On the Solvability of a Mixed Problem for a High-Order Partial Differential Equation with Fractional Derivatives with Respect to Time , with Laplace Operators with Spatial Variables and Nonlocal Boundary Conditions in Sobolev Classes

In this paper, we study the solvability of a mixed problem for a high-order partial differential equation with fractional derivatives with respect to time, and with Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes.


On the Solvability of a Mixed Problem for a High-Order Partial Differential Equation with Fractional Derivatives with Respect to Time, with Laplace Operators with Spatial Variables and Nonlocal Boundary Conditions in Sobolev Classes
Onur Alp İlhan 1, * , Shakirbay G. Kasimov 2 , Shonazar Q. Otaev 2 and Haci Mehmet Baskonus 3

Introduction
The spectral theory of operators finds numerous uses in various fields of mathematics and their applications.
An important part of the spectral theory of differential operators is the distribution of their eigenvalues.This classical question was studied for a second-order operator on a finite interval by Liouville and Sturm.Later, G.D. Birkhoff [1][2][3] studied the distribution of eigenvalues for an ordinary differential operator of arbitrary order on a finite interval with regular boundary conditions.
For quantum mechanics, it is especially interesting to distribute the eigenvalues of operators defined throughout the space and having a discrete spectrum.E.C. Titchmarsh [4][5][6][7][8][9] was the first to rigorously establish the formula for the distribution of the number of eigenvalues for a one-dimensional Sturm-Liouville operator on the whole axis with potential growing at infinity.He also first strictly established the distribution formula for the Schrödinger operator.B.M. Levitan [10][11][12] deserves much credit for the improvement of E.C. Titchmarsh's method.
In solving many mathematical physics problems, the need arises for the expansion of an arbitrary function in a Fourier series with respect to Sturm-Liouville eigenvalues.The so-called regular case of the Sturm-Liouville problem corresponding to a finite interval and a continuous coefficient of the equation has been studied for a relatively long time and is usually described in detail in the manuals on the equations of mathematical physics and integral equations.
The Sturm-Liouville problem for the so-called singular case, as well as with nonlocal boundary conditions, is much less known.
As it is known, so-called fractal media are studied in solid-state physics and, in particular, diffusion phenomena in them.In one of the models studied in [13], diffusion in a strongly porous (fractal) medium is described by an equation of the type of heat-conduction equation, but with a fractional derivative with respect to time coordinate The formulation of initial-boundary value problems for Equation (1), similar to the problems for parabolic differential equations, makes sense if by a regularized fractional derivative: Study of the form equations where A is an elliptic operator (in [14][15][16]).In recent years, many authors studied fractional differential equations in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].

Problem Formulation
In this work, we consider the equation of the form with initial conditions lim and boundary conditions where (x, t) = (x 1 , . . ., x j , . . ., . ., l are functions that can be expanded in terms of the system of eigenfunctions {v n (x), n ∈ Z N } of the spectral problem: Here, for α < 0,, fractional integral D α has the form ) for α = 0, and for l − 1 < α ≤ l, l ∈ N, the fractional derivative has the form In [17], Problems ( 4)-( 6) and, accordingly, spectral Problems ( 7) and ( 8) in the case ν = 1, were considered.
For µ > 0, the general solution of Problem (9) has the form From boundary conditions, we have: Hence, the nontrivial solutions of Problems ( 9) and ( 10) are only possible in the case of Therefore, λπ = arccos −2αβ Further, Thus, the eigenvalues and eigenfunctions of Problems ( 9) and (10) are respectively, where we obtain The norm in space W s 2 (0, π) is introduced as follows: .
Let ε n = ε −n .Then, system of vectors forms the complete orthonormal system in W s 2 (0, π).The following lemma holds.
Lemma 1.Let {a n } be a finite system of complex numbers.Then, inequalities are valid where Using properties of the norm, we have Thus, denoting Lemma 2. Let {a n } be a finite system of complex numbers.Then, inequalities using properties of the norm, we have Thus, inequalities Using Lemmas 1 and 2, we obtain Lemma 3. Let {a n } be a finite system of complex numbers.Then the following inequality Lemma 4. Let α = 0, β = 0, |α| = |β| be real numbers, and Then, eigenfunction system , n ∈ Z, of spectral Problems ( 9) and ( 10) forms the Riesz basis in the space W s 2 (0, π).
Proof.Vector system , n ∈ Z forms the complete orthonormal system in Hilbert space W s 2 (0, π),, and vector system , n ∈ Z by virtue of Lemma 3 satisfying the theorem conditions by R. Paley and N. Wiener (see p. 224, [39]).This theorem implies that system of vectors {y n (x)} n∈Z forms the Riesz basis in space W s 2 (0, π).
Proof.Symmetry of operator L implies that eigenfunctions {y n (x)} n∈Z of operator L, corresponding to the different eigenvalues, are orthogonal in classes L 2 (0, π).System of functions {D α y n (x)} n∈Z is also the system of eigenfunctions of a similar operator corresponding to different eigenvalues, which implies that functions of system {D α y n (x)} n∈Z are orthogonal in classes L 2 (0, π).
As a result, we see that system of eigenfunctions {y n (x)} n∈Z of operator L, corresponding to different eigenvalues, are orthogonal in the Sobolev classes W s 2 (0, π).It is known that, if a sequence of vectors {ψ n (x)} n∈Z forms the Riesz basis in a Hilbert space H, then system of vectors , n ∈ Z also forms the Riesz basis in H (see p. 374, [42]).By virtue of Lemma 4, system of eigenvectors {y n (x)} n∈Z forms the Riesz basis in space W s 2 (0, π).The orthogonality of this system implies that {y n (x)} n∈Z is a complete orthonormal system in the Sobolev classes W s 2 (0, π).
Theorem 1 and the Sobolev embedding theorem imply the following corollaries.
The scalar product in space W s 1 ,s 2 ,...,s N 2 (Π) is introduced in the following way: (D Respectively, the norm in this space is introduced as follows: .
Using the method of mathematical induction and Lemma 6, we obtain the following: m N (x N )} are complete orthonormal systems in spaces W s 1 2 (0, π), . . ., W s N 2 (0, π), respectively, then system of all products Let us apply Lemma 7 to our orthonormal systems.In space form the complete orthonormal system.Here, y Thus, the following statement is valid: of spectral Problems ( 7) and ( 8) forms the complete orthonormal system in Sobolev classes W s 1 ,s 2 ,...,s N 2 (Π).
The proof of Corollary 3 is carried out using Theorem 2 and the Sobolev embedding theorem.
The following are true:

Main Results
In this section, we give the most general case of the works done in [17].
By virtue of Problems ( 4) and ( 5), unknown functions T m 1 ...m N (t) must satisfy equation with initial conditions The solution of Cauchy Problems ( 13) and ( 14) has the form where coefficients are determined as follows: After substituting Problem (15) into Problem (12), we obtain the unique solution of Problems ( 4)-( 6) in the form of Series (8).

Conclusions
In this paper, we considered questions on the unique solvability of a mixed problem for a partial differential equation of high order with fractional Riemann-Liouville derivatives with respect to time, and with Laplace operators with spatial variables and with nonlocal boundary conditions in Sobolev classes.The solution was found in the form of a series of expansions in eigenfunctions of the Laplace operator with nonlocal boundary conditions.Initial and boundary problems with fractional Riemann-Liouville derivatives with respect to time have many applications [13].In connection to this, we chose the fractional Riemann-Liouville derivative, although we could consider other types of fractional derivatives.