Обратная задача об источнике для уравнения смешанного параболо-гиперболического типа с дробной производной по времени в цилиндрической области
- Авторы: Дурдиев Д.К.1,2
-
Учреждения:
- Бухарское отделение Института математики им. В. И. Романовского АН Республики Узбекистан
- Бухарский государственный университет
- Выпуск: Том 26, № 2 (2022)
- Страницы: 355-367
- Раздел: Краткие сообщения
- Статья получена: 25.04.2022
- Статья одобрена: 02.08.2022
- Статья опубликована: 30.06.2022
- URL: https://journals.eco-vector.com/1991-8615/article/view/106724
- DOI: https://doi.org/10.14498/vsgtu1921
- ID: 106724
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Аннотация
Исследуется обратная задача об источнике для уравнения смешанного типа с дробным уравнением диффузии в параболической части и волновым уравнением в гиперболической части цилиндрической области. Решение задачи получено в виде ряда Фурье–Бесселя с использованием ортогонального множества функций Бесселя. Доказаны теоремы единственности и существования решения.
Полный текст
1. Formulation of Problem
The importance of considering equations of mixed type, when an equation of parabolic type is given on one part of the domain and an equation of hyperbolic type on the other, was first pointed out by I. M. Gelfand in 1959 [1]. The study of electrical oscillations in wires leads to a problem for a mixed parabolic-hyperbolic type of equations. In a homogeneous medium, in the case of its low conductivity, the strength of the electromagnetic field satisfies the wave equation, but in the case of relatively high conductivity, when displacement currents can be neglected in comparison with conduction currents, the mentioned value satisfies the heat equation (see [2, pp. 443-447]). Problems of this kind are also encountered in studying the motion of a fluid in a channel surrounded by a porous medium; so, in a channel, the hydrodynamic pressure of a liquid satisfies the wave equation, and in a porous medium it satisfies the filtration equation, which in this case coincides with the diffusion equation [3]. In this case, some matching conditions are satisfied at the channel boundary. Such equations arise in a number of other areas of natural science.
Direct problems for mixed parabolic-hyperbolic equation types were studied in [4–8]. Inverse problems about determining the right side or the initial function in the initial-boundary value problems for the equation of mixed parabolic-hyperbolic type in a rectangular domain were considered in the monograph [9] (see also references there). On the basis of the spectral method, criteria for uniqueness and existence are established.
In this paper, we study direct and inverse problems related to finding a solution to an initial-boundary value problem for a mixed equation, when on one part of the domain the fractional diffusion equation and on another part the wave equation are given, and the unknown right-hand side of this equation in a cylindrical domain.
Consider in a cylinder , the equation of mixed type
(1)
where , are given positive numbers, is the Gerasimov–Caputo fractional derivative of order ( ) in the time variable and it is defined by formula (see [10, p. 90]):
is the Laplace operator in variables and .
We pose the following problem: find in the domain the functions and satisfying the equation (1) and conditions
(2)
(3)
(4)
Here is scalar product of vectors and ; and are given sufficiently smooth functions.
Denote , .
Definition 1. The solution of problem (1)–(4) are the functions and from the classes and respectively, satisfying relations (1)–(4) and the following conjugation conditions:
(5)
Here
If , then conditions (5) mean the continuity of the solution and its derivative with respect to on the line of change of equation type .
In the parabolic part of the domain, the function satisfies the fractional diffusion equation (1). Fractional differential equations become an important tool in mathematical modeling many problems arising in applications. The time fractional diffusion equations can be used to describe superdiffusion and subdiffusion phenomena [11–13] (see also references there). Direct problems, i.e. well-posed initial value problems (Cauchy problem), initial boundary value problems for one time-fractional diffusion equations and various inverse problems, have attracted much more attention in recent years. For instance, on some uniqueness and existence results we refer readers to works [14–17] on direct and inverse source problems (see also references in [17]), and on direct and inverse coefficient problems to [18–23].
The paper organized as follows. Section 2 provides some definitions and known results that will be used later in this article. In Section 3, by using the Fourier method a formal solution of the inverse problem is obtained. In Section 4, the existence and uniqueness of a solution to the inverse problem are proved. Finally, a conclusion and a list of references are given.
2. Preliminaries
In this section, we provide some definitions and results that will be used later in this article.
The classical Mittag–Leffler function with one parameter is defined by the following series:
where , with . This function and its generalizations play an important role in describing solutions to fractional-order differential equations. The Mittag–Leffler function has been studied by many authors who have proposed and studied various generalizations and applications. A very interesting work that has received many results on this function is due to Haubold et al. [24].
If , with and , then
Moreover, The Mittag–Leffler function is bounded [24]:
(6)
Here and throughout this article, denotes a positive constant.
In studying the problem under consideration, we also need the Bessel function and the conditions for the convergence of the Fourier–Bessel series. The linear differential Bessel equation (or the equation of cylindrical functions) with a parameter of order or index with respect to the function of the real variable has the form [25, ch. 8]:
(7)
The solution of Equation (7), except for very particular values , is not expressed in terms of elementary functions (in the final form) and leads to the so-called Bessel functions, which have large applications in the natural sciences [26]. When is an integer number, then Equation (7) has the following solution:
where and are the Bessel functions of the first and second kind of order , respectively. Bessel functions of the second kind are not bounded near the point , so for a bounded solution near zero it is necessary that , i.e. solution (7) has the following form:
Furthermore, if the boundary condition is imposed, then the parameter must satisfy , i.e. the values of are the zeros of the Bessel function , which has the following asymptotic representation [25, p. 213]:
where the function is bounded for . Therefore, for any large , the zeros of are given by the expression [25, p. 214]:
We define the Fourier–Bessel expansion of the given function as follows: for any function , absolutely integrable on , one can compose a Fourier series in the system , or, in briefly, the Fourier–Bessel series
(8)
where the constants are determined by the formula:
and are called the Fourier–Bessel coefficients.
Let us give without proof the most important criteria for the convergence of the Fourier–Bessel series to the function for which it is composed. These criteria are similar to those known to us for the convergence of trigonometric Fourier series.
Theorem 1. [25, p. 282]. If and for all sufficiently large we have the estimate
where and are constants, then series (8) converges absolutely and uniformly on .
Theorem 2. [25, pp. 289-291]. Let the function is defined and times continuously differentiable on the interval and
- is bounded (this derivative may not exist at some points),
Then, for the Fourier–Bessel coefficients of the function the inequality is valid:
We now turn to the study of the problem (1)–(4).
3. Formal Construction of the Solution
Note that since the right-hand side of equation (4) and the functions of (6) and (7) depend on the distance , then , i.e. we have an axisymmetric case. Then the operator Laplace on the function in polar coordinate systems will not depend on the angle and has the form:
Therefore, equation (4) in these coordinate systems is written as follows:
(9)
Conditions (2)–(4) take the following form:
(10)
(11)
(12)
Thus, the inverse problem (2)–(4) is reduced to the problem definitions of the functions , from equalities (9)–(12).
According to the Fourier method, searching partial solutions of equation (9) for the case in form
,
we get the following relations:
Therefore, separating the variables, we have
where is an arbitrary real parameter. Hence, to find the function we get the problem of the equation
with boundary conditions
(13)
which is a self-adjoint problem.
The solutions of equation (10) are the following zero-order Bessel functions of the first kind:
They also are eigenfunctions. We find the eigenvalues using the second boundary condition of (13) (the validity of the first boundary condition in (13) is obvious), positive roots of the equation . As noted in the previous section, they look like:
Expand now all functions in a Fourier–Bessel series in terms of eigenfunctions i.e.
(14)
(15)
where
Substituting (14), (15) into (9), we obtain
It is not difficult to find that these differential equations have general solutions:
(16)
where is the Mittag–Leffler function; , , are arbitrary constants.
To find the coefficients , , we use conditions
which follow from conditions (5). In view of this, from (16) we have
From the initial and additional conditions (11), (12), we get:
where , are Fourier–Bessel coefficients of functions , , respectively:
Substituting the values found through into the previous equations and solving the resulting system with respect to and we find
(17)
Introduce the notation
(18)
4. Existence and Uniqueness of the Solution
We find the values of and for which (18) takes values not equal to zero. To do this, we rewrite (18) in the following form:
(18')
where Obtain the values of for which It equals
We now find the values of and for which the following condition is met:
(19)
For this, we calculate
If , , then
According to (6) we have for all then As , it can be taken as the largest of all possible such constants.
Thus, we have obtained the following uniqueness criterion:
Theorem A. If there exists a solution to problem (1)–(4), then it is unique for the values for any
We now investigate the existence of a solution. To this end, we prove the following assertion:
Theorem B. Assume that and, in addition, condition (19) and the equalities
are satisfied.
Then there is a unique solution to problem (1)–(4), which is defined (20)–(22), where are th derivatives of the functions and are the Fourier–Bessel coefficients of the functions and respectively.
To prove the theorem, substituting the found values of the coefficients in (16), (17), we find and :
Taking into account these relations, from (14) and (15) we obtain the formal solution of problem in the form of series:
(20)
(21)
(22)
To prove the existence of a solution, we need to show that the series in (20)–(22) and the series obtained as a result of the action of fractional differentiation , by differentiating with respect to twice, in domain and by differentiating twice in in domain converge uniformly. To this end, we calculate by formally performing differentiation under the signs of sums. Using properties of the Bessel functions, namely (see [24]) , from formulas (20), (21) we obtain the following:
(23)
(24)
(25)
(26)
Let the functions and satisfy the conditions of Theorem 2 with some (we define the number later). Then for the Fourier–Bessel coefficients of these functions are true the following estimates:
Now we will evaluate the expressions at Bessel functions on the right-hand sides of equalities (20)–(26). In this case, the expressions in (20), (21) are estimated as follows:
,
where are positive constants.
Similarly, it is established that the expressions in (22), (23), (25) are less than and the expressions in (24), (26) are less than are positive constants.
It follows from these estimates that if then, according to Theorem 1, the series in (23)–(26) and the series obtained as a result of the action of fractional differentiation , by differentiating with respect to twice, in domain and by differentiating twice in in domain converge uniformly. Thus, Theorem B is proved.
5. Conclusion
This paper concerns the existence and uniqueness of a solution to the inverse source problem for a mixed-type equation with a fractional diffusion equation in the parabolic part and a wave equation in the hyperbolic part of a cylindrical domain. The solution is obtained in the form of Fourier-Bessel series expansion using an orthogonal set of Bessel functions.
Competing interests. I declare that I have no competing interests.
Authors’ contributions and responsibilities. I take full responsibility for submitting the final manuscript for printing. I approve the final version of the manuscript. FundingNot applicable.
Об авторах
Дурдимурод Каландарович Дурдиев
Бухарское отделение Института математики им. В. И. Романовского АН Республики Узбекистан; Бухарский государственный университет
Автор, ответственный за переписку.
Email: durdiev65@mail.ru
ORCID iD: 0000-0002-6054-2827
Scopus Author ID: 16411517300
http://www.mathnet.ru/person29112
доктор физико-математических наук, профессор, заведующий отделением, проф. кафедры дифференциальных уравнений
Узбекистан, 705018, Бухара, ул. Мухаммад Икбол, 11; 705018, Бухара, ул. Мухаммад Икбол, 11Список литературы
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