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
(2.1)Ia+αg(t)=1Γ(α)∫atg(s)(t-s)1-αds,
where Γ(·) is the Gamma function defined for any complex number z as
(2.2)Γ(z)=∫0∞tz-1e-tdt.

Definition 2.2.

The Caputo fractional derivative of order α>0 of a continuous function g:(a,b)→R is defined by
(2.3)Dca+αg(t)=1Γ(n-α)∫atg(n)(s)(t-s)α-n+1ds,
where n=[α]+1, (the notation [α] stands for the largest integer not greater than α).

Lemma 2.3 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Let p,q≥0, f(t)∈L1[0,T]. Then,
(2.4)I0+pI0+qf(t)=I0+p+qf(t)=I0+qI0+pf(t)
is satisfied almost everywhere on [0,T]. Moreover, if f(t)∈C[0,T], then (2.4) is true and Dc0+αI0+αf(t)=f(t) for all t∈[0,T] and α>0.

Theorem 2.4 (see [<xref ref-type="bibr" rid="B31">47</xref>, page 123]).

Let f(t)∈L1(0,T). Then, the integral equation
(2.5)z(t)=f(t)+λ∫0t(t-τ)α-1Γ(α)z(τ)dτ
has a unique solution z(t) defined by the following formula:
(2.6)z(t)=f(t)+λ∫0t(t-τ)α-1Eα,α(λ(t-τ)α)f(τ)dτ,
where Eα,β(z)=∑k=0∞(zk/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
(2.7)Bz(t)=λ∫0t(t-τ)α-1Γ(α)z(τ)dτ.
Then,
(2.8)z(t)=∑k=0m-1Bkf(t)+Bmz(t),m=1,…,n.
The proof of this theorem is based on formula (2.8) and
(2.9)Bmz(t)=λm∫0t(t-τ)mα-1Γ(mα)z(τ)dτ,
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
(2.10)Bmz(t)=λm-1∫0t(t-s)(m-1)α-1Γ((m-1)α)Bz(s)ds=λm-1∫0t(t-s)(m-1)α-1Γ((m-1)α)λ∫0s(s-τ)α-1z(τ)dτds=λmΓ(α)Γ((m-1)α)∫0t∫0s(t-s)(m-1)α-1(s-τ)α-1z(τ)dτds=λmΓ(α)Γ((m-1)α)∫0t∫τt(t-s)(m-1)α-1(s-τ)α-1dsz(τ)dτ.
Performing the change of variables s-τ=(t-τ)p, we get
(2.11)∫τt(t-s)(m-1)α-1(s-τ)α-1ds=(t-τ)mα-1∫01(1-p)(m-1)α-1pα-1dp=(t-τ)mα-1B((m-1)α,α)=(t-τ)mα-1Γ(mα)Γ((m-1)α)Γ(α).
Then,
(2.12)Bmz(t)=λm∫0t(t-τ)mα-1Γ(mα)z(τ)dτ.
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:
(2.13)(-1)kDc0+αu+∂2ku∂x2k+∂2ku∂y2k=f(x,y,t),0<x<p,0<y<q,0<t<T,∂2mu(0,y,t)∂x2m=∂2mu(p,y,t)∂x2m=0,m=0,1,…,k-1,0≤y≤q,0≤t≤T,∂2mu(x,0,t)∂y2m=∂2mu(x,q,t)∂y2m=0,m=0,1,…,k-1,0≤x≤p,0≤t≤T,u(x,y,0)=φ(x,y),ut(x,y,0)=ψ(x,y),0≤x≤p,0≤y≤q
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 (<xref ref-type="disp-formula" rid="EEq7">2.13</xref>)

We seek a solution of problem (2.13) in the form of Fourier series:
(3.1)u(x,y,t)=∑n,m=1∞unm(t)vnm(x,y),
expanded along a complete orthonormal system:
(3.2)vnm(x,y)=2pqsinnπpxsinmπqy,1≤n,m<∞.
We denote
(3.3)Ω0=Ω¯∩(t=0)={(x,y,0):0≤x≤p,0≤y≤q},nπp=νn,mπq=μm,νn2k+μm2k=λnm2k,1≤n,m<∞.
We expand the given function f(x,y,t) in the form of a Fourier series along the functions vnm(x,y),1≤n,m<∞:
(3.4)f(x,y,t)=∑n,m=1∞fnm(t)vnm(x,y),
where
(3.5)fnm(t)=∫0p∫0qf(x,y,t)vnm(x,y)dydx,1≤n,m<∞.
Substituting (3.1) and (3.4) into (2.13), we obtain
(3.6)(-1)kDc0+αunm(t)+(-1)kλnm2kunm(t)=fnm(t).
By Lemma 2.3, we have that
(3.7)Dc0+αunm(t)=I0+2-αunm′′(t),
where
(3.8)I0+αf(t)=1Γ(α)∫0t(t-τ)α-1f(τ)dτ
is Riemann-Liouville integral of fractional order α. Using (3.6) and (3.7), we get the following equation:
(3.9)I0+2-αunm′′(t)+λnm2kunm(t)=(-1)kfnm(t).
Applying the operator I0+α to this equation, we get the following Volterra integral equation of the second kind:
(3.10)unm(t)=-λnm2kΓ(α)∫0t(t-τ)α-1unm(τ)dτ+unm(0)+tunm′(0)+(-1)kI0+αfnm(t).
According to the Theorem 2.4, (3.10) has a unique solution unm(t) defined by the following formula:
(3.11)unm(t)=(-1)kΓ(α)∫0t(t-τ)α-1fnm(τ)dτ+unm(0)[1-λnm2k∫0t(t-τ)α-1Eα,α(-λnm2k(t-τ)α)dτ]+unm′(0)[t-λnm2k∫0t(t-τ)α-1Eα,α(-λnm2k(t-τ)α)τdτ]-λnm2kΓ(α)∫0t(t-η)α-1Eα,α(-λnm2k(t-η)α)dη∫0η(η-τ)α-1fnm(τ)dτ.

Using the formula (see, e.g., [27, page 118] and [47, page 120])
(3.12)1Γ(β)∫0ztμ-1Eα,μ(λtα)(z-t)β-1dt=zμ+β-1Eα,μ+β(λzα),1Γ(μ)+zEα,α+μ(z)=Eα,μ(z),
we get
-λnm2kΓ(α)∫0t(t-η)α-1Eα,α(-λnm2k(t-η)α)dη∫0η(η-τ)α-1fnm(τ)dτ=∫0tfnm(τ){-λnm2kΓ(α)∫τt(t-η)α-1Eα,α(-λnm2k(t-η)α)(η-τ)α-1dη}dτ=∫0tfnm(τ){-λnm2kΓ(α)∫0t-τzα-1Eα,α(-λnm2kzα)(t-τ-z)α-1dz}dτ=-∫0tfnm(τ)λnm2k(t-τ)2α-1Eα,2α(-λnm2k(t-τ)α)dτ=∫0t(t-τ)α-1fnm(τ){-1Γ(α)+Eα,α(-λnm2k(t-τ)α)}dτ,-λnm2k∫0t(t-τ)α-1Eα,α(-λnm2k(t-τ)α)dτ=-λnm2k∫0tzα-1Eα,α(-λnm2kzα)(t-z)1-1dz=Γ(1)λnm2ktαEα,α+1(-λnm2ktα)=Eα,1(-λnm2ktα)-1,-λnm2k∫0t(t-τ)α-1Eα,α(-λnm2k(t-τ)α)τdτ=-λnm2k∫0tzα-1Eα,α(-λnm2kzα)(t-z)2-1dz=Γ(2)λnm2ktα+1Eα,α+2(-λnm2ktα)=tEα,2(-λnm2ktα)-t.
From these three formulas and (3.11), it follows that
(3.14)unm(t)=unm(0)Eα,1(-λnm2ktα)+tunm′(0)Eα,2(-λnm2ktα)+(-1)k∫0t(t-τ)α-1Eα,α(-λnm2k(t-τ)α)fnm(τ)dτ.
For unm(0) and unm′(0), we expand the given functions φ(x,y) and ψ(x,y) in the form of a Fourier series along the functions vnm(x,y),1≤n,m<∞:
(3.15)φ(x,y)=∑n,m=1∞φnmvnm(x,y),ψ(x,y)=∑n,m=1∞ψnmvnm(x,y),
where
(3.16)φnm=∫0p∫0qφ(x,y)vnm(x,y)dydx,ψnm=∫0p∫0qψ(x,y)vnm(x,y)dydx.
Using (2.13), (3.14), (3.16), we obtain
(3.17)unm(t)=Eα,1(-λnm2ktα)φnm+tEα,2(-λnm2ktα)ψnm+(-1)k∫0t(t-τ)α-1Eα,α(-λnm2k(t-τ)α)fnm(τ)dτ.
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])
(3.18)|Eα,β(z)|≤M1+|z|,M=const>0,Rez<0,
we get the following inequality:
(3.19)|unm(t)|≤C0(|φnm|+|ψnm|+(∫0t|fnm(t)|2dt)1/2)
for the solution of (3.10) for any t,t∈[0,T]. Here, C0=max{M,TM,M(Tα-1/2/2α-1)}.

4. Solvability of (<xref ref-type="disp-formula" rid="EEq7">2.13</xref>) in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M76"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> 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
(4.1)‖u‖L2(Ω)≤C1(‖φ‖L2(Ω0)+‖ψ‖L2(Ω0)+‖f‖L2(Ω))
holds, where C1 does not depend on φ(x,y),ψ(x,y), and f(x,y,t).

Proof.

We consider the sum:
(4.2)uN(x,y,t)=∑n,m=1Nunm(t)vnm(x,y),
where N is a natural number. For the positive integer number L, we have that
(4.3)‖uN+L-uN‖L2(Ω)2=‖∑n,m=N+1N+Lunm(⋅)vnm(⋅,⋅)‖L2(Ω)2=∑n,m=N+1N+L∫0T|unm(t)|2dt.
Applying (3.19), we get
(4.4)∑n,m=1∞∫0T|unm(t)|2dt≤3C02(∑n,m=1∞|φnm|2+∑n,m=1∞|ψnm|2+∑n,m=1∞∫0T|fnm(t)|2dt)=C2(‖φ‖L2(Ω0)2+‖ψ‖L2(Ω0)2+‖f‖L2(Ω)2),
where C2=3TC02. Therefore, ∑n,m=N+1N+L∫0T|unm(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 (<xref ref-type="disp-formula" rid="EEq7">2.13</xref>)

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

Lemma 5.1.

Suppose φ(x,y)∈C1(Ω¯0), φxy(x,y)∈L2(Ω0), ψ(x,y)∈C1(Ω¯0), ψxy(x,y)∈L2(Ω0), φ(x,y)=0 on ∂Ω0, ψ(x,y)=0 on ∂Ω0, f(x,y,t)∈C2(Ω¯),fxxy(x,y,t)∈C(Ω¯)fxyy(x,y,t)∈C(Ω¯0),fxxyy(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
(5.1)|unm(t)|≤C1(|φnm|νnkμmk+|ψnm|νnkμmk+1νnk+1μmk+1+1νnk+1-εμmk+1+1νnk+1μmk+1-ε),(5.2)λnm2k|unm(t)|≤C2(|φnm(1,1)|νnμm+|ψnm(1,1)|νnμm+1νn2μm2+1νn2-εμm2+1νn2μm2-ε)
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
(5.3)φnm=1νnμmφnm(1,1),(5.4)ψnm=1νnμmψnm(1,1),(5.5)fnm(t)=1νnμmfnm(1,1,0)(t),(5.6)fnm(t)=1νn2μm2fnm(2,2,0)(t),
where
(5.7)φnm(1,1)=∫0p∫0q∂2φ(x,y)∂x∂yvnm(x,y)dydx,ψnm(1,1)=∫0p∫0q∂2ψ(x,y)∂x∂yvnm(x,y)dydx,fnm(1,1,0)(t)=∫0p∫0q∂2f(x,y,t)∂x∂yvnm(x,y)dydx,fnm(2,2,0)(t)=∫0p∫0q∂4f(x,y,t)∂x2∂y2vnm(x,y)dydx.
Under the assumptions of Lemma 5.1, it follows that the functions fnm(1,1,0)(t) and fnm(2,2,0)(t) are bounded, that is,
(5.8)|fnm(1,1,0)(t)|≤N1,|fnm(2,2,0)(t)|≤N2,
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:
(5.9)lnλnmε<λnmε<νnε+μmε,0<ε<1,1+λnm2kTα<2λnm2kTα.

Using (3.16), (5.8), (5.9), and (3.17), we get
(5.10)|unm(t)|≤M(12t0α|φnm|νnkμmk+12t0α-1|ψnm|νnkμmk-N1ανnk+1μmk+1∫0td(1+λnm2k(t-τ)α)1+λnm2k(t-τ)α)≤M(12t0α|φnm|νnkμmk+12t0α-1|ψnm|νnkμmk+N1(ln2Tα+(2k/ε)lnλnmε)ανnk+1μmk+1)≤C1(|φnm|νnkμmk+|ψnm|νnkμmk+1νnk+1μmk+1+1νnk+1-εμmk+1+1νnk+1μmk+1-ε),
where C1=max{M/2t0α,M/2t0α-1,MN1ln2Tα/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
(5.11)λnm2k|unm(t)|≤M(|φnm|tα+|ψnm|tα-1+λnm2k∫0t(t-τ)α-1fnm(τ)1+λnm2k(t-τ)αdτ)≤M(1t0α|φnm(1,1)|νnμm+|ψnm(1,1)|t0α-1νnμm-N2α∫0td(1+λnm2k(t-τ)α)νn2μm2(1+λnm2k(t-τ)α))≤Mt0α|φnm(1,1)|νnμm+Mt0α-1|ψnm(1,1)|νnμm+2MN2lnTαανn2μm2+2kMN2αενn2-εμm2+2kMN2αενn2μm2-ε≤C2(|φnm(1,1)|νnμm+|ψnm(1,1)|νnμm+1νn2μm2+1νn2-εμm2+1νn2μm2-ε),
where C2=max{M/t0α,M/t0α-1,MN2ln2Tα/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
(5.12)∂2ku(x,y,t)∂x2k=∑n,m=1∞(-1)kνn2kunm(t)vnm(x,y),(5.13)∂2ku(x,y,t)∂x2k=∑n,m=1∞(-1)kμn2kunm(t)vnm(x,y),(5.14)Dc0+αu(x,y,t)=-∑n,m=1∞(-1)kλnm2kunm(t)vnm(x,y)+∑n,m=1∞fnm(t)vnm(x,y).
The series
(5.15)∑n,m=1∞|unm(t)|
is majorant for the series (3.1). From (5.1), it follows that the series (5.15) uniformly converges. Actually,
(5.16)∑n,m=1∞|unm(t)|≤C∑n,m=1∞(|φnm|νnkμmk+|ψnm|νnkμmk+1νnkμnk+1νnk+1-εμnk+1+1νnk+1μnk+1-ε).
Applying the Cauchy-Schwarz inequality and the Parseval equality, we obtain
(5.17)∑n,m=1∞|φnm|νnkμmk≤(∑n,m=1∞1νn2kμm2k)1/2(∑n,m=1∞|φnm|2)1/2=pkqkπ2k(∑n=1∞1n2k∑m=1∞1m2k)1/2‖φ‖L2(Ω0).
Analogously, we get
(5.18)∑n,m=1∞|ψnm|νnkμmk≤pkqkπ2k(∑n=1∞1n2k∑m=1∞1m2k)1/2‖ψ‖L2(Ω0).
Since 2k≥2, then the series ∑n=1∞(1/n2k),∑m=1∞(1/m2k) converges by the integral test. Further, k+1-ε>1, then the series
(5.19)∑n,m=1∞1νnk+1μmk+1,∑n,m=1∞1νnk+1-εμmk+1,∑n,m=1∞1νnk+1μmk+1-ε
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 Ωt0=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 νn2k<λnm2k, μm2k<λnm2k, then the series
(5.20)∑n,m=1∞λnm2k|unm|
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
(5.21)∑n,m=1∞|φnm(1,1)|νnμm≤(∑n=1∞1νn2∑m=1∞1m2)1/2(∑n,m=1∞|φnm(1,1)|2)1/2=pq6‖∂2φ∂x∂y‖L2(Ω0).
Analogously, we conclude that
(5.22)∑n,m=1∞|ψnm(1,1)|νnμm≤pq6‖∂2ψ∂x∂y‖L2(Ω0).
The series
(5.23)∑n,m=1∞(1νn2μm2+1νn2-εμm2+1νn2μm2-ε)
converges for any ε∈(0,1) according to the integral test. The series
(5.24)∑n,m=1∞|fnm(t)|
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,
(5.25)∑n,m=1∞|fnm(t)|=∑n,m=1∞1νn2μm2|fnm2,2,0(t)|≤N2∑n,m=1∞1νn2μm2=N2p2q236.
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 vnm(x,y). Simple computations show that
(5.26)limt→0Eα,1(-λnm2ktα)=1,limt→0ddtEα,1(-λnm2ktα)=0,limt→0Eα,2(-λnm2ktα)=1,limt→0tddtEα,2(-λnm2ktα)=0.
Consequently, limt→0unm(t)=φnm, limt→0unm′(t)=ψnm. 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:
(6.1)(-1)kDc0+αu+∂2ku∂x2k+∂2ku∂y2k+u=f(x,y,t),0<x<p,0<y<q,0<t<T,∂2m+1u(0,y,t)∂x2m=∂2m+1u(p,y,t)∂x2m=0,m=0,1,…,k-1,0≤y≤q,0≤t≤T,∂2m+1u(x,0,t)∂y2m=∂2m+1u(x,q,t)∂y2m=0,m=0,1,…,k-1,0≤x≤p,0≤t≤T,u(x,y,0)=φ(x,y),ut(x,y,0)=ψ(x,y),0≤x≤p,0≤y≤q
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.

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