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Journal of Mathematics
Volume 2018, Article ID 5395124, 29 pages
https://doi.org/10.1155/2018/5395124
Research Article

Boundary Value Problems Governed by Superdiffusion in the Right Angle: Existence and Regularity

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, G. Batyuka St. 19, 84100 Sloviansk, Ukraine

Correspondence should be addressed to Nataliya Vasylyeva; moc.oohay@v_yilatan

Received 21 June 2018; Accepted 5 September 2018; Published 2 December 2018

Academic Editor: Reza Ezzati

Copyright © 2018 Ramzet Dzhafarov and Nataliya Vasylyeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For , we analyze a stationary superdiffusion equation in the right angle in the unknown : where is the Caputo fractional derivative. The classical solvability in the weighted fractional Hölder classes of the associated boundary problems is addressed.

1. Introduction

Fractional partial differential equations (FPDE) play a key role in the description of the so-called anomalous phenomena in nature and in the theory of complex systems (see, e.g., [1]). In particular, these equations provide a more faithful representation of the long-memory and nonlocal dependence of many anomalous processes. The signature of an anomalous diffusion species scales as a nonlinear power law in time; i.e., , . When , this is referred to as superdiffusion.

Superdiffusion is used in modelling turbulent flow [2, 3], chaotic dynamics of classical conservative systems [4], model solute transport in underground aquifers [57], and rivers [810], biophysics [11], and physical and chemical models described by the Lévy processes [12, 13]. In the present paper, we focus on the boundary value problems to the stationary superdiffusion equation.

Let be the first quarter with a boundary , , . For a fixed , we consider the linear equation in the unknown function ,subject either to the Dirichlet boundary condition (DBC),or to the Neumann boundary condition (NBC),where the functions , , , , are prescribed.

Here, the symbols and stand for the Caputo fractional derivative of order with respect to and .

Introducing the function,we define the Caputo derivative as (see, e.g., (2.4.1) in [1]) where is the ceiling function of (i.e., the smallest integer greater than or equal to ), being the Euler Gamma-function. An equivalent definition in the case of reads (see (2.4.15) in [1]) In the limit case , the Caputo fractional derivatives of boils down .

Elliptic boundary value problems, , in domain with conical and dihedral singularities have been extensively studied, starting with Kondratiev’s famous paper [14]. In this field of researches, it should be also noted the works of Kondariev and Oleinik [15], Borsuk and Kondratiev [16], Grisvard [17], and Maz’ya and Plamenevskii [18]. The results of those investigations are formulated in both and spaces and corresponding weighted classes, where the desired functions together with their derivatives are bounded with some power weights in the or norms.

Recently, a great attention in the literature has been devoted to the study of boundary value problems to the fractional Laplacian operator with order (see, e.g., [1924] and references therein). Boundary value problems for the equation with the main part,are studied with various approaches, such as spectral technique, method of potential theory, and Fourier method (see [2528] and references therein). We also quote the works [29, 30], where the authors presented a Galerkin finite element approximation for variational solution to the steady state fractional advection dispersion equation:where , are left and right fractional integral operators, , .

The goal of the present paper is the proof of the well-posedness and the regularity of solutions to boundary value problems (1)-(3) in weighted Hölder classes. It is worth noting that these classes allow one to control the behavior of the solution near the boundary including the corner point and at the infinity.

Outline of the Paper. In the next section, we introduce necessary functional spaces and state the main results (Theorems 3 and 4) along with the general assumptions. In Section 3, we construct the integral representation to in the case of homogenous boundary conditions. To this end, we apply Mellin transform and reduce problem (1)-(3) to the linear nonhomogenous difference equation of the first-order with variables coefficients in the two-dimensional case. Section 4 is devoted to some auxiliary results which will play a key role in the investigation. In Section 5, we estimate the seminorms of the minor derivatives , , and in Section 6 we evaluate the Hölder coefficients of the major derivatives , . In Section 7, using these estimates, we provide the proofs of Theorems 3 and 4. Moreover, in Remark 27 we show how results of Theorems 3 and 4 can be extend to the more general equation compared to (1).

2. Functional Setting and Main Results

Throughout this work, the symbol will denote a generic positive constant, depending only on the structural quantities of the problem. We will carry out our analysis in the framework of the weighted Hölder spaces. Let be arbitrary fixed. We denote by and the distance from a point to the origin and to the boundary , correspondingly. Then for every and from we define and . Note that if , then and .

For fixed , we introduce the Banach spaces and of the functions with the normswherefor .

Definition 1. A function belongs to the class , for , if , , , and the norms below are finite, if , and in the case of .

Definition 2. A function belongs to the class , for , if , , , and the norms below are finite, if , and in the case of .

In a similar way we introduce the spaces and . As for and , these spaces concave with and .

and defined above are Banach spaces. Indeed, the fact that they are normed spaces is easily seen, whereas the completeness follows from Theorem 2.7 [31] together with standard arguments (see, e.g., Remark 3.1.3 in [32]).

We begin to stipulate the general assumptions.

h1 (condition on the parameters): we require

h2 (conditions on the right-hand sides of boundary conditions): we demand

h3 (conditions on the right-hand side of the equation): we suppose

We are now in the position to state our main results.

Theorem 3. Let assumptions (h1), (h2), and (17) hold and moreover for every points and andThen, (1) subject either to the DBC (2) or to the NBC (3), admits a unique classical solution on , satisfying the regularityfor . Besides, the following estimates hold:in the case of DBC (2), andif NBC (3) hold.

Theorem 4. Let assumptions (h1), (h2), and (18) hold. Then, boundary value problems (1)-(3) admit a unique classical solution on , satisfying the regularityfor . Besides, the following estimates hold:in the case of DBC (2), andif NBC (3) hold.

Remark 5. It is easily apparent that the functions,satisfy conditions (h2) and (17), (19), and (20).

The remainder of the paper is devoted to the proof of Theorems 3 and 4 in the DBC case. The proof of these theorems for NBC is almost identical and is left to the interested reader.

3. Integral Representation for in the Special Case

We first dwell on the special case where and is a finite function. Namely (2), (17), and (19) are replaced by the simpler conditionsfor some given positive and

We denote by the Mellin transform of the function . Due to conditions (30)-(32) and assumptions of Theorem 3, we can apply, at least formally, the Mellin transformation to problem (1) and (2) (see for details § 1.4 and § 2.5 in [1]). Then simple calculations lead to the equation

Introducing new variablesand new functions we rewrite the equation in the more compact formThus, we transform problem (1) and (2) to the linear nonhomogeneous difference equation of the first-order with variable coefficients. In order to solve this equation, we adapt the technique from Section 3 in [33] to our case.

Proposition 6. Let denote Euler-Mascheroni constant (see, e.g., Definition in [34]),and let and be arbitrary analytic functions such thatThen the functionsolves homogenous equation (36) (i.e., ) and does not have any poles.
If, in addition, the functions and have no zeros, then the function does not have any poles and the following estimate holds:for and for every fixed and satisfying inequalities above.

Proof. In order to verify that the function solves homogenous equation (36), it is enough to substitute in (36) and take into account the properties of the Gamma-function: . Besides, the properties of the function are simple consequences of the well-known properties of Gamma function:(i) has simple poles in the points(ii)The Stirling asymptotic formula holds:as while remains bounded

We are now in the position to construct the solution of nonhomogeneous equation (36).

Till the end of the paper, we assume that is an arbitrary fixed quantity and define the contour in the complex plane as(i) (ii)If , then consists of three parts:the half-circle ,the intervals and with the small positive number , .(iii)The contour () is obtained from after its shifting to the right on .

Introducing the periodic function with period 1,we assert the following results.

Proposition 7. Let , ; conditions (32) and (31) hold, and letThen the solution of inhomogeneous equation (36) is given bywhere is defined in (39) with , .

Proof. First, we prove this statement in the case ; i.e., . To this end, we will seek the solution of (36) as a product where the unknown function solves the first-order difference inhomogeneous equation with the constant coefficientsThen adapting the technique from [35] to our case, we deduce thatif condition (45) holds.
Indeed, substituting to (48) and applying Cauchy’s residue theorem arrive at To reach these equalities, we essentially used the following properties of the functions , , and :(i) has no poles if and , satisfy (45).(ii) is periodic of period with simple poles at and multiple poles at .Besides, the following inequality holds:for (iii)There is the asymptotic representationfor the bounded and and if , meet requirement (45).Note that statements (i) and (ii) are simple consequences of Proposition 6 and (32). As for assessment (iii), it can be easily drawn from (i) and (ii).
After that, we return to the representation of and obtain solution (36) asThis proves Proposition 7 in the case .
Recasting the arguments above in the case and applying Cauchy theorem allow us to prove Proposition 7 in the case of arbitrary contour , . It completes the proof of Proposition 7.

We are now in the position to obtain integral representation of the solution . Indeed, Propositions 6 and 7 providewhereNote that, conditions on hold, for example, in the domain

At last, we carry out the inverse Mellin transform to derive an explicit integral representation for ,with for and , where and , , are chosen that

In the light of (56), we can pick up and as

4. Some Technical Results

First we introduce some equivalent norms.

Proposition 8. Let and be any nonnegative numbers. Then for any functions and , , and , we have the following norm equivalence:

The proof of this statement follows with direct calculations. LetNext we represent certain estimates for the functionswhich will be frequently used to evaluate the functions and ,

Proposition 9. Let be positive number, . Then there are estimates.

Proof. For simplicity consideration, we put . Here we prove this proposition for the function . The case of is considered in the similar way. We start our consideration with the case of positive (i.e., ).
The Stirling asymptotic formula for the Gamma-function,provides that the integrand , , has the order at the infinity. Moreover, the function with has no poles in the domain Thus, by the residue theorem, we get Then, following [33], we decompose the plane into 15 subdomains, as shown in Figure 1In this decomposition is a sufficiently large number such that the terms in (65) can be neglected in all regions .
Taking into account this decomposition, we represent the function asIt is apparent that, if the function is estimated analogous to the function if . Thus, we consider here just case . To this end, we apply Stirling formula (65) and the well-known properties of the Gamma-function. Introducing the functions we arrive at the representation to the function in every domain , , and ,where the functions , , are uniformly bounded with respect to and together with .
Moreover,for belongs to one of the corresponding domains , , , It is worth mentioning that representation (70) holds for each .
If , representation (70) with aid (65) and (71) arrives at the inequalityThese relations guarantee the first estimate in Proposition 9 for .
Further, we verify the estimate if . Representation (70) and straightforward calculations lead to the inequalityTo get the same estimate in the domains , , and , it is enough to integrate by parts. Namely, let us consider the case . Based on representation (70), we can writeIt is apparent that Thus, we can enhance estimate (74), taking into account (71), so as to getRecasting the arguments above in the case of domains , , and yields the same estimate to .
After that, we combine inequalities (73) and (76) and obtain the required estimate to if .
At last, we are left to produce suitable estimate of . Standard direct calculations lead to