R E S E A R C H
Open Access
Uniqueness result for a fractional diffusion
coefficient identification problem
Fadhel Jday
1,2and Ridha Mdimagh
1,3**Correspondence:
1Inverse Problems, LAMSIN, Ecole
Nationale d’Ingénieurs de Tunis, University of Tunis El Manar, Tunis Belvédère, Tunisia
3Mathematics Department, Faculty
of Science and Arts, Khulais, University of Jeddah, Jeddah, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
In this paper, we establish an identifiability result for the coefficient identification problem in a fractional diffusion equation in a bounded domain from the observation of the Cauchy data on particular subsets of the boundary.
MSC: 35R30; 35R11; 58J99
Keywords: Inverse problems; Fractional diffusion equation; Partial data; Uniqueness result
1 Introduction
LetΩbe a smooth bounded domain ofRd(d≥3) with boundary∂Ω∈C∞, and letT> 0
be a fixed real number. Our inverse problem consists in determining the potentialq:=q(x) via the fractional diffusion equation from given Cauchy data on particular subsets of the boundary. This problem is presented by the following equations:
Pαq,f
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂α
t u–xu+q(x)u= 0 inΩT:=Ω×(0,T),
u(x, 0) =u0(x) inΩ,
u=f onΣT:=∂Ω×(0,T),
where∂α
t urepresents the fractional Caputo time derivative of order 0 <α< 1 defined by
equation (3.7). We assume that the coefficientq∈L∞(Ω¯),f ∈C1([0;T];H32(∂Ω)), and
u0∈L2(Ω).
The classical diffusion–advection equation does not often describe the abnormal diffu-sion (e.g., for the problem of the soil scatter field data [1]) and that the fractional diffusion equation is used as a model equation for this problem and for many physical phenom-ena such as the diffusion of material in heterogeneous media, the diffusion of fluid flows in inhomogeneous anisotropic porous media, turbulent plasma, the diffusion of carriers in amorphous photoconductors, the diffusion in a turbulent medium flow, a percolation model in porous media, various biological phenomena, and finance problems (see [9]).
The inverse coefficients problems associated with the system (Pα
q,f) were investigated by
many authors for the parabolic case ([8,12,13], . . .) whenα= 1 and for the hyperbolic case ([3–6,23], . . .) whenα= 2. Few works exist dealing with this type of problem in the fractional case (0 <α< 1). In [11] the authors, in the one-dimensional case, by Dirich-let boundary measurements determined the fractional orderα and a time-independent
coefficient. Using pointwise measurements of the solution over the entire time span, the authors in [16] identified the fractional orderα for the cased≥2. Using a specifically designed Carleman estimate, the authors in [10,25] derived a stability estimate of a zero-order time-independent coefficient, with respect to partial internal observation of the so-lution in the particular case whered= 1 andα=12. In [14] the authors proved a unique determination of a time-dependent parameter appearing in the source term or in the zero-order coefficient from pointwise measurements of the solution over the whole time inter-val. By performing a single measurement of the Cauchy data on the accessible boundary, the authors in [15] gave identifiability and local Lipschitz stability results to solve the in-verse problem of identifying fractional sources. The authors in [19] proved a uniqueness result for the time-independent coefficients using the Dirichlet-to-Neumann operator ob-tained by probing the system with inhomogeneous Dirichlet boundary conditions of the formλ(t)g(x), whereλwas a fixed real-analytic positive function of the time variable. In [17] the authors proved the uniqueness in the inverse problem of determining the smooth manifold (up to an isometry) and various time-independent smooth coefficients appear-ing in their equation from measurements of the solution on a subset of the boundary at fixed time.
In this work, we establish a uniqueness result of the coefficient identification problem using Carleman estimates and particular complex geometrical solutions of the fractional diffusion equation. These techniques are inspired by [7], where the authors proved the uniqueness of a coefficient identification problem for the Schrödinger equation.
In Sect.2, we state the forward problem. In Sect.3, we recall some definitions and prop-erties of the fractional derivatives, and in Sect.4, we give the proof of our main result.
2 Forward problem
In this work, our fundamental question is to prove the uniqueness of the potentialqvia the fractional diffusion equation from the knowledge of the Cauchy data. The forward problem is to find the solutionufrom the following system of equations:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂α
tu–xu+q(x)u= 0 inΩT:=Ω×(0,T),
u(x, 0) =u0(x) inΩ,
u=f onΣT:=∂Ω×(0,T),
(2.1)
where the coefficientq∈L∞(Ω¯),f ∈C1([0;T];H32(∂Ω)), andu0∈L2(Ω) are given. Re-ferring to [17,20], we can chooseG∈C1([0;T];H2(Ω)) satisfyingG=f onΣ
T, and by
settingw=u–G,wis a solution of
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂α
tw–xw+q(x)w=F inΩT,
w(x, 0) =a(x) inΩ,
w= 0 onΣT,
(2.2)
whereF= –(∂α
Theorem 2.1 Let a∈L2(Ω)and F∈L∞(0,T,L2(Ω)).Then problem(2.2)has a unique
weak solution
w∈C[0,T];L2(Ω)∩C(0,T];H2(Ω)∩H01(Ω)∩L2[0,T];H2(Ω)∩H01(Ω).
Proof We split problem (2.2) into the following two problems by takingw=w1+w2, where
w1is the solution of
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂α
tu–xu+q(x)u= 0 inΩT,
u(x, 0) =a inΩ,
u= 0 onΣT,
(2.3)
andw2is the solution of
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂α
tu–xu+q(x)u=F inΩT,
u(x, 0) = 0 inΩ,
u= 0 onΣT.
(2.4)
From Sakamoto et al. [22] we have
1. Problem (2.3) has a unique weak solution
w1∈C
[0,T];L2(Ω)∩C(0,T];H2(Ω)∩H01(Ω)
satisfying
u(·,t)H2(Ω)+∂ α
t u(·,t)L2(Ω)≤C1t–αaL2(Ω). (2.5)
2. Problem (2.4) has a unique weak solutionw2∈L2([0,T];H2(Ω)∩H01(Ω))satisfying
uL2([0,T];H2(Ω))+∂tαu
L2(Ω
T)≤C2FL2(ΩT). (2.6)
3 Preliminaries
We start this section by giving some definitions and fundamental properties of fractional integrals and fractional derivatives, which can be found in [18,21].
Letα> 0, letnbe the integer satisfyingn– 1≤α<n, and leta,b∈R.
Definition 3.1 Letg: [a,b]→Rbe a function, and letΓ be the Euler gamma function.
1. The left and right Riemann–Liouville fractional integrals of orderαare defined respectively by
aItαg(t) :=
1 Γ(α)
t
a
(t–s)α–1g(s)ds (3.1)
and
tIαbg(t) :=
1 Γ(α)
b
t
2. The left and right Riemann–Liouville fractional derivatives of orderαare defined respectively by
aDαtg(t) :=
dn
dtn aI n–α
t g(t) =
1 Γ(n–α)
dn
dtn t
a
(t–s)n–α–1g(s)ds (3.3)
and
tDαbg(t) := (–1)n
dn
dtn tI n–α
b g(t) =
(–1)n
Γ(n–α)
dn
dtn b
t
(s–t)n–α–1g(s)ds. (3.4)
In particular, ifα= 0, then
aD0tg(t) =tD0bg(t) =g(t),
and ifα=k∈N, then
aDktg(t) =tDbkg(t) =g(k)(t).
3. The left and right Caputo fractional derivatives of orderαare defined by
c
aDαtg(t) :=aItn–αg(n)(t) =
1 Γ(n–α)
t
a
(t–s)n–α–1g(n)(s)ds (3.5)
and
c tD
α
bg(t) := (–1)ntIn–
α
b g(n)(t) =
(–1)n
Γ(n–α)
b
t
(s–t)n–α–1g(n)(s)ds. (3.6)
In particular, if0 <α< 1, then we denote
∂tαg(t) :=0cDαtg(t) = 1 Γ(1 –α)
t
0
(t–s)–αg(s)ds. (3.7)
Lemma 3.2([18]) If g∈ACn[a,b],then the Riemann–Liouville fraction derivative and the
Caputo fractional derivative are connected with each other by the following relations:
aDαtg(t) =c0D α
tg(t) + n–1
k=0
g(k)(a)
Γ(1 +k–α)(t–a)
k–α (3.8)
and
tDαbg(t) =c0D α
bg(t) + n–1
k=0
(–1)kg(k)(b) Γ(1 +k–α)(b–t)
k–α. (3.9)
Here
ACn[a,b] =
g: [a,b]→Rsuch that d
n–1
dxn–1(g)∈AC[a,b]
g∈AC[a,b] ⇔ there existsϕ∈L(a,b)such that g(x) =c+
b
a
ϕ(t)dt, c∈R,
and L(a,b)is the set of Lebesgue-measurable complex-valued functions on[a,b].
Remark3.3 If 0 <α< 1, then
0Dαtu(x,t) =c0D α
tu(x,t) +
u(x, 0) Γ(1 –α)t
–α. (3.10)
In addition, ifu(x, 0) = 0, then0Dαtu(x,t) =c0D α
tu(x,t).
Definition 3.4([18])
1. The Mittag-Leffler function is
Eα,β(z) =
∞
k=0
zk
Γ(αk+β), z∈C, (3.11)
whereα> 0andβare arbitrary constants. 2. Theα-exponential function is defined by
eλz
α =z α–1E
α,α(λz), z∈C\ {0}, (3.12)
whereα> 0andλ∈C.
Lemma 3.5([18])
1. For0 <α< 1andλ> 0,we have
∂tαEα,1
–λtα= –λEα,1
–λtα, t> 0.
2. For0 <α< 1andλ∈C,we have
0Dαte
λt
α =λe λt
α, t> 0.
Lemma 3.6([21]) For s>|λ|α1,we have
∞
0
e–sttαk+β–1Eα(k,)β
–λtαdt= k!s α–β
(sα–λ)k+1.
4 The main result
In this work, we focus on the uniqueness of the solutionqof the inverse problem. Let
H(ΩT) =
u∈D(ΩT)/u∈L2(ΩT),u∈L2(ΩT)
.
Forx= (x1, . . . ,xd) andy= (y1, . . . ,yd)∈Cd,x,y:= d
i=1xiyi.
Fixξ∈Sd–1, whereSd–1={x∈Rd,|x|= 1}, and forε> 0, denote
∂Ω+(ξ) =
∂Ω–(ξ) =
x∈∂Ω,ν(x),ξ< 0,
∂Ω+,ε(ξ) =
x∈∂Ω,ν(x),ξ>ε,
∂Ω–,ε(ξ) =
x∈∂Ω,ν(x),ξ<ε,
whereνrepresents the unit outer normal to the boundary∂Ω,
Cq,ε=
u(·, 0),u|ΣT,∂νu|∂Ω–,ε(ξ)×(0,T)
/u∈H(ΩT),
∂tα–+qu= 0 inΩT
.
In the following theorem, we give a global uniqueness result for the fractional diffusion equation in which the Cauchy data are given only on part of the boundary.
Theorem 4.1 Let qi∈L∞(Ω),i= 1, 2.Givenξ∈Sd–1andε> 0,assume that Cq1,ε=Cq2,ε.
Then q1=q2.
To prove this result, we need the following results.
Lemma 4.2([7]) Letρ∈Cdbe such thatρ,ρ= 0,whereρ=τ(ξ +iη)withξ,η∈Sd–1.
Suppose that f(·,ρ/|ρ|)∈W2,∞(Ω)satisfies∂ξf =∂ηf = 0,where∂ξdenotes the directional
derivative in the directionξ.Then there is a solution tou–qu= 0inΩof the form
u(x,ρ) =ex,ρfx,ρ/|ρ|+ψ(x,ρ),
where
ψ|∂Ω–(ξ)= 0,
and
ψ(·,ρ)L2(Ω)≤
M
τ (4.1)
for some M> 0andτ ≥τ0> 0.
Corollary 4.3 Letρ∈Cdbe such thatρ,ρ= 0,whereρ=τ(ξ+iη)withξ,η∈Sd–1,and
λ> 0.
1. There is a solutionwto–w+ (q–λ)w= 0inΩof the form
uq,λ(x,ρ) :=ex,ρ
1 +ψq(x,ρ)
, (4.2)
where
ψq|∂Ω–(ξ)= 0
and
ψq(·,ρ)L2(Ω)≤
M
τ , τ≥τ0,
2. The function(x,t)→Eα,1(–λtα)uq,λ(x,ρ)is a solution of(∂tα–+q)w= 0inΩT. 3. The function(x,t)→e–λt
α uq,λ(x,ρ)is a solution of(0Dαt –+q)w= 0inΩT.
Proof 1. Applying Lemma4.2forf = 1 andq:=q–λ, we obtain the result.
2. Using Proposition3.5and item1, it is easy to prove items2and3.
Lemma 4.4(Carleman estimates [7]) For q∈L∞(Ω),there existτ0> 0and C> 0such that
for all u∈C2(Ω),u|∂Ω= 0,andτ≥τ0,we have the estimate
Cτ2 Ω
e–τx,ξu2dx+τ ∂Ω+
ν,ξe–τx,ξ∂νu2dS
≤ Ω
e–τx,ξ(–q)u2dx–τ ∂Ω–
ν,ξe–τx,ξ∂νu2dS. (4.3)
Lemma 4.5 If u∈H(ΩT)and u|ΣT= 0,then u∈L
2(0,T;H2(Ω)).
Proof Suppose first thatu∈C2(Ω¯
T). From the first Green formula we obtain
2 ΩT
|∇u|2dx= –2 ΩT
uu dx≤
ΩT
|u|2+|u|2dx=u2H(ΩT)<∞.
From the formula of Kadlec [24],
i,j ΩT
|∂i∂ju|2dx≤
ΩT
|u|2dx+C0 ΣT
|∂νu|2dx,
sinceΣ
T|∂νu|
2dx≤C 1
ΩT|u|
2dx, we get
uL2(0,T;H2(Ω))≤C2uH(ΩT).
Using standard density arguments, we obtain
uL2(0,T;H2(Ω))≤C2uH(ΩT), ∀u∈H(ΩT), andu|ΣT= 0.
Thusu∈L2(0,T;H2(Ω)).
Lemma 4.6 Let ui:=uαqi,fi,i= 1, 2,be solutions of the fractional diffusion equations(P
α
qi,fi), qi∈L∞(Ω)and fi∈L2(ΣT).Then we have
∂tαu1u2=u10Dαtu2, (4.4)
whereis the convolution product.
Proof From [2] we have
b
a
g(s)caDαsf(s)ds=
b
a
f(s)sDαbg(s)ds+ n–1
j=0
sD
α+j–n
b g(s)·sDnb–1–jf(s) b
for 0 <α< 1,a= 0, andb=t, and
t
0
g(s)∂sαf(s)ds=
t
0
f(s)sDαtg(s)ds+
sDαt–1g(s)·f(s) t
0. (4.6)
Then
∂tαu1u2=
t
0
∂sαu1(x,s)u2(x,t–s)ds
=
t
0
u1(x,s)sDαtu2(x,t–s)ds+
sDαt–1u2(x,t–s)u1(x,s)
t
0. (4.7)
Sinceα– 1 < 0, we havesDαt–1u2(x,t–s) =sI1–t αu2(x,t–s) (see [2]) and
sI1–t αu2(x,t–s) = 1 Γ(1 –α)
t
s
(τ–s)–αu
2(x,t–τ)dτ.
Using integration by parts, we get
sI1–t αu2(x,t–s) = 1 Γ(2 –α)
(t–s)1–αu2(x,t–s) +
t
s
(τ–s)1–α∂u2
∂τ (x,t–τ)dτ
.
We deduce thatlims→t sDα–1
t u2(x,t–s) = 0 and
t
0
∂sαu1(x,s)u2(x,t–s)ds=
t
0
u1(x,s)sDαtu2(x,t–s)ds.
By definition
sDαtu2(x,t–s) := 1 Γ(1 –α)
d dt
t
s
(t–τ)–αu2(x,τ–s)dτ.
By change of variableξ=τ–swe obtain
sDαtu2(x,t–s) = 1 Γ(1 –α)
d dt
t–s
0
(t–s–ξ)–α
u2(x,ξ)dξ
=0Dαt–su2(x,t–s), (4.8)
which implies
t
0
∂sαu1(x,s)u2(x,t–s)ds=
t
0
u1(x,s)0Dαt–su2(x,t–s)ds
and
∂tαu1u2=u10Dαtu2.
Remark4.7 Ifu2(x, 0) = 0, then from Remark3.3we obtain
Proof of Theorem4.1 Letξ∈Sd–1. Fixk∈Rdsuch thatk,ξ= 0. By Corollary4.3there
existsu2∈H(ΩT) satisfying (∂tα–+q2)u2= 0 inΩTof the form
u2(x,t) =Eα
–λtαuq2,λ(x,ρ2),
whereuq2,λid defined by (4.2),ρ2=τ ξ–i
k+
2 ,k,=ξ,= 0, and|k+|2= 4τ2(under these conditions,ρ2,ρ2= 0). In dimensiond≥3, we can always choose such a vector.
SinceCq1,ε=Cq2,ε, there existsu1∈H(ΩT) such that
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(∂α
t –+q1)u1= 0 inΩT,
u1(x, 0) =u2(x, 0) inΩ,
u1=u2 onΣT,
∂νu1=∂νu2 on∂Ω–,ε(ξ)×(0,T).
Settingu=u1–u2andq=q1–q2, we have
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(∂α
t –+q1)u= –qu2 inΩT,
u(x, 0) = 0 inΩ,
u= 0 onΣT,
∂νu= 0 on∂Ω–,ε(ξ)×(0,T).
By Lemma4.5, sinceu∈H(ΩT) andu|ΣT= 0, it follows thatu∈L
2(0,T;H2(Ω)). From Green’s formula and Lemma4.6, forv∈H(ΩT), we obtain
Ω
∂tα–+q1
u¯v dx= – Ω
qu2v dx¯
= Ω
u0Dαt –+q1
¯
v dx+ ∂Ω
∂νu¯v ds. (4.9)
We choose
¯
v(x,t) =eα–λtuq1,λ(x,ρ1),
which is a solution of (0Dαt –+q1)v¯= 0, whereuq1,λ(x, ,ρ1) =e
x,ρ1(1 +ψ
q1(x,ρ1)),
ψq1|∂Ω–(ξ)= 0,
and
ψq1(·,ρ1)L2(Ω)≤ ˜
M
τ , τ≥τ0,
whereρ1= –τ ξ–ik–2. We haveρ1+ρ2= –ikand|ρ1|2= 2τ2. Then equation (4.9) becomes
– Ω
qu2v dx¯ = ∂Ω
Letλ> 0 ands>λα1, extendinguby 0 on (T,∞). Applying the Laplace transform to (4.10),
we obtain
– Ω
qu˜2˜¯v dx= ∂Ω
∂νu˜v dS˜¯ , (4.11)
whereu˜:=u˜(·,s) is a solution of the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
u˜– (q1+sα)u˜=qu˜2 inΩ, ˜
u= 0 on∂Ω,
∂νu˜= 0 on∂Ω–,ε(ξ),
and the Laplace transformg˜of a functiongis defined by
˜
g(s) =
∞
0
e–stg(t)dt.
Using Lemma3.6, we obtain
˜
u2(x,s) =
sα–1
sα–λuq2,λ(x,ρ2), ˜¯
v(x,s) = 1
sα–λuq1,λ(x,ρ1).
Equation (4.11) becomes
Ω
quq1,λ(x,ρ1)uq2,λ(x,ρ2)dx= (sα–λ)
sα–1 ∂Ω
∂νuu˜ q1,λ(x,ρ1)dS.
In the following step, we show thatlimτ→∞∂Ω∂νuu˜ q1,λ(x,ρ1)dS= 0. Indeed, since∂νu˜= 0 on∂Ω–,ε(ξ), we have
∂Ω∂νuu˜ q1,λ(x,ρ1)dS
= ∂Ω+,ε(ξ)
∂νue˜ x,ρ11 +ψ
q1(x,ρ1)
dS
≤ ∂Ω+,ε(ξ)
∂νue˜ –τx,ξ1 +ψ
q1(x,ρ1)dS
≤
∂Ω+,ε(ξ)
∂νue˜ –τx,ξ2dS 1
2
∂Ω+,ε(ξ)
1 +ψq1(x,ρ1) 2
dS 1
2
≤
∂Ω+,ε(ξ)
∂νue˜ –τx,ξ2dS 1
2 2
∂Ω+,ε(ξ)
1 +ψq1(x,ρ1) 2
dS 1
2
≤√2
∂Ω+,ε(ξ)
∂νue˜ –τx,ξ 2
dS 1
2
vol(∂Ω) +ψq1 2
L2(∂Ω)
1 2.
From [7, p. 666] we have
and since
ψq1L2(∂Ω)≤vol(∂Ω)ψq1C(∂Ω),
we have
∂Ω∂νuu˜ q1,λ(x,ρ1)dS
≤C√2vol(∂Ω) 1 21 +τ14
∂Ω+,ε(ξ)
∂νue˜ –τx,ξ2dS 1
2 .
From estimates (4.3) and (4.1) we obtain
τ ε ∂Ω+,ε(ξ)
∂νue˜ –τx,ξ 2
ds≤τ ∂Ω+,ε(ξ)
ν,ξ∂νue˜ –τx,ξ 2
dS
≤ Ω
e–τx,ξ–q1+sα
˜
u2dx= Ω
e–τx,ξqu˜22dx
≤2q1∞+q2∞
2
Ω
1 +ψq2(x,ρ2) 2
dx
≤2q1∞+q2∞
2
vol(Ω) +ψq2(x,ρ2) 2
L2(Ω)
≤2q1∞+q2∞
2
vol(Ω) +M τ2
.
Hence we have proved that
∂Ω∂νu˜v ds¯ ≤ ˜C(1 +τ
1 4) √
τ
vol(Ω) +M τ
,
where
˜
C=2C(q1∞+q2∞) √
vol(∂Ω) √
ε ,
and then
lim
τ→∞ ∂Ω∂νu˜v dS˜¯ = 0.
Using equalities (4.1) and (4.11), by tendingτ to∞we have
Ω
q(x)e–ix,kdx= 0.
Changingξ∈Sd–1in a small conic neighborhood and using the fact thatqˆ(k) is analytic, we get thatq= 0, and thenq1=q2.
5 Conclusion
Acknowledgements
Not applicable.
Funding
Not applicable.
Abbreviations
Not applicable.
Availability of data and materials
Not applicable.
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this paper.
Authors’ contributions
All authors contributed equally to this paper. All authors read and approved the final manuscript.
Author details
1Inverse Problems, LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, University of Tunis El Manar, Tunis Belvédère, Tunisia. 2Mathematics Department, Jamoum University College, Umm Al-Qura University, Mecca, Saudi Arabia.3Mathematics Department, Faculty of Science and Arts, Khulais, University of Jeddah, Jeddah, Saudi Arabia.
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Received: 21 February 2019 Accepted: 9 October 2019 References
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