Abstract
In the present work, we study properties of some integrodifferential operators of the HadamardMarchaud type in the class of harmonic functions. As an application of these properties, we consider the question of the solvability of a nonlocal boundary value problem for the Laplace equation in the unit ball.
MSC: 35J05, 35J25, 26A33.
Keywords:
HadamardMarchaud operator; fractional derivative; nonlocal problem1 Introduction
Let
If we prescribe the values of the solution at the boundary ∂Ω of Ω, then we can solve equation (1) uniquely. Of course, one can consider many other boundary conditions such as Neumann’s boundary conditions.
In some applied problems of hydrodynamics [1], it is necessary to prescribe the value of a fractional derivative of the solution on the boundary. Fractional differential equations and boundary value problems involving fractional derivatives appear in many applied problems ranging from the springpot model [2] to geology [3] or from nonlinear circuits [4] to alternative models to differential equations [5].
Hence, in this paper we study the Laplace equation concentrating on some conditions on the boundary involving derivatives of fractional order.
Note that numerous works of authors [613] were dedicated to the research questions of the solvability of boundary value problems for partial differential equations with boundary operators of high (whole and fractional) order. In the paper of A.N. Tikhonov [14] boundary value problems with boundary conditions containing derivatives of higher order have been investigated for the heat equation. Research questions as regards the solvability of similar problems for higherorder equations with boundary operators of whole and fractional order were carried out in [15,16]. Later in [17], these results were generalized for partial differential equations of fractional order. In [1820] questions about the solvability of boundary value problems with boundary operators of high order were studied for the Laplace equation. In the studies of these authors the exact conditions for the solvability have been established and the integral representations of solutions of the studied problems have been found. The cycle of studies by the authors [2126] is devoted to the study of the existence and smoothness of solutions of boundary value problems for the secondorder elliptic equations with boundary operators of fractional order. In the paper mentioned above local boundary value problems with boundary operators of fractional order in the RiemannLiouville or Caputo sense are studied. In this paper we study nonlocal problems with boundary operators of fractionalorder derivatives of Hadamard type. Definitions of Hadamard operators, a statement of the main problems, and the history of the questions on this topic are in Section 3.
The organization of this paper is as follows. In Section 2, we present the operators of integration and differentiation in the Hadamard sense and some modifications. In the third section we provide a formulation of the basic problem of this paper and some historical information as regards nonlocal boundary value problems. In the fourth section we study the properties of integral and differential HadamardMarchaud operators in the class of harmonic functions in the ball. In Section 5 we provide some auxiliary propositions. Finally, Section 6 is devoted to the study of the fundamental problem, where we formulate and prove the main statement of the paper.
2 Definition of Hadamard operators of integration and differentiation and some modifications
In this section, we give a statement on the operators of fractional differentiation in the sense of Hadamard, HadamardMarchaud, and their modifications.
For any positive α, fractional integrals and derivatives of the order α in the sense of Hadamard are defined by the following formulas [27]:
where
If
This operator is said to be the differentiation operator of order α in the sense of HadamardMarchaud.
In [28], the following modification of the HadamardMarchaud operator was considered:
In [18], in the class of harmonic functions in a ball, the properties and applications of the operators in the form of
are considered. Here
Let
Introduce the operators
If
3 Statement of the problem
Let
We assume that the series (7) converges uniformly on ∂Ω.
Further, let
Consider the following boundary value problem:
where
A harmonic function
The abovementioned problem is a simple generalization of BitsadzeSamarskii’s nonlocal problem [29]. For convenience of the reader, we formulate the BitsadzeSamarskii problem.
Let D be a finite simplyconnected domain of the plane of complex variables
We denote by
Formulation of the problem: We are to find a harmonic function
where
Similar problems with operators of integer order were considered in [3032], and for operators of fractional order with fractionalorder derivatives in the sense of RiemannLiouville and Caputo in [3341]. It should also be noted that some questions of solvability of nonlocal problems for fractionalorder equations in the onedimensional case were studied in [4244].
4 Properties of operators
J
μ
α
and
D
μ
α
In this section, we study some properties of the operators
Lemma 1Let
Proof Let
The value of the last integral can easily be calculated with the help of the change
of variables
The equality (10) is proved.
Further, note that the relation
holds for the operator
Now, let us study actions of the operator
Denoting
After the change of variables
which implies
Further, taking into account fulfilling of equality (12), we obtain in the general
case for
The lemma is proved. □
Lemma 2Let
Proof Let
where
Now let us check convergence of the series (13) and (14). The following asymptotical
estimate is valid for
Moreover, the series (13) converges absolutely and uniformly by x at
Therefore, the series (14) converges absolutely and uniformly by x at
Let us study the function
Convergence of this series can be checked as in the case of series (14), and that
is why
Now we show that the function
Lemma 3Let
is valid.
Proof Let
Further, taking into account equalities (10)(11), and the absolute and uniform convergence
of the series (15) by x at
The lemma is proved. □
One can similarly prove the following lemma.
Lemma 4Let
is valid.
Lemma 5Let
Proof Let
By virtue of Lemma 3, the value of the last integral is equal to
To prove the second equality, apply the operator
Then, in the general case,
The lemma is proved. □
5 Some auxiliary propositions
Let
Consider the following problem in the domain Ω:
where
A harmonic function
It should be noted that problem (17)(18) was investigated for the case of
Let us investigate uniqueness for the solution of problem (17)(18). The following statement holds.
Lemma 6Let
and let a solution of problem (17)(18) exist.
Then:
(1) If
then the solution of problem (17)(18) is unique.
(2) If
then the solution of problem (17)(18) is unique up to a constant summand.
Proof Let
Denote
Then if
The boundary condition (18) at
Further, since
Hence,
If now condition (19) is realized , then
Hence, if condition (19) holds, it is necessary that
The last equality is equivalent to the equality
We obtain from this the result that either
Thus, if conditions (19) and (20) are fulfilled, we obtain
If the conditions (21) are fulfilled, then any constant is a solution of the homogeneous
problem (17)(18). In fact, substituting
The lemma is proved. □
Now investigate existence of a solution of problem (17)(18). Let
Introduce the function
and consider the equation
The following statement holds.
Lemma 7Let
(1) If the condition (20) is realized, then problem (17)(18) is uniquely solvable at any
(2) If the condition (21) is realized, then problem (17)(18) is solvable if the following condition is realized:
where the function
Proof Since
Designate
Then equation (25) can be rewritten in the form of
To investigate the solvability of the integral equation (26), we study the properties
of the kernel
In fact, since
Hence, one can apply Fredholm theory to equation (26). Since in the case of
Hence, for any
If the condition (21) is valid, then
6 Study of the basic problem
We now formulate the basic statement.
Theorem 1Let
(1) If the condition (20) is fulfilled, then problem (8)(9) is uniquely solvable at any
(2) If the condition (21) is fulfilled, then the condition (24) is necessary and sufficient for solvability of problem (8)(9) where the function
(3) If a solution of problem (8)(9) exists, then it is represented in the form of
Proof (1) Let a solution
Further, since according to Lemma 5 the equality
with respect to the function
In addition, since
Now, let the conditions (19) and (20) be realized. Then by Lemmas 6 and 7, for any
The first statement of the theorem is proved.
(2) Let now the condition (21) be fulfilled, and let the solution
We show that if the equality (21) is fulfilled, then the condition (24) is also sufficient for the existence of the solution of problem (8)(9).
In fact, if the conditions (21) and (24) are realized, a solution of problem (17)(18)
exists, is unique up to constant summand, and
Remark 1 One can show that in the case of
Example 1 Let
Then
and
Hence, for
If the dimension of the space
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
This work has been supported by the MON Republic of Kazakhstan under Research Grant No. 0713/GF.
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