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

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)∈ℝ²,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^𝛼𝑎+𝑔(𝑡)=1Γ(𝛼)^𝑡𝑎𝑔(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,(2.1) where Γ(⋅) is the Gamma function defined for any complex number 𝑧 as Γ(𝑧)=^∞0𝑡^𝑧−1𝑒^−𝑡𝑑𝑡.(2.2)

Definition 2.2. The Caputo fractional derivative of order 𝛼>0 of a continuous function 𝑔∶(𝑎,𝑏)→𝑅 is defined by ^𝑐𝐷^𝛼𝑎+𝑔(𝑡)=1Γ(𝑛−𝛼)^𝑡𝑎𝑔^(𝑛)(𝑠)(𝑡−𝑠)^{𝛼−𝑛+1}𝑑𝑠,(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)∈𝐿₁[0,𝑇]. Then, 𝐼^𝑝0+𝐼^𝑞0+𝑓(𝑡)=𝐼^𝑝+𝑞0+𝑓(𝑡)=𝐼^𝑞0+𝐼^𝑝0+𝑓(𝑡)(2.4) is satisfied almost everywhere on [0,𝑇]. Moreover, if 𝑓(𝑡)∈𝐶[0,𝑇], then (2.4) is true and ^𝑐𝐷^𝛼0+𝐼^𝛼0+𝑓(𝑡)=𝑓(𝑡) for all 𝑡∈[0,𝑇] and 𝛼>0.

Theorem 2.4 (see [47, page 123]). Let 𝑓(𝑡)∈𝐿₁(0,𝑇). Then, the integral equation 𝑧(𝑡)=𝑓(𝑡)+𝜆^𝑡0(𝑡−𝜏)^𝛼−1Γ(𝛼)𝑧(𝜏)𝑑𝜏(2.5) has a unique solution 𝑧(𝑡) defined by the following formula: 𝑧(𝑡)=𝑓(𝑡)+𝜆^𝑡0(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼(𝜆(𝑡−𝜏)^𝛼)𝑓(𝜏)𝑑𝜏,(2.6) where 𝐸_𝛼,𝛽(𝑧)=∑^∞𝑘=0(𝑧^𝑘/Γ(𝑘𝛼+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 𝐵𝑧(𝑡)=𝜆^𝑡0(𝑡−𝜏)^𝛼−1Γ(𝛼)𝑧(𝜏)𝑑𝜏.(2.7) Then, 𝑧(𝑡)=^𝑚−1_𝑘=0𝐵^𝑘𝑓(𝑡)+𝐵^𝑚𝑧(𝑡),𝑚=1,…,𝑛.(2.8) The proof of this theorem is based on formula (2.8) and 𝐵^𝑚𝑧(𝑡)=𝜆^𝑚^𝑡0(𝑡−𝜏)^𝑚𝛼−1Γ(𝑚𝛼)𝑧(𝜏)𝑑𝜏,(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^𝑚𝑧(𝑡)=𝜆^𝑚−1^𝑡0(𝑡−𝑠)^{(𝑚−1)𝛼−1}=𝜆_𝑚−1Γ((𝑚−1)𝛼)𝐵𝑧(𝑠)𝑑𝑠^𝑡0(𝑡−𝑠)^{(𝑚−1)𝛼−1}Γ((𝑚−1)𝛼)𝜆^𝑠0=𝜆(𝑠−𝜏)^𝛼−1𝑧(𝜏)𝑑𝜏𝑑𝑠_𝑚Γ(𝛼)Γ((𝑚−1)𝛼)^𝑡0^𝑠0=𝜆(𝑡−𝑠)^{(𝑚−1)𝛼−1}(𝑠−𝜏)^𝛼−1𝑧(𝜏)𝑑𝜏𝑑𝑠_𝑚Γ(𝛼)Γ((𝑚−1)𝛼)^𝑡0^𝑡𝜏(𝑡−𝑠)^{(𝑚−1)𝛼−1}(𝑠−𝜏)^𝛼−1𝑑𝑠𝑧(𝜏)𝑑𝜏.(2.10) Performing the change of variables 𝑠−𝜏=(𝑡−𝜏)𝑝, we get ^𝑡𝜏(𝑡−𝑠)^{(𝑚−1)𝛼−1}(𝑠−𝜏)^𝛼−1𝑑𝑠=(𝑡−𝜏)^𝑚𝛼−1¹⁰=(𝑡−𝜏)_𝑚𝛼−1(1−𝑝)^{(𝑚−1)𝛼−1}𝑝^𝛼−1𝑑𝑝=(𝑡−𝜏)_𝑚𝛼−1𝐵((𝑚−1)𝛼,𝛼)Γ(𝑚𝛼)Γ((𝑚−1)𝛼)Γ(𝛼).(2.11) Then, 𝐵^𝑚𝑧(𝑡)=𝜆^𝑚^𝑡0(𝑡−𝜏)^𝑚𝛼−1Γ(𝑚𝛼)𝑧(𝜏)𝑑𝜏.(2.12) 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)^𝑘𝑐𝐷^𝛼0+𝑢+𝜕^2𝑘𝑢𝜕𝑥^2𝑘+𝜕^2𝑘𝑢𝜕𝑦^2𝑘𝜕=𝑓(𝑥,𝑦,𝑡),0<𝑥<𝑝,0<𝑦<𝑞,0<𝑡<𝑇,_2𝑚𝑢(0,𝑦,𝑡)𝜕𝑥^2𝑚=𝜕^2𝑚𝑢(𝑝,𝑦,𝑡)𝜕𝑥^2𝑚𝜕=0,𝑚=0,1,…,𝑘−1,0≤𝑦≤𝑞,0≤𝑡≤𝑇,_2𝑚𝑢(𝑥,0,𝑡)𝜕𝑦^2𝑚=𝜕^2𝑚𝑢(𝑥,𝑞,𝑡)𝜕𝑦^2𝑚𝑢(𝑥,𝑦,0)=𝜑(𝑥,𝑦),𝑢_𝑡=0,𝑚=0,1,…,𝑘−1,0≤𝑥≤𝑝,0≤𝑡≤𝑇,(𝑥,𝑦,0)=𝜓(𝑥,𝑦),0≤𝑥≤𝑝,0≤𝑦≤𝑞(2.13) 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=1𝑢_𝑛𝑚(𝑡)𝑣_𝑛𝑚(𝑥,𝑦),(3.1) expanded along a complete orthonormal system: 𝑣_𝑛𝑚(𝑥,𝑦)=2√𝑝𝑞sin𝑛𝑝𝜋𝑥sin𝑚𝑞𝜋𝑦,1≤𝑛,𝑚<∞.(3.2) We denote 𝑛Ω𝜋₀=Ω∩(𝑡=0)={(𝑥,𝑦,0)∶0≤𝑥≤𝑝,0≤𝑦≤𝑞},𝑝=𝜈^𝑛,𝑚𝜋𝑞=𝜇^𝑚,𝜈^𝑛2𝑘+𝜇^𝑚2𝑘=𝜆^{2𝑘𝑛𝑚},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_𝑛𝑚(𝑥,𝑦),1≤𝑛,𝑚<∞: 𝑓(𝑥,𝑦,𝑡)=^∞_𝑛_,𝑚=1𝑓_𝑛𝑚(𝑡)𝑣_𝑛𝑚(𝑥,𝑦),(3.4) where 𝑓_𝑛𝑚(t)=^𝑝0^𝑞0𝑓(𝑥,𝑦,𝑡)𝑣_𝑛𝑚(𝑥,𝑦)𝑑𝑦𝑑𝑥,1≤𝑛,𝑚<∞.(3.5) Substituting (3.1) and (3.4) into (2.13), we obtain (−1)^𝑘𝑐𝐷^𝛼0+𝑢_𝑛𝑚(𝑡)+(−1)^𝑘𝜆^{2𝑘𝑛𝑚}𝑢_𝑛𝑚(𝑡)=𝑓_𝑛𝑚(𝑡).(3.6) By Lemma 2.3, we have that ^𝑐𝐷^𝛼0+𝑢_𝑛𝑚(𝑡)=𝐼^2−𝛼0+𝑢^{𝑛𝑚}(𝑡),(3.7) where 𝐼^𝛼0+𝑓(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏(3.8) is Riemann-Liouville integral of fractional order 𝛼. Using (3.6) and (3.7), we get the following equation: 𝐼^2−𝛼0+𝑢^{𝑛𝑚}(𝑡)+𝜆^{2𝑘𝑛𝑚}𝑢_𝑛𝑚(𝑡)=(−1)^𝑘𝑓_𝑛𝑚(𝑡).(3.9) Applying the operator 𝐼^𝛼0+ to this equation, we get the following Volterra integral equation of the second kind: 𝑢_𝑛𝑚(𝑡)=−𝜆^{2𝑘𝑛𝑚}Γ(𝛼)^𝑡0(𝑡−𝜏)^𝛼−1𝑢_𝑛𝑚(𝜏)𝑑𝜏+𝑢_𝑛𝑚(0)+𝑡𝑢^𝑛𝑚(0)+(−1)^𝑘𝐼^𝛼0+𝑓_𝑛𝑚(𝑡).(3.10) According to the Theorem 2.4, (3.10) has a unique solution 𝑢_𝑛𝑚(𝑡) defined by the following formula: 𝑢_𝑛𝑚(𝑡)=(−1)^𝑘Γ(𝛼)^𝑡0+𝑢_𝑛𝑚(𝑡−𝜏)^𝛼−1𝑓_𝑛𝑚(𝜏)𝑑𝜏(0)1−𝜆^{2𝑘𝑛𝑚}^𝑡0+𝑢_𝑛𝑚(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝑑𝜏(0)𝑡−𝜆^{2𝑘𝑛𝑚}^𝑡0−𝜆(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝜏𝑑𝜏_{2𝑘𝑛𝑚}Γ(𝛼)^𝑡0(𝑡−𝜂)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜂)^𝛼𝑑𝜂^𝜂0(𝜂−𝜏)^𝛼−1𝑓_𝑛𝑚(𝜏)𝑑𝜏.(3.11)

Using the formula (see, e.g., [27, page 118] and [47, page 120]) 1Γ(𝛽)^𝑧0𝑡^𝜇−1𝐸1_𝛼,𝜇(𝜆𝑡^𝛼)(𝑧−𝑡)^𝛽−1𝑑𝑡=𝑧^{𝜇+𝛽−1}𝐸_{𝛼,𝜇+𝛽}(𝜆𝑧^𝛼),Γ(𝜇)+𝑧𝐸^{𝛼,𝛼+𝜇}(𝑧)=𝐸_𝛼,𝜇(𝑧),(3.12) we get −𝜆^{2𝑘𝑛𝑚}Γ(𝛼)^𝑡0=(𝑡−𝜂)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜂)^𝛼𝑑𝜂^𝜂0(𝜂−𝜏)^𝛼−1𝑓_𝑛𝑚(𝜏)𝑑𝜏_𝑡0𝑓_𝑛𝑚(𝜏)−𝜆^{2𝑘𝑛𝑚}Γ(𝛼)^𝑡𝜏=(𝑡−𝜂)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜂)^𝛼(𝜂−𝜏)^𝛼−1𝑑𝜂𝑑𝜏_𝑡0𝑓_𝑛𝑚(𝜏)−𝜆^{2𝑘𝑛𝑚}Γ(𝛼)^0𝑡−𝜏=−_𝑡0𝑧^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}𝑧^𝛼(𝑡−𝜏−𝑧)^𝛼−1𝑑𝑧𝑑𝜏=𝑓_𝑛𝑚(𝜏)𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^2𝛼−1𝐸_𝛼,2𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝑑𝜏_𝑡0(𝑡−𝜏)^𝛼−1𝑓_𝑛𝑚(𝜏)−1Γ(𝛼)+𝐸^𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}−𝜆_{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝑑𝜏,^𝑡0=−𝜆_{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝑑𝜏^𝑡0=Γ(1)𝜆_{2𝑘𝑛𝑚}𝑧^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}𝑧^𝛼(𝑡−𝑧)¹⁻¹𝑑𝑧−𝜆_{2𝑘𝑛𝑚}𝑡^𝛼𝐸_𝛼,𝛼+1−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼=𝐸_𝛼,1−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼−1,^𝑡0=−𝜆_{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝜏𝑑𝜏^𝑡0=Γ(2)𝜆_{2𝑘𝑛𝑚}𝑧^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}𝑧^𝛼(𝑡−𝑧)²⁻¹𝑑𝑧𝑡^𝛼+1𝐸_𝛼,𝛼+2−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼=𝑡𝐸_𝛼,2−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼−𝑡. From these three formulas and (3.11), it follows that 𝑢_𝑛𝑚(𝑡)=+𝑢(_𝑛𝑚−(10))_𝑘𝐸_𝛼,1−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼+𝑡𝑢^𝑛𝑚(0)𝐸_𝛼,2−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼^𝑡0(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝑓_𝑛𝑚(𝜏)𝑑𝜏.(3.14) For 𝑢_𝑛𝑚(0) and 𝑢^𝑛𝑚(0), we expand the given functions 𝜑(𝑥,𝑦) and 𝜓(𝑥,𝑦) in the form of a Fourier series along the functions 𝑣_𝑛𝑚(𝑥,𝑦),1≤𝑛,𝑚<∞: 𝜑(𝑥,𝑦)=^∞_𝑛_,𝑚=1𝜓(𝑥,𝑦)=_∞𝜑_𝑛𝑚𝑣_𝑛𝑚(𝑥,𝑦),_𝑛_,𝑚=1𝜓_𝑛𝑚𝑣_𝑛𝑚(𝑥,𝑦),(3.15) where 𝜑_𝑛𝑚=^𝑝0^𝑞0𝜓𝜑(𝑥,𝑦)𝑣_𝑛𝑚(𝑥,𝑦)𝑑𝑦𝑑𝑥,_𝑛𝑚=^𝑝0^𝑞0𝜓(𝑥,𝑦)𝑣_𝑛𝑚(𝑥,𝑦)𝑑𝑦𝑑𝑥.(3.16) Using (2.13), (3.14), (3.16), we obtain 𝑢_𝑛𝑚(𝑡)=𝐸_𝛼,1−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼𝜑_𝑛𝑚+𝑡𝐸_𝛼,2−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼𝜓_𝑛𝑚+(−1)^𝑘^𝑡0(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼−𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝑓_𝑛𝑚(𝜏)𝑑𝜏.(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).

Applying the formula (3.17), the Cauchy-Schwarz inequality, and the estimate (see [13, page 136]) ||𝐸_𝛼,𝛽(𝑧)||≤𝑀1+|𝑧|,𝑀=const>0,Re𝑧<0,(3.18) we get the following inequality: ||𝑢_𝑛𝑚(𝑡)||≤𝐶₀||𝜑^𝑛𝑚||+||𝜓_𝑛𝑚||+^𝑡0||𝑓_𝑛𝑚(𝑡)||²𝑑𝑡^1/2(3.19) for the solution of (3.10) for any 𝑡,𝑡∈[0,𝑇]. Here, 𝐶₀=max{𝑀,𝑇𝑀,𝑀(𝑇^𝛼−1/2/√2𝛼−1)}.

4. Solvability of (2.13) in 𝐿₂(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)∈𝐿₂(Ω₀),𝜓(𝑥,𝑦)∈𝐿₂(Ω₀), and 𝑓(𝑥,𝑦,𝑡)∈𝐿₂(Ω), then the series (3.1) converges in 𝐿₂(Ω) to 𝑢∈𝐿₂(Ω) and for the solution of problem (2.13), the following stability inequality ‖𝑢‖_𝐿2(Ω)≤𝐶₁‖𝜑‖_𝐿2(Ω0)+‖𝜓‖_𝐿2(Ω0)+‖𝑓‖_𝐿2(Ω)(4.1) holds, where 𝐶₁ 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=1𝑢_𝑛𝑚(𝑡)𝑣_𝑛𝑚(𝑥,𝑦),(4.2) where 𝑁 is a natural number. For the positive integer number 𝐿, we have that ‖‖𝑢_𝑁+𝐿−𝑢_𝑁‖‖^2𝐿_2(Ω)=‖‖‖‖^𝑁+𝐿_𝑛,𝑚_=𝑁+1𝑢_𝑛𝑚(⋅)𝑣_𝑛𝑚(⋅,⋅)‖‖‖‖^2𝐿_2(Ω)=^𝑁+𝐿_𝑛,𝑚_=𝑁+1^𝑇0||𝑢_𝑛𝑚(𝑡)||²𝑑𝑡.(4.3) Applying (3.19), we get ^∞_𝑛,𝑚=1^𝑇0||𝑢_𝑛𝑚(𝑡)||²𝑑𝑡≤3𝐶²⁰^∞_𝑛,𝑚=1||𝜑_𝑛𝑚||²+^∞_𝑛_,𝑚=1||𝜓_𝑛𝑚||²+^∞_𝑛_,𝑚=1^𝑇0=𝐶₂||𝑓_𝑛𝑚(𝑡)||²𝑑𝑡‖𝜑‖^2𝐿_2Ω0+‖𝜓‖^2𝐿_2Ω0+‖𝑓‖^2𝐿_2(Ω),(4.4) where 𝐶²=3𝑇𝐶²⁰. Therefore, ∑^{𝑁+𝐿𝑛,𝑚=𝑁+1}∫^𝑇0|𝑢_𝑛𝑚(𝑡)|²𝑑𝑡→0 as 𝑁→∞. Consequently, the series (3.1) converges in 𝐿₂(Ω) to 𝑢(𝑥,𝑦,𝑡)∈𝐿₂(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)∈𝐶¹(Ω₀), 𝜑_𝑥𝑦(𝑥,𝑦)∈𝐿₂(Ω₀), 𝜓(𝑥,𝑦)∈𝐶¹(Ω₀), 𝜓_x𝑦(𝑥,𝑦)∈𝐿₂(Ω₀), 𝜑(𝑥,𝑦)=0 on 𝜕Ω₀, 𝜓(𝑥,𝑦)=0 on 𝜕Ω₀, 𝑓(𝑥,𝑦,𝑡)∈𝐶²(Ω),𝑓_𝑥𝑥𝑦(𝑥,𝑦,𝑡)∈𝐶(Ω) 𝑓_𝑥𝑦𝑦(𝑥,𝑦,𝑡)∈𝐶(Ω₀),𝑓_{𝑥𝑥𝑦𝑦}(𝑥,𝑦,𝑡)∈𝐶(Ω₀), and 𝑓(𝑥,𝑦,𝑡)=0 on 𝜕Ω×[0,𝑇]. Then, for any 𝜀∈(0,1), the following estimates ||𝑢_𝑛𝑚(𝑡)||≤𝐶₁||𝜑^𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+||𝜓^𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+1𝜈^𝑛𝑘+1𝜇^𝑚𝑘+1+1𝜈^{𝑛𝑘+1−𝜀}𝜇^𝑚𝑘+1+1𝜆𝜈^𝑛𝑘+1𝜇^{𝑚𝑘+1−𝜀},(5.1)_{2𝑘𝑛𝑚}||𝑢_𝑛𝑚(𝑡)||≤𝐶₂||𝜑^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚+||𝜓^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚+1𝜈^2𝑛𝜇^2𝑚+1𝜈^𝑛2−𝜀𝜇^2𝑚+1𝜈^2𝑛𝜇^𝑚2−𝜀(5.2) hold, where 𝐶₁ and 𝐶₂ 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_𝑛𝑚=1𝜈_𝑛𝜇_𝑚𝜓𝜑^{(1,1)𝑛𝑚},(5.3)_𝑛𝑚=1𝜈_𝑛𝜇_𝑚𝑓𝜓^{(1,1)𝑛𝑚},(5.4)_𝑛𝑚(𝑡)=1𝜈_𝑛𝜇_𝑚𝑓𝑓^{(1,1,0)𝑛𝑚}(𝑡),(5.5)_𝑛𝑚(𝑡)=1𝜈^2𝑛𝜇^2𝑚𝑓^{(2,2,0)𝑛𝑚}(𝑡),(5.6) where 𝜑^{(1,1)𝑛𝑚}=^𝑝0^𝑞0𝜕²𝜑(𝑥,𝑦)𝜕𝑥𝜕𝑦𝑣^𝑛𝑚𝜓(𝑥,𝑦)𝑑𝑦𝑑𝑥,_{(1,1)𝑛𝑚}=^𝑝0^𝑞0𝜕²𝜓(𝑥,𝑦)𝜕𝑥𝜕𝑦𝑣^𝑛𝑚𝑓(𝑥,𝑦)𝑑𝑦𝑑𝑥,_{(1,1,0)𝑛𝑚}(𝑡)=^𝑝0^𝑞0𝜕²𝑓(𝑥,𝑦,𝑡)𝜕𝑥𝜕𝑦𝑣^𝑛𝑚𝑓(𝑥,𝑦)𝑑𝑦𝑑𝑥,_{(2,2,0)𝑛𝑚}(𝑡)=^𝑝0^𝑞0𝜕⁴𝑓(𝑥,𝑦,𝑡)𝜕𝑥²𝜕𝑦²𝑣_𝑛𝑚(𝑥,𝑦)𝑑𝑦𝑑𝑥.(5.7) Under the assumptions of Lemma 5.1, it follows that the functions 𝑓^{(1,1,0)𝑛𝑚}(𝑡) and 𝑓^{(2,2,0)𝑛𝑚}(𝑡) are bounded, that is, ||𝑓^{(1,1,0)𝑛𝑚}(𝑡)||≤𝑁₁,||𝑓^{(2,2,0)𝑛𝑚}(𝑡)||≤𝑁₂,(5.8) where 𝑁₁=const>0, 𝑁₂=const>0. Let 0<𝑡₀≤𝑡≤𝑇, where 𝑡₀ is a sufficiently small number. For sufficiently large 𝑛 and 𝑚, the following inequalities are true: ln𝜆^𝜀𝑛𝑚<1𝜆+^𝜀𝑛𝑚𝜆<_{2𝑘𝑛𝑚}𝜈^𝜀𝑛+𝜇^𝜀𝑚,0<𝜀<1,𝑇^𝛼<2𝜆^{2𝑘𝑛𝑚}𝑇^𝛼.(5.9)
Using (3.16), (5.8), (5.9), and (3.17), we get ||𝑢_𝑛𝑚(𝑡)||≤𝑀12𝑡^𝛼0||𝜑_𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+12𝑡^0𝛼−1||𝜓_𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚−𝑁¹𝛼𝜈^𝑛𝑘+1𝜇^𝑚𝑘+1^𝑡0𝑑1+𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼≤𝑀11+𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼2𝑡^𝛼0||𝜑_𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+12𝑡^0𝛼−1||𝜓_𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+𝑁¹ln2𝑇^𝛼+(2𝑘/𝜀)ln𝜆^𝜀𝑛𝑚≤𝐶₁𝛼𝜈^𝑛𝑘+1𝜇^𝑚𝑘+1||𝜑^𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+||𝜓^𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+1𝜈^𝑛𝑘+1𝜇^𝑚𝑘+1+1𝜈^{𝑛𝑘+1−𝜀}𝜇^𝑚𝑘+1+1𝜈^𝑛𝑘+1𝜇^{𝑚𝑘+1−𝜀},(5.10) where 𝐶₁=max{𝑀/2𝑡^𝛼0,𝑀/2𝑡^0𝛼−1,𝑀𝑁₁ln2𝑇^𝛼/𝛼,2𝑘𝑀𝑁₁/𝛼𝜀}. 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^{2𝑘𝑛𝑚}||𝑢_𝑛𝑚(𝑡)||≤𝑀||𝜑^𝑛𝑚||𝑡^𝛼+||𝜓^𝑛𝑚||𝑡^𝛼−1+𝜆^{2𝑘𝑛𝑚}^𝑡0(𝑡−𝜏)^𝛼−1𝑓_𝑛𝑚(𝜏)1+𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼≤𝑀1𝑑𝜏𝑡^𝛼0||𝜑^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚+||𝜓^{(1,1)𝑛𝑚}||𝑡^0𝛼−1𝜈_𝑛𝜇_𝑚−𝑁²𝛼^𝑡0𝑑1+𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼≤𝑀𝜈^2𝑛𝜇^2𝑚1+𝜆^{2𝑘𝑛𝑚}(𝑡−𝜏)^𝛼𝑡^𝛼0||𝜑^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚+𝑀𝑡^0𝛼−1||𝜓^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚+2𝑀𝑁²ln𝑇^𝛼𝛼𝜈^2𝑛𝜇^2𝑚+2𝑘𝑀𝑁²𝛼𝜀𝜈^𝑛2−𝜀𝜇^2𝑚+2𝑘𝑀𝑁²𝛼𝜀𝜈^2𝑛𝜇^𝑚2−𝜀≤𝐶₂||𝜑^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚+||𝜓^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚+1𝜈^2𝑛𝜇^2𝑚+1𝜈^𝑛2−𝜀𝜇^2𝑚+1𝜈^2𝑛𝜇^𝑚2−𝜀,(5.11) where 𝐶₂=max{𝑀/𝑡^𝛼0,𝑀/𝑡^0𝛼−1,𝑀𝑁₂ln2𝑇^𝛼/𝛼,2𝑘𝑀𝑁₂/𝛼𝜀}. 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 𝜕^2𝑘𝑢(𝑥,𝑦,𝑡)𝜕𝑥^2𝑘=^∞_𝑛_,𝑚=1𝜕(−1)^𝑘𝜈^𝑛2𝑘𝑢_𝑛𝑚(𝑡)𝑣_𝑛𝑚(𝑥,𝑦),(5.12)_2𝑘𝑢(𝑥,𝑦,𝑡)𝜕𝑥^2𝑘=^∞_𝑛_,𝑚=1(−1)^𝑘𝜇^𝑛2𝑘𝑢_𝑛𝑚(𝑡)𝑣_𝑛𝑚(𝑥,𝑦),(5.13)_𝑐𝐷^𝛼0+𝑢(𝑥,𝑦,𝑡)=−^∞_𝑛_,𝑚=1(−1)^𝑘𝜆^{2𝑘𝑛𝑚}𝑢_𝑛𝑚(𝑡)𝑣_𝑛𝑚(𝑥,𝑦)+^∞_𝑛_,𝑚=1𝑓_𝑛𝑚(𝑡)𝑣_𝑛𝑚(𝑥,𝑦).(5.14) The series ^∞_𝑛,𝑚=1||𝑢_𝑛𝑚(𝑡)||(5.15) is majorant for the series (3.1). From (5.1), it follows that the series (5.15) uniformly converges. Actually, ^∞_𝑛,𝑚=1||𝑢_𝑛𝑚(𝑡)||≤𝐶^∞_𝑛_,𝑚=1||𝜑^𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+||𝜓^𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚+1𝜈^𝑘𝑛𝜇^𝑘𝑛+1𝜈^{𝑛𝑘+1−𝜀}𝜇^𝑛𝑘+1+1𝜈^𝑛𝑘+1𝜇^{𝑛𝑘+1−𝜀}.(5.16) Applying the Cauchy-Schwarz inequality and the Parseval equality, we obtain ^∞_𝑛,𝑚=1||𝜑_𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚≤^∞_𝑛_,𝑚=11𝜈^𝑛2𝑘𝜇^𝑚2𝑘^1/2^∞_𝑛,𝑚=1||𝜑_𝑛𝑚||²^1/2=𝑝^𝑘𝑞^𝑘𝜋^2𝑘^∞_𝑛=11𝑛^∞2𝑘_𝑚=11𝑚^2𝑘^1/2‖𝜑‖_𝐿2(Ω0).(5.17) Analogously, we get ^∞_𝑛,𝑚=1||𝜓_𝑛𝑚||𝜈^𝑘𝑛𝜇^𝑘𝑚≤𝑝^𝑘𝑞^𝑘𝜋^2𝑘^∞_𝑛=11𝑛^∞2𝑘_𝑚=11𝑚^2𝑘^1/2‖𝜓‖_𝐿2(Ω0).(5.18) 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 ^∞_𝑛,𝑚=11𝜈^𝑛𝑘+1𝜇^𝑚𝑘+1,^∞_𝑛_,𝑚=11𝜈^{𝑛𝑘+1−𝜀}𝜇^𝑚𝑘+1,^∞_𝑛_,𝑚=11𝜈^𝑛𝑘+1𝜇^{𝑚𝑘+1−𝜀}(5.19) 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_𝑡0=Ω×[𝑡₀,𝑇] for any 𝑡₀∈(0,𝑇). At 𝑡=0, the series (3.1) converges and has a sum equal to 𝜑(𝑥,𝑦). Since 𝜈^𝑛2𝑘<𝜆^{2𝑘𝑛𝑚}, 𝜇^𝑚2𝑘<𝜆^{2𝑘𝑛𝑚}, then the series ^∞_𝑛,𝑚=1𝜆^{2𝑘𝑛𝑚}||𝑢_𝑛𝑚||(5.20) 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)𝑛𝑚}||𝜈_𝑛𝜇_𝑚≤^∞_𝑛=11𝜈^2𝑛∞_𝑚=11𝑚²^1/2^∞_𝑛,𝑚=1||𝜑^{(1,1)𝑛𝑚}||²^1/2=𝑝𝑞6‖‖‖𝜕²𝜑𝜕𝑥𝜕𝑦‖‖‖_𝐿2(Ω0).(5.21) Analogously, we conclude that ^∞_𝑛,𝑚=1||𝜓^{(1,1)𝑛𝑚}||𝜈_𝑛𝜇_𝑚≤𝑝𝑞6‖‖‖𝜕²𝜓𝜕𝑥𝜕𝑦‖‖‖_𝐿2(Ω0).(5.22) The series ^∞_𝑛,𝑚=1𝜈^2𝑛1𝜇^2𝑚+1𝜈^𝑛2−𝜀𝜇^2𝑚+1𝜈^2𝑛𝜇^𝑚2−𝜀(5.23) converges for any 𝜀∈(0,1) according to the integral test. The series ^∞_𝑛,𝑚=1||𝑓_𝑛𝑚(𝑡)||(5.24) 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_𝑛𝑚(𝑡)||=^∞_𝑛_,𝑚=11𝜈^2𝑛𝜇^2𝑚||𝑓^{2,2,0𝑛𝑚}(𝑡)||≤𝑁_2∞𝑛_,𝑚=11𝜈^2𝑛𝜇^2𝑚=𝑁²𝑝²𝑞²36.(5.25) 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_𝑛𝑚(𝑥,𝑦). Simple computations show that lilmim_{𝑡→0𝑡→0}𝐸_𝛼,1−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼=1,𝑑𝑑𝑡𝐸^𝛼,1−𝜆^{2𝑘𝑛𝑚}lim_𝑡→0𝑡^𝛼=0,lim_𝑡→0𝐸_𝛼,2−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼=1,𝑡𝑑𝑑𝑡𝐸^𝛼,2−𝜆^{2𝑘𝑛𝑚}𝑡^𝛼=0.(5.26) Consequently, lim_𝑡→0𝑢_𝑛𝑚(𝑡)=𝜑_𝑛𝑚, lim_𝑡→0𝑢^𝑛𝑚(𝑡)=𝜓_𝑛𝑚. 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)^𝑘𝑐𝐷^𝛼0+𝑢+𝜕^2𝑘𝑢𝜕𝑥^2𝑘+𝜕^2𝑘𝑢𝜕𝑦^2𝑘𝜕+𝑢=𝑓(𝑥,𝑦,𝑡),0<𝑥<𝑝,0<𝑦<𝑞,0<𝑡<𝑇,_2𝑚+1𝑢(0,𝑦,𝑡)𝜕𝑥^2𝑚=𝜕^2𝑚+1𝑢(𝑝,𝑦,𝑡)𝜕𝑥^2𝑚𝜕=0,𝑚=0,1,…,𝑘−1,0≤𝑦≤𝑞,0≤𝑡≤𝑇,_2𝑚+1𝑢(𝑥,0,𝑡)𝜕𝑦^2𝑚=𝜕^2𝑚+1𝑢(𝑥,𝑞,𝑡)𝜕𝑦^2𝑚𝑢(𝑥,𝑦,0)=𝜑(𝑥,𝑦),𝑢_𝑡=0,𝑚=0,1,…,𝑘−1,0≤𝑥≤𝑝,0≤𝑡≤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.