Initial-Boundary Value Problem for Fractional Partial Differential Equations of Higher Order

The initial-boundary value problem for partial di ﬀ erential equations of higher-order involving the Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established. Abstract value problem for partial di ﬀ erential equations of higher order with the Caputo fractional derivative was studied in the case when the order of the fractional derivative belongs to the interval (cid:5) 0,1 (cid:6) . In the paper (cid:3) 46 (cid:4) the initial-boundary value problem in plane domain for partial di ﬀ erential equations of fourth order with the fractional derivative in the sense of Caputo was studied in the case when the order of fractional derivative belongs to the interval (cid:5) 1,2 (cid:6) . The present paper generalizes results of (cid:3) 46 (cid:4) in the case of space domain for partial di ﬀ erential equations of higher order with a fractional derivative in the sense of Caputo. The organization of this paper is as follows. In Section 2, we provide the necessary background and formulation of problem. In Section 3, the formal solution of problem is presented. In Sections 4 and 5, the solvability and the regular solvability of the problem are studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established. Finally, Section 6 is conclusion.


Introduction
Many problems in viscoelasticity 1-3 , dynamical processes in self-similar structures 4 , biosciences 5 , signal processing 6 , system control theory 7 , electrochemistry 8 , diffusion processes 9 , and linear time-invariant systems of any order with internal point delays 10 lead to differential equations of fractional order. For more details of fractional calculus, see 11-15 . The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional differential equations have been studied extensively in a large cycle works see, e.g.,  and the references therein .
In the paper 43 , Cauchy problem in a half-space { x, y, t : x, y ∈ R 2 , t > 0} for partial pseudodifferential equations involving the Caputo fractional derivative was studied. The existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation were established.
In the paper 44 , the initial-boundary value problem for heat conduction equation with the Caputo fractional derivative was studied. Moreover, in 45 , the initial-boundary value problem for partial differential equations of higher order with the Caputo fractional derivative was studied in the case when the order of the fractional derivative belongs to the interval 0,1 .
In the paper 46 , the initial-boundary value problem in plane domain for partial differential equations of fourth order with the fractional derivative in the sense of Caputo was studied in the case when the order of fractional derivative belongs to the interval 1,2 . The present paper generalizes results of 46 in the case of space domain for partial differential equations of higher order with a fractional derivative in the sense of Caputo.
The organization of this paper is as follows. In Section 2, we provide the necessary background and formulation of problem. In Section 3, the formal solution of problem is presented. In Sections 4 and 5, the solvability and the regular solvability of the problem are studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established. Finally, Section 6 is conclusion.

Preliminaries
In this section, we present some basic definitions and preliminary facts which are used throughout the paper.
Definition 2.1. If g t ∈ C a, b and α > 0, then the Riemann-Liouville fractional integral is defined by where Γ · is the Gamma function defined for any complex number z as Definition 2.2. The Caputo fractional derivative of order α > 0 of a continuous function g : where n α 1, the notation α stands for the largest integer not greater than α .
has a unique solution z t defined by the following formula: For the convenience of the reader, we give the proof of Theorem 2.4, applying the fixed-point iteration method. We denote Then, The proof of this theorem is based on formula 2.8 and for any m ∈ N. Let us prove 2.9 for any m ∈ N. For m 1, it follows from 2.7 directly. Assume that 2.9 holds for some m − 1 ∈ N. Then, applying 2.7 and 2.9 for m − 1 ∈ N, we get

2.11
Then, So, identity 2.9 holds for m ∈ N. Therefore, by induction identity 2.9 holds for any m ∈ N.
In the space domain, Ω { x, y, t : 0 < x < p, 0 < y < q, 0 < t < T}, we consider the initial-boundary value problem: for partial differential equations of higher order with the fractional derivative order α ∈ 1, 2 in the sense of Caputo. Here, k k ≥ 1 is a fixed positive integer number.

The Construction of the Formal Solution of 2.13
We seek a solution of problem 2.13 in the form of Fourier series: expanded along a complete orthonormal system: Abstract and Applied Analysis 5 We denote

3.3
We expand the given function f x, y, t in the form of a Fourier series along the functions v nm x, y , 1 ≤ n, m < ∞: Substituting 3.1 and 3.4 into 2.13 , we obtain By Lemma 2.3, we have that where is Riemann-Liouville integral of fractional order α. Using 3.6 and 3.7 , we get the following equation: Applying the operator I α 0 to this equation, we get the following Volterra integral equation of the second kind: Abstract and Applied Analysis According to the Theorem 2.4, 3.10 has a unique solution u nm t defined by the following formula:

3.11
Using the formula see, e.g., 27, page 118 and 47, page 120 From these three formulas and 3.11 , it follows that

3.17
So, the unique solution of 3.10 is defined by 3.17 . Consequently, the unique solution of problem 2.13 is defined by 3.1 .

Abstract and Applied Analysis
Applying the formula 3.17 , the Cauchy-Schwarz inequality, and the estimate see 13, page 136 we get the following inequality: for the solution of 3.10 for any t, t ∈ 0, T . Here, C 0 max{M, T M, M T α−1/2 / √ 2α − 1 }.

Solvability of 2.13 in L 2 Ω Space
Now, we will prove that the solution u x, y, t of problem 2.13 continuously depends on ϕ x, y , ψ x, y , and f x, y, t .

Theorem 4.1.
Suppose ϕ x, y ∈ L 2 Ω 0 , ψ x, y ∈ L 2 Ω 0 , and f x, y, t ∈ L 2 Ω , then the series 3.1 converges in L 2 Ω to u ∈ L 2 Ω and for the solution of problem 2.13 , the following stability inequality holds, where C 1 does not depend on ϕ x, y , ψ x, y , and f x, y, t .
Proof. We consider the sum: where N is a natural number. For the positive integer number L, we have that

4.3
Abstract and Applied Analysis 9 Applying 3.19 , we get Consequently, the series 3.1 converges in L 2 Ω to u x, y, t ∈ L 2 Ω . Inequality 4.1 for the solution of problem 2.13 follows from the estimate 4.4 . Theorem 4.1 is proved.

The Regular Solvability of 2.13
In this section, we will study theregular solvability of problem 2.13 .

5.17
Analogously, we get Since 2k ≥ 2, then the series ∞ n 1 1/n 2k , ∞ m 1 1/m 2k converges by the integral test. Further, k 1 − ε > 1, then the series converges also by the integral test for any k ≥ 1 and ε ∈ 0, 1 . Consequently, the series 3.1 absolutely and uniformly converges in the domain Ω t 0 Ω × t 0 , T for any t 0 ∈ 0, T . At t 0, the series 3.1 converges and has a sum equal to ϕ x, y . Since ν 2k n < λ 2k nm , μ 2k m < λ 2k nm , then the series

5.26
Consequently, lim t → 0 u nm t ϕ nm , lim t → 0 u nm t ψ nm . Hence, we conclude that the solution 3.1 satisfies initial conditions. Theorem 5.2 is proved.

Conclusion
In this paper, the initial-boundary value problem 2.13 for partial differential equations of higher order involving the Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established. Of course, such type of results have been established for the initial-boundary value problem: ∂ 2k u ∂y 2k u f x, y, t , 0 < x < p, 0 < y < q, 0 < t < T, ∂ 2m 1 u 0, y, t ∂x 2m ∂ 2m 1 u p, y, t ∂x 2m 0, m 0, 1, . . . , k − 1, 0 ≤ y ≤ q, 0 ≤ t ≤ T, ∂ 2m 1 u x, 0, t ∂y 2m ∂ 2m 1 u x, q, t ∂y 2m 0, m 0, 1, . . . , k − 1, 0 ≤ x ≤ p, 0 ≤ t ≤ T, u x, y, 0 ϕ x, y , u t x, y, 0 ψ x, y , 0 ≤ x ≤ p, 0 ≤ y ≤ q 6.1 for partial differential equations of higher order with a fractional derivative of order α ∈ 1, 2 in the sense of Caputo. Here, k k ≥ 1 is a fixed positive integer number. Moreover, applying the result of the papers 15, 23 , the first order of accuracy difference schemes for the numerical solution of nonlocal boundary value problems 2.13 and 6.1 can be presented. Of course, the stability inequalities for the solution of these difference schemes have been established without any assumptions about the grid steps τ in t and h in the space variables.