About this Journal Submit a Manuscript Table of Contents 
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 973102, 16 pages
http://dx.doi.org/10.1155/2012/973102
Research Article

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

Djumaklych Amanov1 and Allaberen Ashyralyev2,3

1Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences, 29 Do'rmon yo'li street, Tashkent 100047, Uzbekistan
2Department of Mathematics, Fatih University, 34500 Buyucekmece, Istanbul, Turkey
3ITTU, Ashgabat 74400, Turkmenistan

Received 30 March 2012; Revised 25 April 2012; Accepted 12 May 2012

Academic Editor: Valery Covachev

Copyright © 2012 Djumaklych Amanov and Allaberen Ashyralyev. 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

The initial-boundary value problem 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.

1. 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., [16–42] 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.

2. 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 𝐼 𝛼 𝑎 + 1 𝑔 ( 𝑡 ) = Γ ( 𝛼 ) 𝑡 𝑎 𝑔 ( 𝑠 ) ( 𝑡 − 𝑠 ) 1 − 𝛼 𝑑 𝑠 , ( 2 . 1 ) where Γ ( ⋅ ) is the Gamma function defined for any complex number 𝑧 as Γ ( 𝑧 ) = ∞ 0 𝑡 𝑧 − 1 𝑒 − 𝑡 𝑑 𝑡 . ( 2 . 2 )

Definition 2.2. The Caputo fractional derivative of order 𝛼 > 0 of a continuous function 𝑔 ∶ ( 𝑎 , 𝑏 ) → 𝑅 is defined by 𝑐 𝐷 𝛼 𝑎 + 1 𝑔 ( 𝑡 ) = Γ ( 𝑛 − 𝛼 ) 𝑡 𝑎 𝑔 ( 𝑛 ) ( 𝑠 ) ( 𝑡 − 𝑠 ) 𝛼 − 𝑛 + 1 𝑑 𝑠 , ( 2 . 3 ) where 𝑛 = [ 𝛼 ] + 1 , (the notation [ 𝛼 ] stands for the largest integer not greater than 𝛼 ).

Lemma 2.3 (see [13]). Let 𝑝 , 𝑞 ≥ 0 ,   𝑓 ( 𝑡 ) ∈ 𝐿 1 [ 0 , 𝑇 ] . Then, 𝐼 𝑝 0 + 𝐼 𝑞 0 + 𝑓 ( 𝑡 ) = 𝐼 𝑝 + 𝑞 0 + 𝑓 ( 𝑡 ) = 𝐼 𝑞 0 + 𝐼 𝑝 0 + 𝑓 ( 𝑡 ) ( 2 . 4 ) is satisfied almost everywhere on [ 0 , 𝑇 ] . Moreover, if 𝑓 ( 𝑡 ) ∈ 𝐶 [ 0 , 𝑇 ] , then (2.4) is true and 𝑐 𝐷 𝛼 0 + 𝐼 𝛼 0 + 𝑓 ( 𝑡 ) = 𝑓 ( 𝑡 ) for all 𝑡 ∈ [ 0 , 𝑇 ] and 𝛼 > 0 .

Theorem 2.4 (see [47, page 123]). Let 𝑓 ( 𝑡 ) ∈ 𝐿 1 ( 0 , 𝑇 ) . Then, the integral equation 𝑧 ( 𝑡 ) = 𝑓 ( 𝑡 ) + 𝜆 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 Γ ( 𝛼 ) 𝑧 ( 𝜏 ) 𝑑 𝜏 ( 2 . 5 ) has a unique solution 𝑧 ( 𝑡 ) defined by the following formula: 𝑧 ( 𝑡 ) = 𝑓 ( 𝑡 ) + 𝜆 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 ( 𝜆 ( 𝑡 − 𝜏 ) 𝛼 ) 𝑓 ( 𝜏 ) 𝑑 𝜏 , ( 2 . 6 ) where 𝐸 𝛼 , 𝛽 ∑ ( 𝑧 ) = ∞ 𝑘 = 0 ( 𝑧 𝑘 / Γ ( 𝑘 𝛼 + 𝛽 ) ) is a Mittag-Leffler type function.

For the convenience of the reader, we give the proof of Theorem 2.4, applying the fixed-point iteration method. We denote 𝐵 𝑧 ( 𝑡 ) = 𝜆 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 Γ ( 𝛼 ) 𝑧 ( 𝜏 ) 𝑑 𝜏 . ( 2 . 7 ) Then, 𝑧 ( 𝑡 ) = 𝑚 − 1 𝑘 = 0 𝐵 𝑘 𝑓 ( 𝑡 ) + 𝐵 𝑚 𝑧 ( 𝑡 ) , 𝑚 = 1 , … , 𝑛 . ( 2 . 8 ) The proof of this theorem is based on formula (2.8) and 𝐵 𝑚 𝑧 ( 𝑡 ) = 𝜆 𝑚 𝑡 0 ( 𝑡 − 𝜏 ) 𝑚 𝛼 − 1 Γ ( 𝑚 𝛼 ) 𝑧 ( 𝜏 ) 𝑑 𝜏 , ( 2 . 9 ) 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 𝐵 𝑚 𝑧 ( 𝑡 ) = 𝜆 𝑚 − 1 𝑡 0 ( 𝑡 − 𝑠 ) ( 𝑚 − 1 ) 𝛼 − 1 Γ ( ( 𝑚 − 1 ) 𝛼 ) 𝐵 𝑧 ( 𝑠 ) 𝑑 𝑠 = 𝜆 𝑚 − 1 𝑡 0 ( 𝑡 − 𝑠 ) ( 𝑚 − 1 ) 𝛼 − 1 𝜆 Γ ( ( 𝑚 − 1 ) 𝛼 ) 𝑠 0 ( 𝑠 − 𝜏 ) 𝛼 − 1 = 𝜆 𝑧 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 𝑚 Γ ( 𝛼 ) Γ ( ( 𝑚 − 1 ) 𝛼 ) 𝑡 0 𝑠 0 ( 𝑡 − 𝑠 ) ( 𝑚 − 1 ) 𝛼 − 1 ( 𝑠 − 𝜏 ) 𝛼 − 1 = 𝜆 𝑧 ( 𝜏 ) 𝑑 𝜏 𝑑 𝑠 𝑚 Γ ( 𝛼 ) Γ ( ( 𝑚 − 1 ) 𝛼 ) 𝑡 0 𝑡 𝜏 ( 𝑡 − 𝑠 ) ( 𝑚 − 1 ) 𝛼 − 1 ( 𝑠 − 𝜏 ) 𝛼 − 1 𝑑 𝑠 𝑧 ( 𝜏 ) 𝑑 𝜏 . ( 2 . 1 0 ) Performing the change of variables 𝑠 − 𝜏 = ( 𝑡 − 𝜏 ) 𝑝 , we get 𝑡 𝜏 ( 𝑡 − 𝑠 ) ( 𝑚 − 1 ) 𝛼 − 1 ( 𝑠 − 𝜏 ) 𝛼 − 1 𝑑 𝑠 = ( 𝑡 − 𝜏 ) 𝑚 𝛼 − 1 1 0 ( 1 − 𝑝 ) ( 𝑚 − 1 ) 𝛼 − 1 𝑝 𝛼 − 1 𝑑 𝑝 = ( 𝑡 − 𝜏 ) 𝑚 𝛼 − 1 = 𝐵 ( ( 𝑚 − 1 ) 𝛼 , 𝛼 ) ( 𝑡 − 𝜏 ) 𝑚 𝛼 − 1 Γ ( 𝑚 𝛼 ) Γ ( ( 𝑚 − 1 ) 𝛼 ) Γ ( 𝛼 ) . ( 2 . 1 1 ) Then, 𝐵 𝑚 𝑧 ( 𝑡 ) = 𝜆 𝑚 𝑡 0 ( 𝑡 − 𝜏 ) 𝑚 𝛼 − 1 Γ ( 𝑚 𝛼 ) 𝑧 ( 𝜏 ) 𝑑 𝜏 . ( 2 . 1 2 ) 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: ( − 1 ) 𝑘 𝑐 𝐷 𝛼 0 + 𝜕 𝑢 + 2 𝑘 𝑢 𝜕 𝑥 2 𝑘 + 𝜕 2 𝑘 𝑢 𝜕 𝑦 2 𝑘 𝜕 = 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) , 0 < 𝑥 < 𝑝 , 0 < 𝑦 < 𝑞 , 0 < 𝑡 < 𝑇 , 2 𝑚 𝑢 ( 0 , 𝑦 , 𝑡 ) 𝜕 𝑥 2 𝑚 = 𝜕 2 𝑚 𝑢 ( 𝑝 , 𝑦 , 𝑡 ) 𝜕 𝑥 2 𝑚 𝜕 = 0 , 𝑚 = 0 , 1 , … , 𝑘 − 1 , 0 ≤ 𝑦 ≤ 𝑞 , 0 ≤ 𝑡 ≤ 𝑇 , 2 𝑚 𝑢 ( 𝑥 , 0 , 𝑡 ) 𝜕 𝑦 2 𝑚 = 𝜕 2 𝑚 𝑢 ( 𝑥 , 𝑞 , 𝑡 ) 𝜕 𝑦 2 𝑚 = 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 ( 2 . 1 3 ) 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.

3. The Construction of the Formal Solution of (2.13)

We seek a solution of problem (2.13) in the form of Fourier series: 𝑢 ( 𝑥 , 𝑦 , 𝑡 ) = ∞ 𝑛 , 𝑚 = 1 𝑢 𝑛 𝑚 ( 𝑡 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , ( 3 . 1 ) expanded along a complete orthonormal system: 𝑣 𝑛 𝑚 2 ( 𝑥 , 𝑦 ) = √ 𝑝 𝑞 s i n 𝑛 𝜋 𝑝 𝑥 s i n 𝑚 𝜋 𝑞 𝑦 , 1 ≤ 𝑛 , 𝑚 < ∞ . ( 3 . 2 ) We denote Ω 0 = Ω ∩ ( 𝑡 = 0 ) = { ( 𝑥 , 𝑦 , 0 ) ∶ 0 ≤ 𝑥 ≤ 𝑝 , 0 ≤ 𝑦 ≤ 𝑞 } , 𝑛 𝜋 𝑝 = 𝜈 𝑛 , 𝑚 𝜋 𝑞 = 𝜇 𝑚 , 𝜈 𝑛 2 𝑘 + 𝜇 𝑚 2 𝑘 = 𝜆 2 𝑘 𝑛 𝑚 , 1 ≤ 𝑛 , 𝑚 < ∞ . ( 3 . 3 ) We expand the given function 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) in the form of a Fourier series along the functions 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , 1 ≤ 𝑛 , 𝑚 < ∞ : 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) = ∞ 𝑛 , 𝑚 = 1 𝑓 𝑛 𝑚 ( 𝑡 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , ( 3 . 4 ) where 𝑓 𝑛 𝑚 ( t ) = 𝑝 0 𝑞 0 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) 𝑑 𝑦 𝑑 𝑥 , 1 ≤ 𝑛 , 𝑚 < ∞ . ( 3 . 5 ) Substituting (3.1) and (3.4) into (2.13), we obtain ( − 1 ) 𝑘 𝑐 𝐷 𝛼 0 + 𝑢 𝑛 𝑚 ( 𝑡 ) + ( − 1 ) 𝑘 𝜆 2 𝑘 𝑛 𝑚 𝑢 𝑛 𝑚 ( 𝑡 ) = 𝑓 𝑛 𝑚 ( 𝑡 ) . ( 3 . 6 ) By Lemma 2.3, we have that 𝑐 𝐷 𝛼 0 + 𝑢 𝑛 𝑚 ( 𝑡 ) = 𝐼 2 − 𝛼 0 + 𝑢 𝑛 𝑚 ( 𝑡 ) , ( 3 . 7 ) where 𝐼 𝛼 0 + 1 𝑓 ( 𝑡 ) = Γ ( 𝛼 ) 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝑓 ( 𝜏 ) 𝑑 𝜏 ( 3 . 8 ) is Riemann-Liouville integral of fractional order 𝛼 . Using (3.6) and (3.7), we get the following equation: 𝐼 2 − 𝛼 0 + 𝑢 𝑛 𝑚 ( 𝑡 ) + 𝜆 2 𝑘 𝑛 𝑚 𝑢 𝑛 𝑚 ( 𝑡 ) = ( − 1 ) 𝑘 𝑓 𝑛 𝑚 ( 𝑡 ) . ( 3 . 9 ) Applying the operator 𝐼 𝛼 0 + to this equation, we get the following Volterra integral equation of the second kind: 𝑢 𝑛 𝑚 ( 𝑡 ) = − 𝜆 2 𝑘 𝑛 𝑚 Γ ( 𝛼 ) 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝑢 𝑛 𝑚 ( 𝜏 ) 𝑑 𝜏 + 𝑢 𝑛 𝑚 ( 0 ) + 𝑡 𝑢 𝑛 𝑚 ( 0 ) + ( − 1 ) 𝑘 𝐼 𝛼 0 + 𝑓 𝑛 𝑚 ( 𝑡 ) . ( 3 . 1 0 ) According to the Theorem 2.4, (3.10) has a unique solution 𝑢 𝑛 𝑚 ( 𝑡 ) defined by the following formula: 𝑢 𝑛 𝑚 ( 𝑡 ) = ( − 1 ) 𝑘 Γ ( 𝛼 ) 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝑓 𝑛 𝑚 ( 𝜏 ) 𝑑 𝜏 + 𝑢 𝑛 𝑚 ( 0 ) 1 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 𝑑 𝜏 + 𝑢 𝑛 𝑚 ( 0 ) 𝑡 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 − 𝜆 𝜏 𝑑 𝜏 2 𝑘 𝑛 𝑚 Γ ( 𝛼 ) 𝑡 0 ( 𝑡 − 𝜂 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜂 ) 𝛼 𝑑 𝜂 𝜂 0 ( 𝜂 − 𝜏 ) 𝛼 − 1 𝑓 𝑛 𝑚 ( 𝜏 ) 𝑑 𝜏 . ( 3 . 1 1 )

Using the formula (see, e.g., [27, page 118] and [47, page 120]) 1 Γ ( 𝛽 ) 𝑧 0 𝑡 𝜇 − 1 𝐸 𝛼 , 𝜇 ( 𝜆 𝑡 𝛼 ) ( 𝑧 − 𝑡 ) 𝛽 − 1 𝑑 𝑡 = 𝑧 𝜇 + 𝛽 − 1 𝐸 𝛼 , 𝜇 + 𝛽 ( 𝜆 𝑧 𝛼 1 ) , Γ ( 𝜇 ) + 𝑧 𝐸 𝛼 , 𝛼 + 𝜇 ( 𝑧 ) = 𝐸 𝛼 , 𝜇 ( 𝑧 ) , ( 3 . 1 2 ) we get − 𝜆 2 𝑘 𝑛 𝑚 Γ ( 𝛼 ) 𝑡 0 ( 𝑡 − 𝜂 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜂 ) 𝛼 𝑑 𝜂 𝜂 0 ( 𝜂 − 𝜏 ) 𝛼 − 1 𝑓 𝑛 𝑚 = ( 𝜏 ) 𝑑 𝜏 𝑡 0 𝑓 𝑛 𝑚 − 𝜆 ( 𝜏 ) 2 𝑘 𝑛 𝑚 Γ ( 𝛼 ) 𝑡 𝜏 ( 𝑡 − 𝜂 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜂 ) 𝛼 ( 𝜂 − 𝜏 ) 𝛼 − 1 = 𝑑 𝜂 𝑑 𝜏 𝑡 0 𝑓 𝑛 𝑚 − 𝜆 ( 𝜏 ) 2 𝑘 𝑛 𝑚 Γ ( 𝛼 ) 0 𝑡 − 𝜏 𝑧 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 𝑧 𝛼 ( 𝑡 − 𝜏 − 𝑧 ) 𝛼 − 1 𝑑 𝑧 𝑑 𝜏 = − 𝑡 0 𝑓 𝑛 𝑚 ( 𝜏 ) 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 2 𝛼 − 1 𝐸 𝛼 , 2 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 = 𝑑 𝜏 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝑓 𝑛 𝑚 ( − 1 𝜏 ) Γ ( 𝛼 ) + 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 𝑑 𝜏 , − 𝜆 2 𝑘 𝑛 𝑚 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 𝑑 𝜏 = − 𝜆 2 𝑘 𝑛 𝑚 𝑡 0 𝑧 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 𝑧 𝛼 ( 𝑡 − 𝑧 ) 1 − 1 𝑑 𝑧 = Γ ( 1 ) 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 𝐸 𝛼 , 𝛼 + 1 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 = 𝐸 𝛼 , 1 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 − 1 , − 𝜆 2 𝑘 𝑛 𝑚 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 𝜏 𝑑 𝜏 = − 𝜆 2 𝑘 𝑛 𝑚 𝑡 0 𝑧 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 𝑧 𝛼 ( 𝑡 − 𝑧 ) 2 − 1 𝑑 𝑧 = Γ ( 2 ) 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 + 1 𝐸 𝛼 , 𝛼 + 2 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 = 𝑡 𝐸 𝛼 , 2 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 − 𝑡 . From these three formulas and (3.11), it follows that 𝑢 𝑛 𝑚 ( 𝑡 ) = 𝑢 𝑛 𝑚 ( 0 ) 𝐸 𝛼 , 1 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 + 𝑡 𝑢 𝑛 𝑚 ( 0 ) 𝐸 𝛼 , 2 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 + ( − 1 ) 𝑘 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 𝑓 𝑛 𝑚 ( 𝜏 ) 𝑑 𝜏 . ( 3 . 1 4 ) For 𝑢 𝑛 𝑚 ( 0 ) and 𝑢 𝑛 𝑚 ( 0 ) , we expand the given functions 𝜑 ( 𝑥 , 𝑦 ) and 𝜓 ( 𝑥 , 𝑦 ) in the form of a Fourier series along the functions 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , 1 ≤ 𝑛 , 𝑚 < ∞ : 𝜑 ( 𝑥 , 𝑦 ) = ∞ 𝑛 , 𝑚 = 1 𝜑 𝑛 𝑚 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , 𝜓 ( 𝑥 , 𝑦 ) = ∞ 𝑛 , 𝑚 = 1 𝜓 𝑛 𝑚 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , ( 3 . 1 5 ) where 𝜑 𝑛 𝑚 = 𝑝 0 𝑞 0 𝜑 ( 𝑥 , 𝑦 ) 𝑣 𝑛 𝑚 𝜓 ( 𝑥 , 𝑦 ) 𝑑 𝑦 𝑑 𝑥 , 𝑛 𝑚 = 𝑝 0 𝑞 0 𝜓 ( 𝑥 , 𝑦 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) 𝑑 𝑦 𝑑 𝑥 . ( 3 . 1 6 ) Using (2.13), (3.14), (3.16), we obtain 𝑢 𝑛 𝑚 ( 𝑡 ) = 𝐸 𝛼 , 1 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 𝜑 𝑛 𝑚 + 𝑡 𝐸 𝛼 , 2 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 𝜓 𝑛 𝑚 + ( − 1 ) 𝑘 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝐸 𝛼 , 𝛼 − 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 𝑓 𝑛 𝑚 ( 𝜏 ) 𝑑 𝜏 . ( 3 . 1 7 ) 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).

Applying the formula (3.17), the Cauchy-Schwarz inequality, and the estimate (see [13, page 136]) | | 𝐸 𝛼 , 𝛽 | | ≤ 𝑀 ( 𝑧 ) 1 + | 𝑧 | , 𝑀 = c o n s t > 0 , R e 𝑧 < 0 , ( 3 . 1 8 ) we get the following inequality: | | 𝑢 𝑛 𝑚 | | ( 𝑡 ) ≤ 𝐶 0 | | 𝜑 𝑛 𝑚 | | + | | 𝜓 𝑛 𝑚 | | + 𝑡 0 | | 𝑓 𝑛 𝑚 | | ( 𝑡 ) 2 𝑑 𝑡 1 / 2 ( 3 . 1 9 ) for the solution of (3.10) for any 𝑡 , 𝑡 ∈ [ 0 , 𝑇 ] . Here, 𝐶 0 = m a x { 𝑀 , 𝑇 𝑀 , 𝑀 ( 𝑇 𝛼 − 1 / 2 / √ 2 𝛼 − 1 ) } .

4. 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 ‖ 𝑢 ‖ 𝐿 2 ( Ω ) ≤ 𝐶 1 ‖ 𝜑 ‖ 𝐿 2 ( Ω 0 ) + ‖ 𝜓 ‖ 𝐿 2 ( Ω 0 ) + ‖ 𝑓 ‖ 𝐿 2 ( Ω ) ( 4 . 1 ) holds, where 𝐶 1 does not depend on 𝜑 ( 𝑥 , 𝑦 ) , 𝜓 ( 𝑥 , 𝑦 ) , and 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) .

Proof. We consider the sum: 𝑢 𝑁 ( 𝑥 , 𝑦 , 𝑡 ) = 𝑁 𝑛 , 𝑚 = 1 𝑢 𝑛 𝑚 ( 𝑡 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , ( 4 . 2 ) where 𝑁 is a natural number. For the positive integer number 𝐿 , we have that ‖ ‖ 𝑢 𝑁 + 𝐿 − 𝑢 𝑁 ‖ ‖ 2 𝐿 2 ( Ω ) = ‖ ‖ ‖ ‖ 𝑁 + 𝐿 𝑛 , 𝑚 = 𝑁 + 1 𝑢 𝑛 𝑚 ( ⋅ ) 𝑣 𝑛 𝑚 ‖ ‖ ‖ ‖ ( ⋅ , ⋅ ) 2 𝐿 2 ( Ω ) = 𝑁 + 𝐿 𝑛 , 𝑚 = 𝑁 + 1 𝑇 0 | | 𝑢 𝑛 𝑚 ( | | 𝑡 ) 2 𝑑 𝑡 . ( 4 . 3 ) Applying (3.19), we get ∞ 𝑛 , 𝑚 = 1 𝑇 0 | | 𝑢 𝑛 𝑚 | | ( 𝑡 ) 2 𝑑 𝑡 ≤ 3 𝐶 2 0 ∞ 𝑛 , 𝑚 = 1 | | 𝜑 𝑛 𝑚 | | 2 + ∞ 𝑛 , 𝑚 = 1 | | 𝜓 𝑛 𝑚 | | 2 + ∞ 𝑛 , 𝑚 = 1 𝑇 0 | | 𝑓 𝑛 𝑚 | | ( 𝑡 ) 2 𝑑 𝑡 = 𝐶 2 ‖ 𝜑 ‖ 2 𝐿 2 Ω 0 + ‖ 𝜓 ‖ 2 𝐿 2 Ω 0 + ‖ 𝑓 ‖ 2 𝐿 2 ( Ω ) , ( 4 . 4 ) where 𝐶 2 = 3 𝑇 𝐶 2 0 . Therefore, ∑ 𝑁 + 𝐿 𝑛 , 𝑚 = 𝑁 + 1 ∫ 𝑇 0 | 𝑢 𝑛 𝑚 ( 𝑡 ) | 2 𝑑 𝑡 → 0 as 𝑁 → ∞ . 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.

5. The Regular Solvability of (2.13)

In this section, we will study theregular solvability of problem (2.13).

Lemma 5.1. Suppose 𝜑 ( 𝑥 , 𝑦 ) ∈ 𝐶 1 ( Ω 0 ) ,  𝜑 𝑥 𝑦 ( 𝑥 , 𝑦 ) ∈ 𝐿 2 ( Ω 0 ) ,  𝜓 ( 𝑥 , 𝑦 ) ∈ 𝐶 1 ( Ω 0 ) ,  𝜓 x 𝑦 ( 𝑥 , 𝑦 ) ∈ 𝐿 2 ( Ω 0 ) ,  𝜑 ( 𝑥 , 𝑦 ) = 0 on 𝜕 Ω 0 ,  𝜓 ( 𝑥 , 𝑦 ) = 0 on 𝜕 Ω 0 ,  𝑓 ( 𝑥 , 𝑦 , 𝑡 ) ∈ 𝐶 2 ( Ω ) , 𝑓 𝑥 𝑥 𝑦 ( 𝑥 , 𝑦 , 𝑡 ) ∈ 𝐶 ( Ω ) 𝑓 𝑥 𝑦 𝑦 ( 𝑥 , 𝑦 , 𝑡 ) ∈ 𝐶 ( Ω 0 ) , 𝑓 𝑥 𝑥 𝑦 𝑦 ( 𝑥 , 𝑦 , 𝑡 ) ∈ 𝐶 ( Ω 0 ) , and 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) = 0 on 𝜕 Ω × [ 0 , 𝑇 ] . Then, for any 𝜀 ∈ ( 0 , 1 ) , the following estimates | | 𝑢 𝑛 𝑚 | | ( 𝑡 ) ≤ 𝐶 1 | | 𝜑 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + | | 𝜓 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + 1 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 + 1 𝜈 𝑛 𝑘 + 1 − 𝜀 𝜇 𝑚 𝑘 + 1 + 1 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 − 𝜀 , 𝜆 ( 5 . 1 ) 2 𝑘 𝑛 𝑚 | | 𝑢 𝑛 𝑚 | | ( 𝑡 ) ≤ 𝐶 2 | | 𝜑 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 + | | 𝜓 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 + 1 𝜈 2 𝑛 𝜇 2 𝑚 + 1 𝜈 𝑛 2 − 𝜀 𝜇 2 𝑚 + 1 𝜈 2 𝑛 𝜇 𝑚 2 − 𝜀 ( 5 . 2 ) hold, where 𝐶 1 and 𝐶 2 do not depend on 𝜑 ( 𝑥 , 𝑦 ) and 𝜓 ( 𝑥 , 𝑦 ) .

Proof. Integrating by parts with respect to 𝑥 and 𝑦 in (3.5), (3.16), we get 𝜑 𝑛 𝑚 = 1 𝜈 𝑛 𝜇 𝑚 𝜑 ( 1 , 1 ) 𝑛 𝑚 , 𝜓 ( 5 . 3 ) 𝑛 𝑚 = 1 𝜈 𝑛 𝜇 𝑚 𝜓 ( 1 , 1 ) 𝑛 𝑚 , 𝑓 ( 5 . 4 ) 𝑛 𝑚 ( 1 𝑡 ) = 𝜈 𝑛 𝜇 𝑚 𝑓 ( 1 , 1 , 0 ) 𝑛 𝑚 ( 𝑓 𝑡 ) , ( 5 . 5 ) 𝑛 𝑚 1 ( 𝑡 ) = 𝜈 2 𝑛 𝜇 2 𝑚 𝑓 ( 2 , 2 , 0 ) 𝑛 𝑚 ( 𝑡 ) , ( 5 . 6 ) where 𝜑 ( 1 , 1 ) 𝑛 𝑚 = 𝑝 0 𝑞 0 𝜕 2 𝜑 ( 𝑥 , 𝑦 ) 𝑣 𝜕 𝑥 𝜕 𝑦 𝑛 𝑚 𝜓 ( 𝑥 , 𝑦 ) 𝑑 𝑦 𝑑 𝑥 , ( 1 , 1 ) 𝑛 𝑚 = 𝑝 0 𝑞 0 𝜕 2 𝜓 ( 𝑥 , 𝑦 ) 𝑣 𝜕 𝑥 𝜕 𝑦 𝑛 𝑚 𝑓 ( 𝑥 , 𝑦 ) 𝑑 𝑦 𝑑 𝑥 , ( 1 , 1 , 0 ) 𝑛 𝑚 ( 𝑡 ) = 𝑝 0 𝑞 0 𝜕 2 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) 𝑣 𝜕 𝑥 𝜕 𝑦 𝑛 𝑚 𝑓 ( 𝑥 , 𝑦 ) 𝑑 𝑦 𝑑 𝑥 , ( 2 , 2 , 0 ) 𝑛 𝑚 ( 𝑡 ) = 𝑝 0 𝑞 0 𝜕 4 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) 𝜕 𝑥 2 𝜕 𝑦 2 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) 𝑑 𝑦 𝑑 𝑥 . ( 5 . 7 ) Under the assumptions of Lemma 5.1, it follows that the functions 𝑓 ( 1 , 1 , 0 ) 𝑛 𝑚 ( 𝑡 ) and 𝑓 ( 2 , 2 , 0 ) 𝑛 𝑚 ( 𝑡 ) are bounded, that is, | | 𝑓 ( 1 , 1 , 0 ) 𝑛 𝑚 | | ( 𝑡 ) ≤ 𝑁 1 , | | 𝑓 ( 2 , 2 , 0 ) 𝑛 𝑚 | | ( 𝑡 ) ≤ 𝑁 2 , ( 5 . 8 ) where 𝑁 1 = c o n s t > 0 , 𝑁 2 = c o n s t > 0 . Let 0 < 𝑡 0 ≤ 𝑡 ≤ 𝑇 , where 𝑡 0 is a sufficiently small number. For sufficiently large 𝑛 and 𝑚 , the following inequalities are true: l n 𝜆 𝜀 𝑛 𝑚 < 𝜆 𝜀 𝑛 𝑚 < 𝜈 𝜀 𝑛 + 𝜇 𝜀 𝑚 , 0 < 𝜀 < 1 , 1 + 𝜆 2 𝑘 𝑛 𝑚 𝑇 𝛼 < 2 𝜆 2 𝑘 𝑛 𝑚 𝑇 𝛼 . ( 5 . 9 )
Using (3.16), (5.8), (5.9), and (3.17), we get | | 𝑢 𝑛 𝑚 | | 1 ( 𝑡 ) ≤ 𝑀 2 𝑡 𝛼 0 | | 𝜑 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + 1 2 𝑡 0 𝛼 − 1 | | 𝜓 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 − 𝑁 1 𝛼 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 𝑡 0 𝑑 1 + 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 1 + 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 1 ≤ 𝑀 2 𝑡 𝛼 0 | | 𝜑 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + 1 2 𝑡 0 𝛼 − 1 | | 𝜓 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + 𝑁 1 l n 2 𝑇 𝛼 + ( 2 𝑘 / 𝜀 ) l n 𝜆 𝜀 𝑛 𝑚 𝛼 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 ≤ 𝐶 1 | | 𝜑 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + | | 𝜓 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + 1 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 + 1 𝜈 𝑛 𝑘 + 1 − 𝜀 𝜇 𝑚 𝑘 + 1 + 1 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 − 𝜀 , ( 5 . 1 0 ) where 𝐶 1 = m a x { 𝑀 / 2 𝑡 𝛼 0 , 𝑀 / 2 𝑡 0 𝛼 − 1 , 𝑀 𝑁 1 l n 2 𝑇 𝛼 / 𝛼 , 2 𝑘 𝑀 𝑁 1 / 𝛼 𝜀 } . Thus, inequality (5.1) is obtained. Now, we will prove inequality (5.2). Using (5.3), (5.4), (5.6), (5.8), (5.9), and (3.17), we get 𝜆 2 𝑘 𝑛 𝑚 | | 𝑢 𝑛 𝑚 | | | | 𝜑 ( 𝑡 ) ≤ 𝑀 𝑛 𝑚 | | 𝑡 𝛼 + | | 𝜓 𝑛 𝑚 | | 𝑡 𝛼 − 1 + 𝜆 2 𝑘 𝑛 𝑚 𝑡 0 ( 𝑡 − 𝜏 ) 𝛼 − 1 𝑓 𝑛 𝑚 ( 𝜏 ) 1 + 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 1 𝑑 𝜏 ≤ 𝑀 𝑡 𝛼 0 | | 𝜑 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 + | | 𝜓 ( 1 , 1 ) 𝑛 𝑚 | | 𝑡 0 𝛼 − 1 𝜈 𝑛 𝜇 𝑚 − 𝑁 2 𝛼 𝑡 0 𝑑 1 + 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 𝜈 2 𝑛 𝜇 2 𝑚 1 + 𝜆 2 𝑘 𝑛 𝑚 ( 𝑡 − 𝜏 ) 𝛼 ≤ 𝑀 𝑡 𝛼 0 | | 𝜑 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 + 𝑀 𝑡 0 𝛼 − 1 | | 𝜓 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 + 2 𝑀 𝑁 2 l n 𝑇 𝛼 𝛼 𝜈 2 𝑛 𝜇 2 𝑚 + 2 𝑘 𝑀 𝑁 2 𝛼 𝜀 𝜈 𝑛 2 − 𝜀 𝜇 2 𝑚 + 2 𝑘 𝑀 𝑁 2 𝛼 𝜀 𝜈 2 𝑛 𝜇 𝑚 2 − 𝜀 ≤ 𝐶 2 | | 𝜑 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 + | | 𝜓 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 + 1 𝜈 2 𝑛 𝜇 2 𝑚 + 1 𝜈 𝑛 2 − 𝜀 𝜇 2 𝑚 + 1 𝜈 2 𝑛 𝜇 𝑚 2 − 𝜀 , ( 5 . 1 1 ) where 𝐶 2 = m a x { 𝑀 / 𝑡 𝛼 0 , 𝑀 / 𝑡 0 𝛼 − 1 , 𝑀 𝑁 2 l n 2 𝑇 𝛼 / 𝛼 , 2 𝑘 𝑀 𝑁 2 / 𝛼 𝜀 } . Lemma 5.1 is proved.

Theorem 5.2. Suppose that the assumptions of Lemma 5.1 hold. Then, there exists a regular solution of problem (2.13).

Proof. We will prove uniform and absolute convergence of series (3.1) and 𝜕 2 𝑘 𝑢 ( 𝑥 , 𝑦 , 𝑡 ) 𝜕 𝑥 2 𝑘 = ∞ 𝑛 , 𝑚 = 1 ( − 1 ) 𝑘 𝜈 𝑛 2 𝑘 𝑢 𝑛 𝑚 ( 𝑡 ) 𝑣 𝑛 𝑚 ( 𝜕 𝑥 , 𝑦 ) , ( 5 . 1 2 ) 2 𝑘 𝑢 ( 𝑥 , 𝑦 , 𝑡 ) 𝜕 𝑥 2 𝑘 = ∞ 𝑛 , 𝑚 = 1 ( − 1 ) 𝑘 𝜇 𝑛 2 𝑘 𝑢 𝑛 𝑚 ( 𝑡 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) , ( 5 . 1 3 ) 𝑐 𝐷 𝛼 0 + 𝑢 ( 𝑥 , 𝑦 , 𝑡 ) = − ∞ 𝑛 , 𝑚 = 1 ( − 1 ) 𝑘 𝜆 2 𝑘 𝑛 𝑚 𝑢 𝑛 𝑚 ( 𝑡 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) + ∞ 𝑛 , 𝑚 = 1 𝑓 𝑛 𝑚 ( 𝑡 ) 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) . ( 5 . 1 4 ) The series ∞ 𝑛 , 𝑚 = 1 | | 𝑢 𝑛 𝑚 ( | | 𝑡 ) ( 5 . 1 5 ) is majorant for the series (3.1). From (5.1), it follows that the series (5.15) uniformly converges. Actually, ∞ 𝑛 , 𝑚 = 1 | | 𝑢 𝑛 𝑚 ( | | 𝑡 ) ≤ 𝐶 ∞ 𝑛 , 𝑚 = 1 | | 𝜑 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + | | 𝜓 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 + 1 𝜈 𝑘 𝑛 𝜇 𝑘 𝑛 + 1 𝜈 𝑛 𝑘 + 1 − 𝜀 𝜇 𝑛 𝑘 + 1 + 1 𝜈 𝑛 𝑘 + 1 𝜇 𝑛 𝑘 + 1 − 𝜀 . ( 5 . 1 6 ) Applying the Cauchy-Schwarz inequality and the Parseval equality, we obtain ∞ 𝑛 , 𝑚 = 1 | | 𝜑 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 ≤ ∞ 𝑛 , 𝑚 = 1 1 𝜈 𝑛 2 𝑘 𝜇 𝑚 2 𝑘 1 / 2 ∞ 𝑛 , 𝑚 = 1 | | 𝜑 𝑛 𝑚 | | 2 1 / 2 = 𝑝 𝑘 𝑞 𝑘 𝜋 2 𝑘 ∞ 𝑛 = 1 1 𝑛 ∞ 2 𝑘 𝑚 = 1 1 𝑚 2 𝑘 1 / 2 ‖ 𝜑 ‖ 𝐿 2 ( Ω 0 ) . ( 5 . 1 7 ) Analogously, we get ∞ 𝑛 , 𝑚 = 1 | | 𝜓 𝑛 𝑚 | | 𝜈 𝑘 𝑛 𝜇 𝑘 𝑚 ≤ 𝑝 𝑘 𝑞 𝑘 𝜋 2 𝑘 ∞ 𝑛 = 1 1 𝑛 ∞ 2 𝑘 𝑚 = 1 1 𝑚 2 𝑘 1 / 2 ‖ 𝜓 ‖ 𝐿 2 ( Ω 0 ) . ( 5 . 1 8 ) Since 2 𝑘 ≥ 2 , then the series ∑ ∞ 𝑛 = 1 ( 1 / 𝑛 2 𝑘 ) , ∑ ∞ 𝑚 = 1 ( 1 / 𝑚 2 𝑘 ) converges by the integral test. Further, 𝑘 + 1 − 𝜀 > 1 , then the series ∞ 𝑛 , 𝑚 = 1 1 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 , ∞ 𝑛 , 𝑚 = 1 1 𝜈 𝑛 𝑘 + 1 − 𝜀 𝜇 𝑚 𝑘 + 1 , ∞ 𝑛 , 𝑚 = 1 1 𝜈 𝑛 𝑘 + 1 𝜇 𝑚 𝑘 + 1 − 𝜀 ( 5 . 1 9 ) 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 𝑛 2 𝑘 < 𝜆 2 𝑘 𝑛 𝑚 ,  𝜇 𝑚 2 𝑘 < 𝜆 2 𝑘 𝑛 𝑚 , then the series ∞ 𝑛 , 𝑚 = 1 𝜆 2 𝑘 𝑛 𝑚 | | 𝑢 𝑛 𝑚 | | ( 5 . 2 0 ) is majorant for the series (5.12), (5.13) and for the first series from (5.14). From (5.2), it follows that the series (5.20) uniformly converges. Indeed, using the Parseval equality and Cauchy-Schwarz inequality, we get ∞ 𝑛 , 𝑚 = 1 | | 𝜑 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 ≤ ∞ 𝑛 = 1 1 𝜈 2 𝑛 ∞ 𝑚 = 1 1 𝑚 2 1 / 2 ∞ 𝑛 , 𝑚 = 1 | | 𝜑 ( 1 , 1 ) 𝑛 𝑚 | | 2 1 / 2 = 𝑝 𝑞 6 ‖ ‖ ‖ 𝜕 2 𝜑 ‖ ‖ ‖ 𝜕 𝑥 𝜕 𝑦 𝐿 2 ( Ω 0 ) . ( 5 . 2 1 ) Analogously, we conclude that ∞ 𝑛 , 𝑚 = 1 | | 𝜓 ( 1 , 1 ) 𝑛 𝑚 | | 𝜈 𝑛 𝜇 𝑚 ≤ 𝑝 𝑞 6 ‖ ‖ ‖ 𝜕 2 𝜓 ‖ ‖ ‖ 𝜕 𝑥 𝜕 𝑦 𝐿 2 ( Ω 0 ) . ( 5 . 2 2 ) The series ∞ 𝑛 , 𝑚 = 1 1 𝜈 2 𝑛 𝜇 2 𝑚 + 1 𝜈 𝑛 2 − 𝜀 𝜇 2 𝑚 + 1 𝜈 2 𝑛 𝜇 𝑚 2 − 𝜀 ( 5 . 2 3 ) converges for any 𝜀 ∈ ( 0 , 1 ) according to the integral test. The series ∞ 𝑛 , 𝑚 = 1 | | 𝑓 𝑛 𝑚 ( | | 𝑡 ) ( 5 . 2 4 ) is majorant for the second series from (5.14). From (5.6) and (5.8), it follows that the series (5.14) uniformly converges. Indeed, ∞ 𝑛 , 𝑚 = 1 | | 𝑓 𝑛 𝑚 ( | | = 𝑡 ) ∞ 𝑛 , 𝑚 = 1 1 𝜈 2 𝑛 𝜇 2 𝑚 | | 𝑓 2 , 2 , 0 𝑛 𝑚 ( | | 𝑡 ) ≤ 𝑁 2 ∞ 𝑛 , 𝑚 = 1 1 𝜈 2 𝑛 𝜇 2 𝑚 = 𝑁 2 𝑝 2 𝑞 2 . 3 6 ( 5 . 2 5 ) Adding equality (5.12), (5.13), and (5.14), we note that the solution (3.1) satisfies equation (2.13). The solution (3.1) satisfies boundary conditions owing to properties of the functions 𝑣 𝑛 𝑚 ( 𝑥 , 𝑦 ) . Simple computations show that l i m 𝑡 → 0 𝐸 𝛼 , 1 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 = 1 , l i m 𝑡 → 0 𝑑 𝐸 𝑑 𝑡 𝛼 , 1 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 = 0 , l i m 𝑡 → 0 𝐸 𝛼 , 2 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 = 1 , l i m 𝑡 → 0 𝑡 𝑑 𝐸 𝑑 𝑡 𝛼 , 2 − 𝜆 2 𝑘 𝑛 𝑚 𝑡 𝛼 = 0 . ( 5 . 2 6 ) Consequently, l i m 𝑡 → 0 𝑢 𝑛 𝑚 ( 𝑡 ) = 𝜑 𝑛 𝑚 ,  l i m 𝑡 → 0 𝑢 𝑛 𝑚 ( 𝑡 ) = 𝜓 𝑛 𝑚 . Hence, we conclude that the solution (3.1) satisfies initial conditions. Theorem 5.2 is proved.

6. 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: ( − 1 ) 𝑘 𝑐 𝐷 𝛼 0 + 𝜕 𝑢 + 2 𝑘 𝑢 𝜕 𝑥 2 𝑘 + 𝜕 2 𝑘 𝑢 𝜕 𝑦 2 𝑘 𝜕 + 𝑢 = 𝑓 ( 𝑥 , 𝑦 , 𝑡 ) , 0 < 𝑥 < 𝑝 , 0 < 𝑦 < 𝑞 , 0 < 𝑡 < 𝑇 , 2 𝑚 + 1 𝑢 ( 0 , 𝑦 , 𝑡 ) 𝜕 𝑥 2 𝑚 = 𝜕 2 𝑚 + 1 𝑢 ( 𝑝 , 𝑦 , 𝑡 ) 𝜕 𝑥 2 𝑚 𝜕 = 0 , 𝑚 = 0 , 1 , … , 𝑘 − 1 , 0 ≤ 𝑦 ≤ 𝑞 , 0 ≤ 𝑡 ≤ 𝑇 , 2 𝑚 + 1 𝑢 ( 𝑥 , 0 , 𝑡 ) 𝜕 𝑦 2 𝑚 = 𝜕 2 𝑚 + 1 𝑢 ( 𝑥 , 𝑞 , 𝑡 ) 𝜕 𝑦 2 𝑚 = 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 [12, 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.

Acknowledgment

The authors are grateful to Professor Valery Covachev (Sultan Qaboos University, Sultanate of Oman) for his insightful comments and suggestions.

References

  1. R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. G. Sorrentinos, “Fractional derivative linear models for describing the viscoelastic dynamic behavior of polymeric beams,” in Proceedings of IMAS, Saint Louis, Mo, USA, 2006. View at Publisher · View at Google Scholar
  3. G. Sorrentinos, “Analytic modeling and experimental identi?cation of viscoelastic mechanical systems,” in Advances in Fractional Calculus, J. Sabatier, O. P. Agrawal, and J. A Tenreiro Machado, Eds., pp. 403–416, Springer, 2007. View at Publisher · View at Google Scholar
  4. Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, Springer, New York, NY, USA, 1997.
  5. R. L. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 1–104, 2004. View at Publisher · View at Google Scholar · View at Scopus
  6. M. D. Ortigueira and J. A. Tenreiro Machado, “Special issue on Fractional signal processing and applications,” Signal Processing, vol. 83, no. 11, pp. 2285–2286, 2003. View at Publisher · View at Google Scholar · View at Scopus
  7. B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 231–248, 2000. View at Google Scholar · View at Zentralblatt MATH
  8. K. B. Oldham, “Fractional differential equations in electrochemistry,” Advances in Engineering Software, vol. 41, no. 1, pp. 9–12, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. R. Metzler and J. Klafter, “Boundary value problems for fractional diffusion equations,” Physica A, vol. 278, no. 1-2, pp. 107–125, 2000. View at Publisher · View at Google Scholar
  10. M. De la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,” Abstract and Applied Analysis, vol. 2011, Article ID 161246, 25 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
  12. A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Nrtherlands, 2006.
  14. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, London, UK, 1993.
  15. J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240–268, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. C. Yuan, “Two positive solutions for (n-1,1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 930–942, 2012. View at Publisher · View at Google Scholar
  17. M. De la Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a SVEIRS epidemic model under constant and impulsive vaccination,” Advances in Difference Equations, vol. 2011, Article ID 748608, 32 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  18. O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1–10, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,” Georgian Mathematical Journal, vol. 16, no. 3, pp. 401–411, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011. View at Publisher · View at Google Scholar
  23. A. Ashyralyev, F. Dal, and Z. Pınar, “A note on the fractional hyperbolic differential and difference equations,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4654–4664, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations,” AIP Conference Proceeding, vol. 1389, pp. 617–620, 2011. View at Google Scholar
  25. A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176–1180, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. A. A. Kilbas and O. A. Repin, “Analogue of Tricomi's problem for partial differential equations containing diffussion equation of fractional order,” in Proceedings of the International Russian-Bulgarian Symposium Mixed type equations and related problems of analysis and informatics, pp. 123–127, Nalchik-Haber, 2010.
  27. A. A. Nahushev, Elements of Fractional Calculus and Their Applications, Nalchik, Russia, 2010.
  28. A. V. Pshu, Boundary Value Problems for Partial Differential Equations of Fractional and Continual Order, Nalchik, Russia, 2005.
  29. N. A. Virchenko and V. Y. Ribak, Foundations of Fractional Integro-Differentiations, Kiev, Ukraine, 2007.
  30. M. M. Džrbašjan and A. B. Nersesjan, “Fractional derivatives and the Cauchy problem for differential equations of fractional order,” Izvestija Akademii Nauk Armjanskoĭ SSR, vol. 3, no. 1, pp. 3–28, 1968. View at Google Scholar
  31. R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009. View at Google Scholar · View at Zentralblatt MATH
  33. R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,” Advances in Difference Equations, vol. 2010, Article ID 179750, 25 pages, 2010. View at Google Scholar · View at Zentralblatt MATH
  34. R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 3532–3554, 2010. View at Publisher · View at Google Scholar
  35. D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis, vol. 69, no. 11, pp. 3692–3705, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis, vol. 70, no. 5, pp. 1873–1876, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. G. M. Mophou and G. M. N'Guérékata, “Mild solutions for semilinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2009, no. 21, 9 pages, 2009. View at Google Scholar · View at Zentralblatt MATH
  38. G. M. Mophou and G. M. N'Guérékata, “Existence of the mild solution for some fractional differential equations with nonlocal conditions,” Semigroup Forum, vol. 79, no. 2, pp. 315–322, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis, vol. 69, no. 10, pp. 3337–3343, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. V. Lakshmikantham and J. V. Devi, “Theory of fractional differential equations in a Banach space,” European Journal of Pure and Applied Mathematics, vol. 1, no. 1, pp. 38–45, 2008. View at Google Scholar · View at Zentralblatt MATH
  41. V. Lakshmikantham and A. S. Vatsala, “Theory of fractional differential inequalities and applications,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 395–402, 2007. View at Google Scholar · View at Zentralblatt MATH
  42. A. S. Berdyshev, A. Cabada, and E. T. Karimov, “On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator,” Nonlinear Analysis, vol. 75, no. 6, pp. 3268–3273, 2011. View at Google Scholar
  43. R. Gorenflo, Y. F. Luchko, and S. R. Umarov, “On the Cauchy and multi-point problems for partial pseudo-differential equations of fractional order,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 249–275, 2000. View at Google Scholar · View at Zentralblatt MATH
  44. B. Kadirkulov and K. H. Turmetov, “About one generalization of heat conductivity equation,” Uzbek Mathematical Journal, no. 3, pp. 40–45, 2006 (Russian). View at Google Scholar
  45. D. Amanov, “Solvability of boundary value problems for equation of higher order with fractional derivatives,” in Boundary Value Problems for Differential Equations, The Collection of Proceedings no. 17, pp. 204–209, Chernovtsi, Russia, 2008. View at Google Scholar
  46. D. Amanov, “Solvability of boundary value problems for higher order differential equation with fractional derivatives,” in Problems of Camputations and Applied Mathemaitics, no. 121, pp. 55–62, Tashkent, Uzbekistan, 2009. View at Google Scholar
  47. M. M. Djrbashyan, Integral Transformations and Representatation of Functions in Complex Domain, Moscow, Russia, 1966.