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0ВЕСТНИК ОШСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА МАТЕМАТИКА, ФИЗИКА, ТЕХНИКА. 2023, №1
УДК 517
https://doi.org/10.52754/16948645 2023 1 250
INVERSE PROBLEMS FOR FRACTIONAL SCHRODINGER AND SUBDIFFUSION EQUATIONS
Ashurov Ravshan Radjabovich, Dr Sc., professor,
ashurovr@gmail.com Shakarova Marjona Dilshod qizi. shakarova2104@gmail.com Institute of Mathematics named after V.I. Romanovsky,
Tashkent, Uzbekistan
Abstract. The inverse problems of determining the right-hand side of the Schrodinger and the sub-diffusion equations with the fractional derivative is considered. In the problem 1, the time-dependent source identification problem for the Schrodinger equation , in a Hilbert space H is investigated. To solve this inverse problem, we take the additional condition B[u(-, t)] = y(t) with an arbitrary bounded linear functional B . In the problem 2, we consider the subdiffusion equation with a fractional derivative of order p e (0,1], and take the abstract operator as the elliptic part. The right-hand side of the equation has the form g (t) f, where g (t) is a given function and the inverse problem of determining element f is considered. The condition u(t0) = у is taken as the overdetermination condition, where t0 is some interior point of the considering domain and у is a given element.
Obtained results are new even for classical diffusion equations. Existence and uniqueness theorems for the solutions to the problems under consideration are proved.
Key words: Schrodinger and subdiffusion equation, equation, the Caputo derivatives, Fourier method.
ОБРАТНЫЕ ЗАДАЧИ ДЛЯ ДРОБНЫХ УРАВНЕНИЙ ШРЕДИНГЕРА И
СУБДИФФУЗИИ
Ашуров Равшан Раджабович, д.ф.-м.н., профессор,
ashurovr@gmail.com Шакарова Маржона Дилшод щзи.
shakarova2104@gmail.com Институт математики имени В.И. Романовский,
Ташкент, Узбекистан
Аннотация. Рассмотрены обратные задачи определения правой части уравнения Шредингера и уравнения субдиффузии с дробной производной. В задаче 1 исследуется нестационарная задача идентификации источника для уравнения Шрёдингера, в гильбертовом пространстве H . Для решения этой обратной задачи возьмем дополнительное условие B[u(-, t)] = y(t) с произвольным ограниченным линейным функционалом B . В задаче 2 мы рассматриваем уравнение субдиффузии с дробной производной порядка p e (0,1], а в качестве эллиптической части берем абстрактный оператор. Правая часть уравнения имеет вид g(t) f, где g (t) - заданная функция и рассматривается обратная задача определения элемента f . В качестве условия переопределенности принимается условие u (t0) = у, где t0 - некоторая внутренняя точка рассматриваемой области, у - заданный элемент. Полученные результаты являются новыми даже для классических уравнений диффузии. Доказаны теоремы существования и единственности решений рассматриваемых задач.
Ключевые слова: уравнение Шредингера и субдиффузии, уравнение, производные Капуто, метод
Фурье.
Introduction
The fractional integration of order < < 0 of the function h(t) defined on [0, да) has the form (see, [1]):
J°h(t) = dt > 0,
provided the right-hand side exists. Here is Euler's gamma function. Using this definition one can define the Caputo fractional derivative of order p ,
Dph(t) = Jp-1 dh(t). dt
If we first integrate and then differentiate, then we get the Riemann-Liouville derivative.
Let H be a separable Hilbert space. Let A: H ^ H be an arbitrary unbounded positive selfadjoint operator in H .
Let t be an arbitrary real number. We introduce the power of operator A , acting in H according to the rule
ATh = ^Athkvk.
k=1
Obviously, the domain of definition of this operator has the form
TO
D( At) = {h e H : ^^W \2 <to}.
k=1
For elements h e D(At) we introduce the norm:
to
\\h\\T =Z^T \hk \2 = \\ ATh\\2
and together with this norm D (A) turns into a Hilbert space.
Problem 1. Let p e (0,1) be a fixed number. Consider the following Cauchy problem
\iD?u(t) + Au(t) = p(t)q + f (t), 0 < t < T, \u ( 0 ) = p,
where a part of the source function p(t) is a scalar function, f (t) e C(H) and p, q e H are known elements of H .
To solve this time-dependent source identification problem one needs an extra condition. Following the papers of A. Ashyralyev et al. [2] we consider the additional condition in a rather general form:
B[u(t)] = w(t), 0 < t < T, (1.2)
where B: H ^ R is a given bounded linear functional, and y(t) is the given scalar function. We call the Cauchy problem (1.1) together with additional condition (1.2) the inverse problem. Problem 2. Let p e (0,1] be a fixed number. Consider the Cauchy problem
rDpu(t) + Au(t) = g(t) f, 0 < t < T,
^ (13)
u ( 0 ) = p.
Here p, f e H are known elements of H and a part of the source function g(t) is a scalar function.
To solve the inverse problem of determining the right-hand side of the equation, we use the following additional condition:
u (O = W, (14)
where t is a given fixed point of the segment (0,T] .
Main results for the Problem 1 Theorem 1. Let Bq^O , q>eH and D/VO) e C[0, T]. Further, let e e (0,1) be any fixed
number and qGD(A1+e) and /(i)eC([0,r];Z)(/)). Then the inverse problem has a unique solution {u(t), p(t)} .
Theorem 2. Let assumptions of Theorem 1 be satisfied and let p e D(A). Then the solution to the inverst problem obeys the stability estimate
II Dfu lie(H) + 11 Au ||C(H) +II pll
where C Bf is a constant, depending only on p,q , B and e .
Similar results hold for the Riemann-Liouville fractional derivative. Main results for the Problem 2
Lemma 1. Let p = 1, g(t) e e*[0,T] and g(t0) * 0 . Then there exists a number k0 such that, starting from the number k > k0, the following estimates hold:
e e
—0 <| b (t) l< eL - <1 bk ,1(t 0)|< - ,
Ak Ak
where
t0
bk,i(to) = fe~4 sg(to -s)ds
0
and constants eo and Q > 0 depend on k0 and t0 .
Lemma 2. Let p e (0,1), g(t) e e*[0,T] and g(0) * 0 . Then there exist numbers m0 > 0 and k0 such that, for all < m0 and k > k0, the following estimates hold:
e e
<| b (t) |< -1
* <| bk,p(t0)|< - ,
where
t0
bk,p(t0) = f g (t0 - s)sPXp(-VP)ds 0
and constants e and e > 0 depend on m and k .
Let N = .K" , where N is the set of all natural numbers. K and K() p are sets such
that: if b^ (i0) * 0, k e K , otherwise, if b^ (i0) = 0, then k e K0 .
Theorem 3. Let pe (0,1], p eH , e D(A) . Moreover let function g(t) e e[0,T] and g(t) * 0, t e [0, T]. Then there exists a unique solution of the inverse problem (1.3)-(1.4).
Theorem 3 proves the existence and uniqueness of a solution to the inverse problem (1.3)-(1.4) under condition g(t) e e[0,T] and g(t) * 0, t e [0, T], i.e., g(t) does not change sign. Article [3], Example 3.1, shows the non-uniqueness result if g(t) changes its sign. It is proved that if function g (t) does not change sign, then the solution of the inverse problem is unique. Naturally, questions arise: 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 1 and 2 proved above allow us to answer these questions. Let us formulate the corresponding result.
Theorem 4. Let p e H , ye D (A) . Further, we will assume that for p = 1 the conditions of Lemma 1 are satisfied, and for pe (0,1), the conditions of Lemma 2 are satisfied and t0 is sufficiently small. If set K0p is empty, i.e. ^ (70) ^ 0, for all k, then there exists a unique solution of the inverse problem (1.3)-(1.4). If set K0 is not empty, then for the existence of a solution to the inverse problem, it is necessary and sufficient that the following conditions
= PkEp (-VoX k e K0,p, (3.1)
be satisfied. In this case, the solution to the problem (1.3)-(1.4) exists, but is not unique.
Remark. For conditions (3.1) to be satisfied, it suffices that the following orthogonality conditions hold:
Pk = vk) = 0,¥k = vk) = 0k e Ko,p-References
1. Pskhu, A.V. Fractional Differential Equations. ^xt]/ Pskhu A.V. // Moscow: NAUKA. 2005 [in Russian].
2. Ashyralyev, A. Time-dependent source identification problem for the Schrodinger equation with nonlocal boundary conditions, ^xt]/ Ashyralyev.A., Urun.M. // In: AIP Conf. Proc, V:2183, 2019.
3. Slodichka, M. Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation^xt]/ Slodichka M., Sishskova K., Bockstal V. //Appl. Math. Letters, V. 91, pp. 15-21, 2019.
