27]. We only mention the main points of the proof.
Obviously, (3.2) is a formal solution to problem (3.1), see [1], [33]. Let us show that the operators A = —A and Dpt can be applied term-bv-term to series (3.2) and the resulting series
converges uniformly in (x,t) e (H x (0,T])_ If Sj(x,t) is the partial sum of series (3.2), then
j
—ASj(x,t) = ^2 \kWkEp(-Xktp)vk(x). k=1
Using the identity
A-a vk (x) = \-a vk(x),
with a > ^f and applying Lemma 2,1 for g(x) = -ASj(x,t), we have
||- ASj(x,t)\\2cm
Y. h(fikEp(-\ktp)vk(x)
k=1
2 j
^ c£ >Sa+1)IVkEp(-\kn2.
2(Q) k=l
We apply estimates (2,5) to obtain
||- ASj(x,t)\\2c(n) ^ C£
>l(a+1)Wk |2
=1 |1 + Ak tPl2
^ ct-2p £ xkr|^|2, t> 0.
k=l
Therefore, if (x) satisfies conditions (2,2), then -Ay(x,t) E C(Q x (0,T]). From equation (3,1) one has Dpy(x,t) = Ay(x,t), t > 0, and hence we get Dpy(x,t) E C(Q x (0,T]).
The uniqueness of the solution follows from the completeness of the system {vk(x)} in L2(Q), see [20], We only note that it is important here that the derivatives of the solution involved in the equation are continuous up to the boundary of domain Q, see Definition 1,1, Nevertheless, below we give a proof of the uniqueness of a solution of the inverse problem in detail, see the proof of Theorem 4,1, □
Now we consider the following auxiliary initial-boundary value problem: 'Dpu(x,t) - Au(x,t) = f (x)g(t), (x,t) E Q x (0,T], u(x,t)ldn = 0, (3.3)
u(x, 0) = 0, x E Q.
Theorem 3.2. Let f (x) satisfy conditions (2.2) and g(t) E C[0,T]. Then problem (3.3) has a unique solution
(X,t) = ^ fk
k=1
lEp,p(-\krjp)g(t - V)dV
Vk (x).
(3.4)
where fk = (f,Vk)•
Proof. Again, as in the previous theorem, (3.4) is a formal solution to problem (3.3), see [1], [33].
Let Sj(x,t) be the partial sum of series (3.4) and a > ^ Repeating the above reasoning
based on Lemma 2.1 and using the ParsevaPs identity and Lemma 2.5, we arrive at
J_ t
12
||- ASj(x,t) \\2(n)
Y^k fk vp 1Ep,p(—\k Vp)g(t - 'n)d'qvk (x)
k=1
C(Q)
<
<
A-" E K+1 fk VP~ 1Ep,p(-Xkvp)g(t - rj)drjVk(x)
k=1 0 j }
Y,xZ+1fkJ ^lEp,p(-\kvp)g(t - v)dvvk(x) 0
t
K+1fk i Vp- 1Ep,p(-XkVp)g(t - V)dV
C(Q)
k=1
L2(Q)
k=1 < c £
k=1
t
\l+1lfk | max | g(t)lj Vp~ 1Ep,p(-\k Vp)dV 0
2
iC £
k=1
xi+1\fk\ max \9W\tPEP,P+i{-\ktp)
t > 0.
Lemma 2,2 implies
II- AS, (x, t)\\c{n) ^C max \g (t)\f\\
> 0.
Hence, -Au(x, t) G C(Q x (0, T]) and in particular u(x, t) G C(Q x [0, T]). Then from equation (3,3) one has
j
DpSj(x, t) = ASj(x, t) + ^ fkg(t)vk(x), t > 0.
k=i
Therefore, from the above reasoning, we have Dpu(x, t) G C(Q x (0,T]). The uniqueness of the solution follows from the completeness of the system {vk(x)} in L2(Q). □
We proceed to solving main problem (1.1). We note that if y(x, t) and u(x, t) are solutions of problems (3,1) and (3,3), respectively, then the function u(x, t) = y(x, t) + u(x, t) is a solution to problem (1.1). Therefore, we can use the already proven assertions and obtain the following result.
Theorem 3.3. Let <p(x), f(x) satisfy conditions (2.2) and g(t) g C[0,T]. Then problem (1.1) has a unique solution
u(x, t) = y^
k=i
VkEp(-\ktp) + fk riP-1EP,P(-Xk'riP)g(t - v)dv
Vk (x).
(3.5)
4. Well-posedness of inverse problem (1.1), (1.2)
We apply additional condition (1.2) to equation (3.5) and denote by the Fourier coefficients of function ip(x) : ipk = (ip, vk), Then
5>Mto)Vk(x) = ^2^kVk(x) -¿2<PkEp(-\ktPp)Vk(x)
k=i
k=i
k=i
where
h,p(t) = (t - s)P-1Ep,p(-Xk(t - s)P)g(s)ds.
(4.1)
k
fkh,P(to) = i>k - VkEp(-\ktp).
(4.2)
Of course, the case bk,p(t0) = 0 is critical. This can happen when g(t) changes sign. The
( ) k
see also [8].
Example 1. We consider the following homogeneous inverse problem
'Dptu(x, t) - Au(x, t) = f(x)g(t), (x, t) G Q x (0,T], u(x, t^an = 0, u( x, 0) = 0, x G Q, Ux, to) = 0, x G Q.
a
conditions, i.e. —Av = Xv with i>(x)|an = 0 and set t0 = 1, T(t) = tp(l — tb), b > 0, Then, u( x, ) = T( ) ( x)
f(x) = v(x) and g(t) = DpT (t) + XT (t).
Then, besides the trivial solution (u, f) = (0, 0) to problem (4,3), we also have the following non-trivial solution
u( x, ) = T( ) ( x) , ( x) = ( x) .
It can be shown easily that, for example, for the parameters 6 = 0.1 and p = 0.5, the function ( )
pB (p, 1 — p) (b + p) tpB(b + p, 1 — p) bN
9(t) = ~T(r—7) rUT—J) +Xt (1 — t),
and
g(0) = 0.5r(0.5) = ^> 0, ^ n5r(n5) 0.6B(0.6,0.5) A 6r(0.6) < 0
= 05V(0.5) — nv.5) — T(Hy <
We note that g(t) does not belong to C 1[0,T], see Lemma 4,3 below.
Let us divide the set of natural numbers N into two groups K0pP and Kp\ N = Kp U K0p, while the number k is assigned to K0,p, if bk,p(t0) = 0, and if bkp(t0) = 0, then this number is assigned to Kp. ^^^e that for some t0 the set K0,p can be empty, then Kp = N. For example, if g(t) is sign-preserving, then Kp = N, for all t0.
There arises a natural question about the size of set K0>p, i.e., how many elements does K0,p contain? As the authors of paper [6] noted, at least for p = 1, the set K0i1 can contain infinitely many elements. Indeed, in this case
to
bkAU) = J e(t0-s)g(s)ds, 0
and according to Muntz's theorem (see monograph by S, Kaczmarz and H, Steinhouse [34]), the set K0i 1 for some continuous functions g(t) contains infinitely many elements, see also [35], In the case of the diffusion equation, the criterion for the uniqueness of a solution of the inverse problem was studied in the papers cited above [2], [3], [4], [5], [6], This criterion can be formulated as follows: the inverse problem has a unique solution if and only if
bk,i (t 0) = 0. (4.4)
k
solution of the inverse problem for the subdiffusion equation has a similar form:
bk,p(t0) = 0. (4.5)
Let us establish two-sided estimates for bk,p(t0). First we suppose that g(t) does not change sign, for the diffusion equation, i.e. for bki1(10), see Sabitov et al. [3], [4]. Then K0pP is empty.
Lemma 4.1. Let g(t) G C[0,T] and g(t) = 0, t G [0,T]. Then there are constants C0,C1 > 0, depending on 10, such that for all k:
C ^ |bk,p(t0)1 ^ C.
Xk Xk
Proof. By virtue of the Weierstrass theorem, we have lg(t)l ^ g0 = const > 0, We apply the mean value theorem and Lemma 2,5 to obtain
to
| bk, p( to)\ =
Tf- 1EPtP(-XkVP)g(to - v)dv
0
= 19(Ck)|tp0Ep,p+l(-Xktp0), Ck e [0,to].
It is easy to see that
Ep,p+i(-t ) = t~l(1 — Ep(-t)). Therefore, using Lemma 2,3 and the estimate lg(i)| ^ g0 one has
1 C
\bk,p(to)\ = \g(tk)\T(1 -Ep(-Xktp)) > C0.
Xk Xk
Finally Lemma 2,2 implies
\ n( t )i +p nmax \9(0\ c A9k)\to ^ „0S cS t0 . Cl
\bk,p(to)\ S C^op S c-S ^.
□
,p 1 + Xk tp Xk X
k
Theorem 4.1. Let p e (0, l], g(t) e C[0,T] and g(t) = 0, t e [0,T], Moreover, let the function p(x) satisfy condition (2.2) and ip(x) satisfy condition (2.3). Then there exists a unique solution of inverse problem (1.1)-(1.2):
f(x) = Y K \\ - PkEp(-Xktp)] Vk(X), (4.6)
k, p( 0)
k=l
oo
^ ^ b (t) (x, t) = V pkEp(-Xktp)Vk(x) + V [iPk - PkEp(-Xktp0)] Vk(x). (4.7)
=l k, p( 0)
k=l k=
For the diffusion equation (p = l), this theorem is proved only in eases where Q is an interval on R, see [3], or a rectangle in R2, see [4]. This is a new theorem for subdiffusion equations
(pe (0, l)).
Proof. Since bk>p(t0) = 0 for all k e N, we get the following equations from (4.2):
fk = 7-^-, №k — VkEp(—xktpp)] , (4.8)
Ok, p( to)
Uk(t) = PkEp(-Xktp) + №k - <PkEp(-Xktp0)]. (4.9)
k, p( 0)
With these Fourier coefficients, we have the following series for the unknown functions f(x) and u(x, t):
1
f(x) = Y h (+ ) [^k — <fkEp( — Xk tp)] Vk(x) = fk,l + fk,2] Vk(x), (4.10) k=l k,p( 0) k=l ro ^ b (t)
u(x, t) = y~] pkEp(—Xktp) Vk (x) + Y Lk,p(, ) [^k — VkEp(—Xktp)} Vk (x). (4.11) k=l k=l bk,p(to)
If Fj (x) is the partial sums of series (4.10), then applying Lemma 2.1 as above we have
\\A-aFj(x)\\Cm ^ Y^Xk l fk,l + fk,212 ^ 2 YXkfh + 2 Yxkafi,2 = 2h,j + 2hj, (4.12) k=l k=l k=l
where a > 4. Therefore by Lemma 4,1 one has
1 \2a 3 jj
^ ^ E ITATP I" ^ C T,*?2 ^ T = 2a>-, (4.13)
IOkAto)I 2
k=
2 3
3 T7I I \ j_P\ 2 J j
2 / ^ V^ \Ti 12 ~ j
^3 ^ Et X f ^ IK I2 IK I2, r = 2a>j. (4.14)
k=l
Thus, if p(x) satisfies conditions (2.2) and ^(x) satisfies conditions (2.3), then from estimates of Iij and (4.12) we obtain f(x) E C(Q). Further, the feet that function u{x, t) given by (4.11) is a solution to the inverse problem is proved exactly as the proof of Theorem 3.3. Here we also apply Lemma 4.1.
To prove the uniqueness of the solution, we assume the contrary. Let there exist two different solutions {u1, f\} and {u2, f2] satisfying inverse problem (1,1)-(1,2), We need to show that u = u1 — u2 = 0, f = f1 — f2 = 0. For {u, f] we have the following problem:
(Dpu(x, t) — Au(x, t) = f(x)g(t), (x, t) E Q x (0,T],
u(x, t)Ian = 0,
u(x, 0) = 0, x E Q, '
u(x,tQ) = 0, x E Q, t0 E (0,T].
We take any solution {u, f] and define uk = (u, vk) and fk = (f, vk). Then, due to the self-adjointness of the operator —A and the continuity of the derivatives of the solution up to the
Q
Dptuk(t) = (Dpu, vk) = (Au, vk) + fkg(t) = (u, Avk) + fkg(t) = —Xkuk(t) + fkg(t). Therefore, for uk we obtain the Cauchv problem
Dptuk(t) + Xkuk(t) = fkg(t), t > 0, uk(0) = 0, and the additional condition
uk (to) = 0.
If fk is known, then the unique solution of the Cauchv problem has the form
t
uk(t) = fk j 'qp-1Ep,p(—Xk'qp)g(t — 'q)d'q = fkbk,p(t). o
Apply the additional condition to get
uk (t o) = fkbk,P(t o) = 0.
Since bk,p(t0) = 0 for all k E N, then due to completeness of the set of eigenfunetions {vk] in L2(Q), we finally have f(x) = 0 and u(x, t) = 0. □
Now consider the case when g(t) changes sign. In this case, function bktP(t0) can become zero, and as a result, the set K0,p may turn out to be non-empty. Now we should consider separately the case of diffusion (p = 1) and subdiffusion (0 < p < 1) equations.
Lemma 4.2. Let p=1, g(t) E C 1[0,T] and g(t0) = 0. Then there exists a number k0 such that, starting from the number k ^ k0, the following estimates hold:
C ^ Ibk,1 (to)I < C, (4.16)
Xk Xk
where constants C0 and C1 > 0 depend on k0 and t0.
Proof. Bv integrating by parts and the mean value theorem, we get
to
to . to
bk,i(ta) = j e-XkSg(to - s)ds = - 1g(to - s)e-x*S
- -1 f e-XkSg'(to - s)ds 0 Ak
0 0
=1 [g(to) - g(0)e-x^to] + ^ [e-x^to - l], & e [0, to]. Xk \
Therefore, there exists a constant C0 such that the required lower bound holds. The upper estimate follows from the boundedness of function g(t). □
Corollary 4.1. If conditions of Lemma 4-2 are satisfied, then estimate (4-16) holds for all k e Kx.
Corollary 4.2. If conditions of Lemma 4-2 are satisfied, then set K01 has a finite number elements.
In case of subdiifusion equation (p e (0, l)) we have
Lemma 4.3. Let p e (0, l^ g(t) e C1 [0,T] and g(0) = 0. Then there exist numbers m0 > 0 and k0 such that, for all t0 ^ m0 and k ^ k0, the following estimates hold:
C ^ \bk,p(t0)\< C, (4.17)
C0 C1 > 0 m0 0
Proof. Let p e (0, l). Using equality (2,6) we integrate by parts, then apply the mean value theorem. Then we have
to to
bk,p(t0) = j g(t0 - S)sp-1EPiP(-\kSp)ds = J g(t0 - s)d[spEPiP+i(-\kSp)]
00
to
--g(t0 - s)spEp,p+1(-Xksp)
o
+ g'(t0 - s)spEp,p+i(-\kSp)ds
0 j 0
to
=g(0) tp Ep,p+i(-\ktp) + g') j spEPp+i(-\kSp)ds, & e [0, h].
0
For the last integral formula (2,6) implies
to
J spEPpp+i(-\kSp)ds = tpp+1EPp+2(-\ktpp). 0
We apply the asymptotic estimate of the Mittag-Leffler functions (Lemma 2,4) to get
> ,t) g(0) + g'itk), +.J 1 \
bkAt 0 ] = — + —t0 + 0l Txkw).
If g(0) = 0, then for sufficiently small 10 and sufficiently large k we obtain the required lower estimate. This also implies the required upper bound, □
Corollary 4.3. If conditions of Lemma 4-3 are satisfied, then estimate (4-17) holds for all
0 m0 k e Kp
K0,p has a finite number elements.
Theorem 4,1 proves the existence and uniqueness of a solution to the inverse problem (1.1)-(1,2) under condition g(t) E C[0,T^d g(t) = 0, t E [0,T], i.e., g(t) does not change sign. In Example 1, we saw that if this condition is violated, then the uniqueness of the solution to problem (1,1)-(1,2) is violated. This naturally give rise to the questions: if g(t) changes sign, is uniqueness always violated? What can be said about the existence of a solution? How many solutions can there be?
It should be emphasized that the answers to these questions were not known even for the classical diffusion equation (i.e. p = 1),
Lemmas 4.2 and 4.3 proved above allow us to answer these questions. Let us formulate the corresponding result.
Theorem 4.2. Let g(t) E Cl[0,T], function ip(x) satisfy condition (2.2) and p(x) satisfy condition (2.3). Further, we assume that for p = 1 the conditions of Lemma 4-2 are satisfied, and for p E (0,the conditions of Lemma 4-3 are satisfied and to is sufficiently small.
1) If set K0,p is empty, i.e. bk,p(t0) = 0, for all k, then there exists a unique solution of the inverse problem (1.1)-(1.2):
X 1
f(x) = E TT^T P - VkEp(-\ktp0)] vk(x), (4.18)
— Ok,p(to)
k=l
oo oo
u(x, t) = y" VkEp(-\ktp)Vk(x) + V P - VkEp(-\ktp)] Vk(x). (4.19)
k=i k=i bk'P(to)
2) If set K,P is not empty, then for the existence of a solution to the inverse problem, it is necessary and sufficient that the following conditions
pk = ^kEp(-\ktPp), k E K,p, (4.20)
be satisfied. In this case, the solution to the problem (1.1)-(1.2) exists but is not unique:
f(x) = Y b ) [ipk - PkEp(-Xktp)] Vk(x) + ^ fkVk(x), (4.21)
u(x, t) = Y, [fkEp(-\ktp) + fk}vk(x), (4.22)
k=1
where fk, k E K,p, we arbitrary real numbers.
Proof. The proof of the first part of the theorem is completely analogous to the proof of Theorem 4.1. As regards the proof of the second part of the theorem, we note the following. If k E Kp, then again from (4.2) we have (4.8) and (4.9).
If k E K0,p, i.e. bk,p(t0) = 0, then the solution of equation (4.2) with respect to fk exists if and only if the conditions (4.20) are satisfied. In this case, the solution of the equation can be
k
the theorem, the set K0p E (0,1], contains finitely many elements. □
Note that condition (4.20) is rather difficult to verify. Given relation Ep(-1) = 0, t > 0 (see Lemma 2.3), one can replace this condition with a simpler condition.
Remark 4.1. For conditions (4-20) to be satisfied, it suffices that the following orthogonality conditions hold:
Vk = (<p, Vk) = 0, Ak = (ip, Vk) = 0, kE K,p. In other words, if the symbol H0 denotes a subspace of L2(Q) spanned by a linear combination of eigenfunctions vk(x), k E K0,p then in order for conditions (4-20) to be satisfied, it is sufficient that v and p to be orthogonal to H0.
Let us briefly mention some known results on inverse problems for the diffusion equation (i.e., p =1). In work by D.G, Orlovskii [5] abstract diffusion equations in Banaeh and Hilbert spaces were considered. In the case of a Hilbert space, the elliptic part of the equation is self-adjoint, and the found uniqueness criterion is similar to (4,5), A condition on the function bkt1(T) is found, which ensures the existence of a generalized solution (note that here condition (4,5) is 0= T
In I.V, Tikhonov, Yu.S, Eidel'man [6], abstract diffusion equations in Banach and Hilbert spaces are also considered. In the case of a self-adjoint elliptic part, the uniqueness criterion coincides with (4,5), It is shown that if we consider equations in a Banach space, then condition (4,5) is not a criterion, and an addition to (4,5) is found that turns (4,5) into a uniqueness criterion for equations with a non-conjugate elliptic part.
The elliptic part of the diffusion equation in work A.I. Prilepko, A.B, Kostin [2] is a second-order differential expression. Both non-self-adjoint and self-adjoint elliptic parts are considered. In this paper, g(t) also depends on the spatial variable: g(t) := g(x, t). In the case of a self-adjoint elliptic part, the authors succeeded to find a criterion for the uniqueness of the generalized solution of the inverse problem: the solution is unique if and only if the system
to
Wk (x) = vk (x) j g(x, t) e-Xk (to-t]dt, k = 1, 2, ••• 0
is complete in L2(&). It is easy to see that if g(x, t) is independent of x, then this criterion coincides with (4,5), It should be emphasized that the Fourier method is not applicable to the equation considered in this paper.
The closest to our research are works by K.B, Sabitov and A.E, Zaynullov [3] and [4]. We borrowed some ideas from these works. In work [3] the elliptic part of the equation is uxx defined on an interval (in [4] this was the Laplace operator on the rectangle). Having considered the over-determination condition in form (1.2), it is shown that the criterion for the uniqueness of the classical solution is (4.5). When condition (4.5) is satisfied, a classical solution is constructed by the Fourier method. We note that the existence of a classical solution was not discussed in the works listed above.
5. Conclusion
In this paper, we consider the subdiffusion equation with a fractional derivative of order p e (0,1], and take the Laplass operator as the elliptic part. The right-hand side of the
( x) ( ) ( )
( x)
condition is taken in a more general form. It is proved that the criterion for the uniqueness of the classical solution of the inverse problem for the subdiffusion equations coincides with the analogous condition for the diffusion equations.
In the case when this condition is not satisfied, a necessary and sufficient condition for the
existence of a classical solution is found and all solutions of the inverse problem are constructed
( )
[6]) that the set K0>1 can contain infinitely many elements. In Corollaries 4.2 and 4.4, exact
conditions are found that guarantee the finiteness of the number of elements K0p, p e (0,1] for ( )
the classical diffusion equation.
The results of this work can be generalized to more general subdiffusion equations by replacing the Laplace operator in (1.1) with a high-order self-adjoint elliptic operator with variable coefficients. At the same time, instead of the result of V.A. IPin, similar results by Sh.A. Alimov [36] should be used for a general elliptic operator.
6. Acknowledgement The authors are grateful to Sh.A, Alimov for discussions of these results.
СПИСОК ЛИТЕРАТУРЫ
1. A.V Pskhu. Fractional Differential Equations. Nauka, Moscow (2005) fin Russian].
2. A.I. Prilepko, A.B. Kostin. On certain inverse problems for parabolic equations with final and integral observation // Mat. Sb. 183:4, 49-68 (1992). [Russ. Acad. Sci. Sb. Math. 75:2, 473-490 (1993).]
3. K.B. Sabitov, A.R. Zavnullov. On the theory of the known inverse problems for the heat transfer equation // Uchenve Zapiski Kazanskogo Univ. Ser. Fiz.-Matem. Nauki. 161:2, 274-291 (2019). (in Russian).
4. K.B. Sabitov, A.R. Zavnullov. Inverse problems for a two-dimensional heat equation with unknown right-hand side // Russian Math. 65:3, 75-88 (2021).
5. D.G. Orlovskii. Determination of a parameter of an evolution equation // Differ. Uravn. 26:9, 1614-1621 (1990). [Diff. Equat. 26:9, 1201-1207 (1990).]
6. I.V. Tikhonov, Yu.S. Eidel'man. Uniqueness criterion in an inverse problem for an abstract differential equation with nonstationary inhomogeneous term // Matem. Zamet. 77:2, 273-290 (2005). [Math. Notes. 77:2, 246-262 (2005).]
7. M. Slodichka. Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation // Frac. Calc. Appl. Anal. 23:6, 17031711 (2020).
8. M. Slodichka, K. Sishskova, V. Bockstal. Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation // Appl. Math. Lett. 91, 15-21 (2019).
9. O.A. Ladvzhenskava. Mixed problem for a hyperbolic equation. Gostexizdat, Moscow (1953). (in Russian.)
10. Y. Liu, Z. Li, M. Yamamoto. Inverse problems of determining sources of the fractional partial differential equations //in "Fractional Differential Equations. V. 2." Eds. A. Kochubei, Yu. Luchko, De Gruvter. 411-430 (2019).
11. R. Ashurov, M. Shakarova. Time-dependent source identification problem for fractional Schrodinger type equations // Lobachevskii J. Math. 42:3, 517-525 (2022).
12. R. Ashurov, M. Shakarova. Time-dependent source identification problem for a fractional Schrodinger equation with the Riemann-Liouville derivative // Ukrainian Math. J. 75, 997-1015 (2023).
13. K. Sakamoto, M. Yamamoto. Initial value boundary value problems for fractional diffusion-wave equations and applications to some inverse problems // J. Math. Anal. Appl. 382:1, 426-447 (2011).
14. K. Furati, O. Iviola, M. Kirane. An inverse problem for a generalized fractional diffusion // Appl. Math. Сотр. 249, 24-31 (2014).
15. M. Kirane, A. Malik. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time // Appl. Math. Сотр. 218, 163-170 (2011).
16. M. Kirane, B. Samet, B. Torebek. Determination of an unknown source term and the temperature distribution for the subdiffusion equation at the initial and final data // Electr. J. Diff. Equat. 217, 1-13 (2017).
17. Z. Li, Y. Liu, M. Yamamoto. Initial-boundary value problem for multi-term time-fractional diffusion equation with positive constant coefficients // Appl. Math. Сотр. 257, 381-397 (2015).
18. S. Malik, S. Aziz. An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions // Сотр. Math. Appl. 73:12, 2548-2560 (2017).
19. M. Ruzhanskv, N. Tokmagambetov, B.T. Torebek. Inverse source problems for positive operators I: Hypoelliptic diffusion and subdiffusion equations //J. Inverse Ill-Possed Probl. 27:6, 891-911 (2019).
20. R.R. Ashurov, A.T. Mukhiddinova. Inverse problem of determining the heat source density for the subdijfusion equation // Diff. Equat. 56:12, 1550-1563 (2020).
21. R.R. Ashurov, Yu.E. Favziev. Determination of fractional order and source term in a fractional subdijfusion equation // Eurasian Math. J. 13:1, 19-31 (2022).
22. K.B. Sabitov. Direct and inverse problems for of mixed type parabolic-hyperbolic equations. Nauka, Moscow (2016) fin Russian].
23. S.I. Kabanikhin. Inverse and Ill-Posed Problems. Theory and Applications. De Gruvter, Berlin (2011).
24. V.E. Fedorov, A.V. Nagumanova. Inverse problem for evolutionary equation with the Gerasimov-Caputo fractional derivative in the sectorial case // The Bulletin of Irkutsk State University, Series Mathematics. 28, 123-137 (2019). (in Russian).
25. S. Liu, F. Sun, L. Feng. Regularization of inverse source problem for fractional diffusion equation with Riemann-Liouville derivative // Comp. Appl. Math. 40, id 112 (2021).
26. P. Niu, T. Helin, Z. Zhang. An inverse random source problem in a stochastic fractional diffusion equation // Inver. Probl. 36:4, id 045002 (2020).
27. R. Ashurov, A. Mukhiddinova. Initial-boundary value problem for a time-fractional subdijfusion equation with an arbitrary elliptic differential operator // Lobachevskii J. Math. 42:3, 517-525 (2021).
28. M.A. Krasnoselskii, P.P Zabrevko, E.I. Pustilnik, P.E. Sobolevski. Integral operators in the spaces of integrable functions. Nauka, Moscow (1966) (in Russian).
29. V.A. Il'in. On the solvability of mixed problems for hyperbolic and parabolic equations // Russ. Math. Surv. 15:2, 97-154 (1960).
30. M.M. Dzherbashian. Integral Transforms and Representation of Functions in the Complex Domain. Nauka, Moscow (1966) (in Russian).
31. R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogozin. Mittag-Ieffter Functions, Related Topics and Applications. Springer, Berlin (2014).
32. R. Ashurov, R. Zunnunov. Intial-boundary value and inverse problems for subdijfusion equations in RN // Fract. Diíf. Calc. 10:2, 291-306 (2020).
33. R. Ashurov, A. Cabada, B. Turmetov. Operator method for construction of solutions of linear fractional differential equations with constant coefficients // Fract. Calc. Appl. Anal. 1, 229-252 (2016).
34. S. Kaczmarz, H. Steinhaus. Theorie der Orthogonalreihen, Seminarium Matematvczne Uniwer-svtetu Warszawskiego, Warszawa (1935).
35. A.F. Leont'ev. Exponential series, Nauka, Moscow (1976). (in Russian).
36. Sh.A. Alimov, V.A. Il'in, E.M. Nikishin. Convergence problems of multiple trigonometrical series and spectral decompositions. /// Russ. Math. Surv. 31:6, 28-83 (1976).
Ravshan Radjabovich Ashurov,
Institute of Mathematics, Uzbekistan Academy of Science,
Student Town str,,
100174, Tashkent, Uzbekistan
University of Tashkent for Applied Sciences,
Gavhar str. 1,
100149, Tashkent, Uzbekistan E-mail: ashurovr@gmail.com
Marjona Dilshod qizi Shakarova,
Institute of Mathematics, Uzbekistan Academy of Science, Student Town str., 100174, Tashkent, Uzbekistan E-mail: shakarova2104@gmail. com
