An undetermined time-dependent coefficient in a fractional diffusion equation
Zhidong Zhang




Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA 

zhidong@math.tamu.edu

In this work, we consider a FDE (fractional diffusion equation) 

    u(x,t)-a(t)Ł u(x,t)=F(x,t)
 with a time-dependent diffusion 
coefficient a(t). This is an extension of <cit.>, 
which deals with this FDE in one-dimensional space.  
For the direct problem, given an a(t), 
we establish the existence, uniqueness and some regularity properties 
with a more general domain Ω and right-hand side F(x,t). 
For the inverse problem–recovering a(t), 
we introduce an operator K one of whose fixed points is a(t) 
and show its monotonicity, uniqueness and 
existence of its fixed points. With these properties, a reconstruction 
algorithm for a(t) is created and some numerical results are provided 
to illustrate the theories. 


Keywords: fractional diffusion, fractional inverse problem,  
uniqueness, existence, monotonicity, 
iteration algorithm. 



AMS subject classifications: 35R11, 35R30, 65M32.


An undetermined time-dependent coefficient in a fractional diffusion equation
    Zhidong Zhang
    December 30, 2023
=============================================================================








§ INTRODUCTION


This paper considers the fractional diffusion equation (FDE) with 
a continuous and positive coefficient function a(t):

    u(x,t)-a(t)Ł u(x,t)   =F(x,t),       x∈Ω, t∈ (0,T];
    
       u(x,t)   =0,       (x,t)∈∂Ω× (0,T];
    
       u(x,0)   =u_0(x),       x∈Ω,

 where Ω is a bounded and smooth subset of R^n, n=1,2,3, -Ł is a symmetric uniformly elliptic operator defined as 
 
    -Ł u=-∑_i,j=1^n (a^ij(x)u_x_i)_x_j+c(x)u
 with conditions 
 
    a^ij,c∈ C^2(Ω)  (i,j=1,…,n), ∂Ω is  C^3,
 
 and  is the left-sided Djrbashian-–Caputo α-th order derivative with respect to time t. The definition for  is 
    u(x,t)=1/Γ(n-α)∫_0^t (t-τ)^n-α-1d^n/dτ^nu(x,τ)  dτ

 with Gamma function Γ(·) and the nearest integer n with 
 α≤ n. In this paper, we are assuming a subdiffusion process,  
 i.e. α∈(0,1). This simplifies the definition of  as 
 
    u(x,t)=1/Γ(1-α)∫_0^t (t-τ)^-αd/dτu(x,τ)  dτ.

 This work is an extension of <cit.> 
 from a simple space domain Ω to ℝ^n, considers the 
 more general analysis for the direct problem and contains an existence 
 argument for the inverse problem of recovering a(t). 
 
 
This paper consists of two parts; the direct problem and 
the inverse problem. For the direct problem, we build the 
spectral representation of the weak solution u(x,t;a). The notation u(x,t;a) is used for displaying the dependence of the solution u on the diffusivity a(t). 
Then the existence, uniqueness and regularity results are proved 
with several assumptions on the coefficient function a(t). 
Unlike <cit.>, 
the right hand side function F(x,t) is not of the form f(x)g(t), so 
that the proof of regularity is more delicate. For the inverse problem, we use the single point flux data 

    a(t)∂ u/(x_0,t;a)=g(t), x_0∈∂Ω

to recover the coefficient a(t) 
(We choose the data a(t)∂ u/(x_0,t;a)=g(t) instead of 
the classical flux ∂ u/(x_0,t;a) because 
in practice, a(t)∂ u/(x_0,t;a) is usually  
measured as the flux). 
For the reconstruction, we only consider to recover a 
continuous and positive a(t) to match the assumptions set in the direct 
problem. Acting a flux data, we introduce an operator K 
one of whose fixed points is the coefficient a(t). Using 
the weak maximum principle <cit.>, we establish the monotonicity and uniqueness of the fixed points of operator K, and the proof of uniqueness leads to a numerical reconstruction algorithm. Since we consider a multidimensional domain Ω here, the Sobolev Embedding Theorem yields that we need to add the condition (<ref>) on the operator -Ł to ensure the C^1-regularity of the series representation of u. Then the operator K is well-defined, where the proofs can be seen in section 4. This is a significant difference from  <cit.>.  
Furthermore, an existence argument of the fixed points of K is included by this paper, which <cit.> does not contain. 

The rest of this paper follows the following structure. In section 2, we collect 
some preliminary results about fractional calculus and the eigensystem 
of -Ł. The direct problem 
is discussed in section 3, i.e. we establish the existence, uniqueness 
and some regularity results of the weak solution for FDE (<ref>). 
Then section 4 deals with the inverse problem of recovering a(t). Specifically, an operator 
K is introduced at the beginning of this section, then its monotonicity 
and uniqueness of its fixed points give an algorithm to recover the 
coefficient a(t). In particular, the existence argument of the fixed 
points of K is included by this section. In section 5, some 
numerical results are presented to illustrate the theoretical basis. 





§ PRELIMINARY MATERIAL




 §.§ Mittag-Leffler function

In this part, we describe the Mittag-Leffler function which plays 
an important role in fractional diffusion equations. This  
is a two-parameter function defined as 

    E_α,β(z) = ∑_k=0^∞z^k/Γ(kα+β),
       z∈ℂ.

It generalizes the natural exponential function in the sense that 
E_1,1(z)=e^z. We list some important properties of the 
Mittag-Leffler function for future use.


Let 0<α<2 and β∈ℝ be arbitrary, and απ/2
<μ<min(π,απ). Then there exists a constant C=C(α,β,μ)>0 such
that

    |E_α,β(z)|≤C/1+|z|,  μ≤|arg(z)|≤π.



 This proof can be found in <cit.>.




 For λ>0, α>0 and +, we have 
 
    d^n/dt^nE_α,1(-λ t^α)
     =-λ t^α-nE_α,α-n+1(-λ t^α),
      t>0.

In particular, if we set n=1, 
 then there holds 
 
    d/dtE_α,1(-λ t^α)
     =-λ t^α-1E_α,α(-λ t^α),
      t>0.




 This is <cit.>.



 If 0<α<1 and z>0, then 
 E_α,α(-z)≥ 0.


 This proof can be found in <cit.>.



For 0<α<1, E_α,1(-t^α) is completely monotonic, 
that is,  

    (-1)^n d^n/dt^nE_α,1(-t^α)≥ 0, for t>0
     and n=0,1,2,⋯.




 See <cit.>.





 §.§ Fractional calculus
 

In this part, we collect some results of fractional calculus. 
The next lemma states the extremal principle of .

 Fix 0<α<1 and given f(t)∈ C[0,T] with f ∈ C[0,T]. 
 If f attains its maximum (minimum) over the interval [0,T] at the point 
 t=t_0, t_0∈ (0,T], then 
 ^CD_t_0^α f ≥(≤) 0. 


 Even though the conditions are different from the ones of <cit.>, the maximum case can be proved following the proof of  <cit.>. 
 For the minimum case, we only need to set f=-f. 



The following lemma about the composition between 
 and the fractional integral  is presented in 
<cit.>.


Define the Riemann-–Liouville α-th order integral I_t^α as 

    u=1/Γ(α)∫_0^t (t-τ)^α-1
     u(τ)  dτ.

 For 0<α<1, u(t), u∈ C[0,T], we have

    (∘ u)(t)=u(t),   (∘ u)(t)=u(t)-u(0), t∈[0,T].



 


 §.§ Eigensystem of -Ł

Since -Ł is a symmetric uniformly elliptic operator, 
we denote the eigensystem of -Ł by {(λ_n,ϕ_n):+}. 
Then we have 0<λ_1≤λ_2≤⋯ where finite multiplicity is possible, λ_n→∞ and {ϕ_n:+}⊂ H^2(Ω)∩ 
H_0^1(Ω) forms an orthonormal basis of L^2(Ω). 
Moreover, with the condition (<ref>), for each +, it holds that ϕ_n ∈ H^3(Ω) <cit.>. 
Then by the Sobolev Embedding Theorem, we have ϕ_n∈ C^1(Ω) and  
∂ϕ_n/(x_0) is well-defined for each +. Hence,  without loss of generality, we can suppose 

    ∂ϕ_n/(x_0)≥ 0, for each +.

Otherwise, if ∂ϕ_k/(x_0)< 0 for some 
k∈ℕ^+, we can replace ϕ_k by -ϕ_k. 
-ϕ_k satisfies all the properties we need, such as it is 
an eigenfunction of -Ł corresponding to the eigenvalue λ_k, 
composes an orthonormal basis of L^2(Ω) together with 
{ϕ_n:+,n k} and ∂ (-ϕ_k)/(x_0)≥ 0.
The assumption (<ref>) will be used in Section 4.



§ DIRECT PROBLEM–EXISTENCE, UNIQUENESS AND REGULARITY

Throughout this section, we suppose a(t), u_0(x) and F(x,t) 
satisfy the following assumptions:

 

 
  (a) a(t)∈ C^+[0,T]:={ψ∈ C[0,T]:ψ(t)>0, t∈[0,T]};
 
  (b) F(x,t) ∈ C([0,T];L^2(Ω));
 
  (c) u_0(x) ∈ H_0^1(Ω). 





 §.§ Spectral Representation



   We call u(x,t;a) a weak solution of  
   FDE (<ref>) in L^2(Ω) corresponding to the  
   coefficient a(t) if 
   u(·,t;a)∈ H_0^1(Ω) for t∈(0,T] and 
   for any ψ(x)∈ H^2(Ω)∩ H_0^1(Ω), it holds  
   
    ( u(x,t;a),ψ(x))-(a(t)Ł u(x,t;a),ψ(x))=(F(x,t),ψ(x)),
        t∈(0,T];
       (u(x,0;a),ψ(x))=(u_0(x),ψ(x)),

where (·,·) is the inner product in L^2(Ω).


With the above definition, we give a spectral representation 
for the weak solution in the following lemma. 


 Define b_n:=(u_0(x),ϕ_n(x)), F_n(t)=(F(x,t),ϕ_n(x)),
  +.
 The spectral representation of the weak solution of FDE  
 (<ref>) is 
 
    u(x,t;a)=∑_n=1^∞ u_n(t;a)ϕ_n(x), (x,t)∈Ω× [0,T],

 where 
 u_n(t;a) satisfies the fractional ODE
 
    u_n(t;a)+λ_n a(t) u_n(t;a)=F_n(t),
       u_n(0;a)=b_n, +.





 For each +, multiplying ϕ_n(x) on both sides of 
 FDE (<ref>) and integrating it on x over Ω 
 allow us to deduce that   

    ( u(x,t;a), ϕ_n(x))+λ_n a(t)(u(x,t;a),ϕ_n(x))=F_n(t),

where 
(-Ł u(x,t;a),ϕ_n(x))=(u(x,t;a),-Łϕ_n(x))
=λ_n (u(x,t;a),ϕ_n(x)) follows from the symmetricity of -Ł.
Set u_n(t;a)=(u(x,t;a), ϕ_n(x)) and define 
u(x,t;a)=∑_n=1^∞ u_n(t;a)ϕ_n(x). 
Then (<ref>) and the completeness of 
{ϕ_n(x):+} lead to the desired result.




 §.§ Existence and Uniqueness

In order to show the existence and uniqueness 
of the weak solution (<ref>), we state the following lemma 
<cit.>.


 For the Cauchy-type problem
 
    y=f(y,t), y(0)=c_0,

 if for any continuous y(t), f(y,t)∈ C[0,T], ∃ A>0 
 which is independent of y∈ C[0,T] and t∈[0,T] s.t. 
 | f(t,y_1)-f(t,y_2)|≤ A| y_1-y_2|,
then there exists a unique solution y(t) for the Cauchy-type 
problem, which satisfies y∈ C[0,T].


The theorem of existence and uniqueness for u(x,t;a) follows from 
Lemma <ref>.

 Suppose Assumption <ref> holds. Under 
 Definition <ref>, there exists a unique weak solution 
 u(x,t;a) of FDE (<ref>) with the spectral 
 representation (<ref>) and
 for each +, u_n(t;a)∈ C[0,T] is the unique solution of 
 the fractional ODE (<ref>) with u_n(t;a) ∈ C[0,T]. 



From the spectral representation (<ref>), it 
suffices to show the existence and uniqueness of u_n(t;a),+.
Fix +, Assumption <ref> (a) and (b) yield 
that the fractional ODE (<ref>) satisfies the conditions of 
Lemma <ref>. Hence the existence and uniqueness 
for u_n(t;a) hold.





 §.§ Sign of u_n(t;a)

In this part, we state two properties of u_n(t;a) which play important 
roles in building the regularity of u(x,t;a).

 Given h∈ C^+[0,T], f∈ C[0,T] with f ∈ C[0,T], 
 if f(0)≤ (≥)0 and f+h(t)f(t)≤ (≥)0, then f≤ 
 (≥)0 on [0,T].


 Since f(t)∈ C[0,T], f(t) attains its maximum over [0,T] 
 at some point t_0∈ [0,T]. 
 If t_0=0, then f(t)≤ f(0)≤ 0.
 If t_0∈ (0,T], with Lemma <ref>, we have 
 ^CD_t^α f(t_0)≥ 0, which yields h(t_0)f(t_0)≤ 0, i.e. 
 f(t_0)≤ 0 due to h>0 on [0,T]. 
 The definition of t_0 assures f≤ 0.
 For the case of “≥ 0", let f(t)=-f(t), then the 
 above proof gives f≤ 0, i.e. f≥ 0.



The following corollary, which concerns the sign of u_n(t;a), follows 
from Lemma <ref> directly. 

 Set u_n(t;a) be the unique solution of 
 the fractional ODE (<ref>). Then 
 u_n(t;a)+λ_n a(t) u_n(t;a)≤ (≥) 0 on [0,T] and 
 u_n(0;a) ≤ (≥) 0 imply  u_n(t;a)≤ (≥)0 on [0,T], +. 


 Assumption <ref> gives that λ_na(t)∈ 
 C^+[0,T]. Then the proof is completed by applying Lemma <ref> 
 to the fractional ODE (<ref>). 








 §.§ Regularity


In this part, we establish the regularity of u(x,t;a). 
To this end, we split FDE (<ref>) into 
 
    u(x,t)-a(t)Ł u(x,t)   =F(x,t),       x∈Ω, t∈ (0,T];
    
       u(x,t)   =0,       (x,t)∈∂Ω× (0,T];
    
       u(x,0)   =0,       x∈Ω,

 and
 
    u(x,t)-a(t)Ł u(x,t)   =0,       x∈Ω, t∈ (0,T];
    
       u(x,t)   =0,       (x,t)∈∂Ω× (0,T];
    
       u(x,0)   =u_0(x),       x∈Ω.


Denote the weak solutions of FDEs (<ref>) and 
(<ref>) by u^r(x,t;a) and u^i(x,t;a), respectively
(“r" and “i" denote the initials of “right-hand side" and “initial 
condition"). 
The following lemma about u^r(x,t;a) and u^i(x,t;a) 
follows from Lemma <ref> and 
Theorem <ref>.


Suppose Assumption <ref> holds. Then
 u^r(x,t;a) and u^i(x,t;a) are the unique solutions for FDEs 
 (<ref>) and (<ref>), respectively, with the 
 spectral representations as  

    u^r(x,t;a)=∑_n=1^∞ u^r_n(t;a)ϕ_n(x), 
    u^i(x,t;a)=∑_n=1^∞ u^i_n(t;a)ϕ_n(x),

where u^r_n(t;a), u^i_n(t;a) satisfy the following fractional ODEs 

    u_n^r(t;a)+λ_na(t)u_n^r(t;a)=F_n(t), u_n^r(0;a)=0, +;


    u_n^i(t;a)+λ_na(t)u_n^i(t;a)=0, u_n^i(0;a)=b_n, +.

Moreover, Theorem <ref> ensures the weak 
solution u(x,t;a) of FDE (<ref>) 
can be written as u(x,t;a)=u^r(x,t;a)+u^i(x,t;a), i.e. 
u_n(t;a)=u^r_n(t;a)+u^i_n(t;a), +.




  §.§.§ Regularity of u^r

For each +, define 

    F^+_n(t)=
      F_n(t),    if F_n(t)≥0;
    
      0,    if F_n(t)<0,
        F^-_n(t)=
      F_n(t),   if F_n(t)<0;
    
      0,   if F_n(t)≥0.

It is obvious that F_n=F^+_n+F^-_n, the supports 
of F^+_n and F^-_n are disjoint and 
F^+_n,F^-_n ∈ C[0,T] which follows from F_n∈ C[0,T]. 
Split u_n^r(t;a) as u_n^r(t;a)=u_n^r,+(t;a)+u_n^r,-(t;a), 
where u_n^r,+(t;a), u_n^r,-(t;a) satisfy 

    u_n^r,+(t;a)+λ_na(t)u_n^r,+(t;a)=F^+_n(t), u_n^r,+(0;a)=0, +;


    u_n^r,-(t;a)+λ_na(t)u_n^r,-(t;a)=F^-_n(t), u_n^r,-(0;a)=0, +,

respectively. The existence and uniqueness of 
u_n^r,+(t;a) and u_n^r,-(t;a) hold due to Lemma 
<ref> and we can write 

    u^r(x,t;a)=u^r,+(x,t;a)+u^r,-(x,t;a),

where

    u^r,+(x,t;a)=∑_n=1^∞u_n^r,+(t;a)ϕ_n(x),
     u^r,-(x,t;a)=∑_n=1^∞u_n^r,-(t;a)ϕ_n(x).


Then we state some properties of u_n^r,+(t;a) 
and u_n^r,-(t;a).



For any +,
 u_n^r,+(t;a)≥ 0 and u_n^r,-(t;a)≤ 0 on [0,T]. 


 This proof follows from Corollary <ref> directly.



 Given a_1(t), a_2(t)∈ C^+[0,T] with a_1(t)≤ a_2(t) on [0,T], 
 we have 
 
    0≤ u_n^r,+(t;a_2)≤ u_n^r,+(t;a_1),
     u_n^r,-(t;a_1)≤ u_n^r,-(t;a_2)≤ 0,
       t∈ [0,T], +.




 Pick +, u_n^r,+(t;a_1) and u_n^r,+(t;a_2) satisfy 
 the following system:
 
    u_n^r,+(t;a_1)+λ_na_1(t)u_n^r,+(t;a_1)=F^+_n(t);
     u_n^r,+(t;a_2)+λ_na_2(t)u_n^r,+(t;a_2)=F^+_n(t);
    
       u_n^r,+(0;a_1)=u_n^r,+(0;a_2)=0,

which leads to 

    w+λ_na_1(t)w(t)=λ_n u_n^r,+(t;a_2)(a_2(t)-a_1(t))≥0,
     w(0)=0,
 
where w(t)=u_n^r,+(t;a_1)-u_n^r,+(t;a_2) and the 
last inequality follows from Lemma <ref> and a_1≤ a_2. 
Hence, Corollary <ref> shows that 
w(t)≥ 0, i.e. u_n^r,+(t;a_2)≤ u_n^r,+(t;a_1) and 
Lemma <ref> gives 
0≤ u_n^r,+(t;a_2)≤ u_n^r,+(t;a_1), t∈ [0,T].
Similarly, we have  
u_n^r,-(t;a_1)≤ u_n^r,-(t;a_2)≤ 0, t∈ [0,T], 
completing the proof.


Assumption <ref> (a) implies 
there exists constants q_a, Q_a s.t. 

    0<q_a<a(t)<Q_a on [0,T].

 From Lemma <ref>, we obtain 

    |u_n^r,+(t;a)|≤ |u_n^r,+(t;q_a)|, 
    |u_n^r,-(t;a)|≤ |u_n^r,-(t;q_a)| on t∈ [0,T], +,

where u_n^r,+(t;q_a),u_n^r,-(t;q_a) are the unique solutions of 
fractional ODEs (<ref>) and (<ref>) respectively 
with a(t)≡ q_a on [0,T].
The next two lemmas concern the regularity of 
u^r,+(x,t;a) and u^r,+(x,t;a), respectively.


    u^r,+_L^2(0,T;H^2(Ω))≤ CF_L^2([0,T]×Ω).




Calculating u^r,+(x,t;a)_L^2(0,T;H^2(Ω))^2 directly 
yields

    u^r,+(x,t;a)_L^2(0,T;H^2(Ω))^2
        =∫_0^Tu^r,+(x,t;a)_H^2(Ω))^2  dt
     ≤∫_0^T C(-Ł u^r,+)(x,t;a)_L^2(Ω)^2  dt
       =C∫_0^T∑_n=1^∞λ_nu_n^r,+(t;a)ϕ_n(x)
     _L^2(Ω)^2  dt
       =C∫_0^T∑_n=1^∞λ_n^2 |u_n^r,+(t;a)|^2
      dt
     ≤ C∫_0^T∑_n=1^∞λ_n^2 
     |u_n^r,+(t;q_a)|^2  dt,

where the last inequality is obtained from (<ref>). 
By the Monotone Convergence Theorem, we have 

    u^r,+(x,t;a)_L^2(0,T;H^2(Ω))^2
     ≤ C∫_0^T∑_n=1^∞λ_n^2 
     |u_n^r,+(t;q_a)|^2  dt
     =C ∑_n=1^∞∫_0^T
     |λ_nu_n^r,+(t;q_a)|^2  dt.


For each +, <cit.> gives the explicit  
representation of u_n^r,+(t;q_a)

    u_n^r,+(t;q_a)=∫_0^t F^+_n(τ) (t-τ)^α-1 
     E_α,α(-λ_n q_a(t-τ)^α) dτ,

which together with Young's inequality leads to

    ∫_0^T|λ_nu_n^r,+(t;q_a)|^2  dt
        =F_n^+(t)*(λ_nt^α-1
     E_α,α(-λ_nq_at^α))_L^2[0,T]^2
       ≤F_n^+_L^2[0,T]^2 λ_nt^α-1
     E_α,α(-λ_nq_at^α)_L^1[0,T]^2.

Lemmas <ref>, <ref> and 
<ref> give the bound of λ_nt^α-1
 E_α,α(-λ_nq_at^α)_L^1[0,T]

    λ_nt^α-1
     E_α,α(-λ_nq_at^α)_L^1[0,T]   =∫_0^T |λ_nτ^α-1 
     E_α,α(-λ_n q_aτ^α)| dτ
       =∫_0^T λ_nτ^α-1 
     E_α,α(-λ_n q_aτ^α) dτ
       =-q_a^-1∫_0^T d/dτ 
    E_α,1(-λ_nq_aτ^α) dτ
       =q_a^-1(1-E_α,1(-λ_nq_aT^α))
    ≤ q_a^-1;

while the definition (<ref>) provides the bound of 
F_n^+_L^2[0,T] as 
F_n^+_L^2[0,T]≤F_n_L^2[0,T].
Consequently, it holds ∫_0^T|λ_nu_n^r,+(t;q_a)|^2  dt≤ q_a^-2F_n_L^2[0,T]^2, +, 
 i.e.  

    ∑_n=1^∞∫_0^T|λ_nu_n^r,+(t;q_a)|^2  dt
    ≤ q_a^-2∑_n=1^∞F_n_L^2[0,T]^2,

which together with (<ref>) and the completeness 
of {ϕ_n(x):+} in L^2(Ω) gives  

    u^r,+(x,t;a)_L^2(0,T;H^2(Ω))^2
        ≤ C∑_n=1^∞∫_0^T
     |λ_nu_n^r,+(t;q_a)|^2  dt
       ≤ C∑_n=1^∞F_n_L^2[0,T]^2
     =CF_L^2([0,T]×Ω)^2,

where the constant C only depends on a(t). This completes the proof.



 
    u^r,+_L^2([0,T]×Ω)≤ CF_L^2([0,T]×Ω).



(<ref>), (<ref>), 
definition (<ref>) and the Monotone Convergence Theorem give  

    u^r,+_L^2([0,T]×Ω)^2
         =∫_0^T ∑_n=1^∞ u_n^r,+(·;a) ϕ_n(x)
      _L^2(Ω)^2 dt
       =∑_n=1^∞∫_0^T | u_n^r,+(·;a)|^2  dt
       ≤∑_n=1^∞∫_0^T (2|λ_n a(t) u_n^r,+(t;a)|^2
     +2|F_n^+(t)|^2) dt
       ≤2∑_n=1^∞∫_0^T |λ_n a(t) u_n^r,+(t;a)|^2
      dt +2∑_n=1^∞∫_0^T |F_n(t)|^2 dt.

The estimate of ∑_n=1^∞∫_0^T 
|λ_n a(t) u_n^r,+(t;a)|^2 dt follows from 
(<ref>), (<ref>) and the proof of Lemma 
<ref>

    ∑_n=1^∞∫_0^T|λ_n a(t) u_n^r,+(t;a)|^2 dt
     ≤ Q_a ∑_n=1^∞∫_0^T
     |λ_n u_n^r,+(t;q_a)|^2 dt
     ≤ CF_L^2([0,T]×Ω)^2;

while the completeness of {ϕ_n(x):+} gives 
∑_n=1^∞∫_0^T |F_n(t)|^2 dt
 =F_L^2([0,T]×Ω)^2.
Hence, (<ref>) develops 
u^r,+_L^2([0,T]×Ω)^2
≤ CF_L^2([0,T]×Ω)^2, 
which implies the indicated conclusion.



The following corollary follows immediately from the proofs of 
Lemmas <ref> and <ref>.  


 
    u^r,-_L^2(0,T;H^2(Ω))≤ CF_L^2([0,T]×Ω),
       u^r,-_L^2([0,T]×Ω)≤ CF_L^2([0,T]×Ω).



From Lemmas <ref>, <ref>, 
Corollary <ref> and (<ref>), 
we are able to deduce the regularity for u^r(x,t;a) and 
u^r(x,t;a).


 
    u^r_L^2(0,T;H^2(Ω))
      + u^r_L^2([0,T]×Ω)≤ CF_L^2([0,T]×Ω).



 (<ref>) gives 
 u^r(x,t;a)=u^r,+(x,t;a)+u^r,-(x,t;a), which leads to

    u^r_L^2(0,T;H^2(Ω))
      + u^r_L^2([0,T]×Ω)
    ≤    u^r,+_L^2(0,T;H^2(Ω))
     +u^r,-_L^2(0,T;H^2(Ω))
       + u^r,+_L^2([0,T]×Ω)
     + u^r,-_L^2([0,T]×Ω)
    ≤    CF_L^2([0,T]×Ω).



If we impose a higher regularity on F, we can obtain the regularity estimate of u^r_C([0,T];H^2(Ω)). 

Under Assumption <ref>, if F∈ C^θ([0,T];L^2(Ω)), 0<θ<1, then  
    u^r_C([0,T];H^2(Ω))
    + u^r_C([0,T];L^2(Ω))≤ C  F_C^θ([0,T];L^2(Ω)),
 where C depends on Ω, -Ł and a(t). 


For each t∈[0,T], we have 

    u^r,+(x,t;a)^2_H^2(Ω)   ≤ C -Ł u^r,+^2_L^2(Ω)≤ C ∑_n=1^∞
    |λ_nu^r,+_n(t;a)|^2
       ≤ C ∑_n=1^∞|λ_n ∫_0^t F^+_n(τ) (t-τ)^α-1 
    E_α,α(-λ_n q_a(t-τ)^α) dτ|^2
       ≤ C ∑_n=1^∞|λ_n ∫_0^t |F^+_n(τ) -F^+_n(t)|(t-τ)^α-1 
    E_α,α(-λ_n q_a(t-τ)^α) dτ|^2
          +C ∑_n=1^∞|F_n^+(t)∫_0^t λ_n(t-τ)^α-1 
    E_α,α(-λ_n q_a(t-τ)^α) dτ|^2.

The definition of F_n^+(t) yields that  
|F^+_n(τ) -F^+_n(t)|≤ |F_n(τ) -F_n(t)|;
Lemma <ref> gives 

    0<∫_0^t λ_n(t-τ)^α-1 
    E_α,α(-λ_n q_a(t-τ)^α) dτ=
    q_a^-1(1-E_α,1(-λ_n q_a t^α))<q_a^-1.

Hence, 

    u^r,+(x,t;a)^2_H^2(Ω)   ≤ C ∑_n=1^∞|λ_n ∫_0^t |F_n(τ) -F_n(t)|(t-τ)^α-1 
    E_α,α(-λ_n q_a(t-τ)^α) dτ|^2
          +C ∑_n=1^∞|F_n(t)|^2.

By <cit.>, we have 

    u^r,+(x,t;a)^2_H^2(Ω)≤ C F_C^θ([0,T];L^2(Ω))^2+C F(·,t)_L^2(Ω)^2,
     t∈[0,T],

which gives 

    u^r,+_C([0,T];H^2(Ω))≤ C F_C^θ([0,T];L^2(Ω)),

and the constant C depends on Ω, -Ł and a(t). Similarly, we can show
 u^r,-_C([0,T];H^2(Ω))≤ C F_C^θ([0,T];L^2(Ω)). 


For u^r, by (<ref>), we have 
u^r,+=∑_n=1^∞ [-λ_n a(t) u_n^r,+(t;a)+F_n^+(t)]ϕ_n(x). 
Then for each t∈[0,T],

    u^r,+^2_L^2(Ω)   ≤ 
    C∑_n=1^∞ Q_a^2|λ_n  u_n^r,+(t;a)|^2+C∑_n=1^∞ |F_n(t)|^2 
       ≤ C∑_n=1^∞ |λ_n  u_n^r,+(t;a)|^2+CF(·,t)_L^2(Ω)^2.

From the above proof for u^r,+^2_H^2(Ω), it holds 

    u^r,+^2_L^2(Ω)≤ C F^2_C^θ([0,T];L^2(Ω))+CF(·,t)_L^2(Ω)^2, t∈[0,T],
 
which gives 

    u^r,+_C([0,T];L^2(Ω))≤ C F_C^θ([0,T];L^2(Ω)).

Analogously, we can show 
u^r,-_C([0,T];L^2(Ω))≤ C F_C^θ([0,T];L^2(Ω)).

The estimates of u^r,+, u^r,-,  u^r,+ and u^r,- 
yield the desired result and complete this proof.




  §.§.§ Regularity of u^i

In this part we consider the regularity of u^i. Just as in  
the regularity results for u^r, we first state two lemmas which 
concern the positivity and monotonicity of u^i, respectively.

 With the representation (<ref>) and the fractional ODE
 (<ref>), for each +, b_n ≤ (≥)0 implies that 
 u^i_n(t;a)≤ (≥)0 on [0,T].


 This is a directly result of Corollary <ref>.



 Given a_1, a_2∈ C^+[0,T] with a_1≤ a_2 on [0,T], 
 for each +, we have 
 
    0≤ u_n^i(t;a_2)≤ u_n^i(t;a_1), if b_n≥ 0;
    
       u_n^i(t;a_1)≤ u_n^i(t;a_2) ≤ 0, if b_n≤ 0.




 Fix +, from the fractional ODE (<ref>), 
 the functions u^i_n(t;a_1) and u^i_n(t;a_2) satisfy the following system
 
    u_n^i(t;a_1)+λ_na_1(t)u_n^i(t;a_1)=0;
     u_n^i(t;a_2)+λ_na_2(t)u_n^i(t;a_2)=0;
    
       u_n^i(0;a_1)=u_n^i(0;a_2)=b_n.

This gives 
 
    w+λ_na_1(t)w(t)=λ_nu_n^i(t;a_2)(a_2(t)-a_1(t)),
      w(0)=0,
 
 where w(t)=u_n^i(t;a_1)-u_n^i(t;a_2).
 If b_n≥ 0, Corollary <ref> 
 shows that u_n^i(t;a_1),u_n^i(t;a_2)≥ 0.  
 Also, Lemma <ref> and 
 a_1≤ a_2 ensures the right side of (<ref>) is nonnegative, 
 which together with Corollary <ref> implies w≥ 0,
 i.e. 0≤ u_n^i(t;a_2)≤ u_n^i(t;a_1).
  The similar argument yields u_n^i(t;a_1)≤ u_n^i(t;a_2) ≤ 0 
 for the case b_n≤ 0.




 
    u^i_L^2(0,T;H^2(Ω))+ u^i_L^2([0,T]×Ω)≤ C T^1-α/2u_0_H^1(Ω).




 Given t∈ [0,T], the direct calculation and Lemma <ref>
 yield that 
 
    u^i(x,t;a)_H^2(Ω)^2 
         ≤ C-Ł u^i(x,t;a)_L^2(Ω)^2
      =C∑_n=1^∞λ_nu^i_n(t;a)ϕ_n(x) _L^2(Ω)^2
       =C∑_n=1^∞|λ_nu^i_n(t;a)|^2
      ≤ C∑_n=1^∞|λ_nu^i_n(t;q_a)|^2.

Recall that <cit.> established the representation as
  u^i_n(t;q_a)=b_n E_α,1(-λ_nq_at^α), +.
Hence, by Lemma <ref>, 

    u^i(x,t;a)_H^2(Ω)^2
        ≤ C-Ł u^i(x,t;a)_L^2(Ω)^2
     ≤ C∑_n=1^∞|λ_nb_n 
     E_α,1(-λ_nq_at^α)|^2
       ≤ C∑_n=1^∞ |1/1+λ_nq_at^α|^2λ_n^2b_n^2
     = C∑_n=1^∞|(λ_nq_at^α)^1/2/1+λ_nq_at^α|^2t^-αq_a^-1λ_nb_n^2
       ≤ Ct^-α∑_n=1^∞((-Ł)^1/2u_0,ϕ_n)^2
     ≤ Ct^-αu_0_H^1(Ω)^2,

which leads to
u^i_L^2(0,T;H^2(Ω))^2
 ≤ C∫_0^T t^-αu_0_H^1(Ω)^2  dt
 =CT^1-αu_0_H^1(Ω)^2,
i.e.

    u^i_L^2(0,T;H^2(Ω))≤ C T^1-α/2u_0_H^1(Ω).


For the estimate of u^i(x,t;a), 
(<ref>) and (<ref>) yield  

    u^i(x,t;a)=∑_n=1^∞ u^i_n(t;a)ϕ_n(x)
    =-∑_n=1^∞λ_n a(t) u^i_n(t;a)ϕ_n(x),

which together with (<ref>) gives

    u^i(x,t;a)_L^2(Ω)^2
       ≤ Q_a^2 ∑_n=1^∞ |λ_n u^i_n(t;a)|^2
       =Q_a^2 -Ł u^i(x,t;a)_L^2(Ω)^2
    ≤ Ct^-αu_0_H^1(Ω)^2, t∈[0,T],

where the last inequality follows from (<ref>).
This result implies that 

    u^i(x,t;a)_L^2([0,T]×Ω)^2
     =∫_0^T  u^i(x,t;a)_L^2(Ω)^2  dt
     ≤ CT^1-αu_0_H^1(Ω)^2,

i.e. u^i_L^2([0,T]×Ω)≤  C T^1-α/2u_0_H^1(Ω), which together 
 with (<ref>) completes the proof.


Moreover, with a stronger condition on u_0, such as assuming  
u_0∈ H^2(Ω)∩ H^1_0(Ω),
we can deduce the C-regularity estimate of u^i. 



 With Assumption <ref> and 
 u_0∈ H^2(Ω)∩ H^1_0(Ω), then  
 
    u^i_C([0,T];H^2(Ω))+ u^i_C([0,T];L^2(Ω))≤ Cu_0_H^2(Ω).




Lemma <ref> yields that 

    ∑_n=1^∞|λ_nb_n 
     E_α,1(-λ_nq_at^α)|^2
     ≤ C∑_n=1^∞|λ_nb_n|^2
     =C-Ł u_0_L^2(Ω)^2 ≤ C u_0_H^2(Ω)^2, t∈[0,T];

meanwhile, the following estimates have been shown in 
 the proof of Theorem <ref>

    u^i(x,t;a)_H^2(Ω)^2
     ≤ C-Ł u^i(x,t;a)_L^2(Ω)^2
     ≤ C∑_n=1^∞|λ_nb_n 
     E_α,1(-λ_nq_at^α)|^2,
     u^i(x,t;a)_L^2(Ω)^2
    ≤ Q_a^2 ∑_n=1^∞ |λ_n u^i_n(t;a)|^2
    =C -Ł u^i(x,t;a)_L^2(Ω)^2.

Hence, it holds that 

    u^i(x,t;a)_H^2(Ω)+ u^i(x,t;a)_L^2(Ω)≤ Cu_0_H^2(Ω), t∈[0,T],

which leads to the claimed result.





 §.§ Main theorem for the direct problem


The main theorem for the direct problem follows from Theorem 
<ref>, Lemmas  <ref> and <ref>, 
Corollaries <ref> and <ref>, and the relation 
u(x,t;a)=u^r(x,t;a)+u^i(x,t;a).


 Let Assumption <ref> be valid, then under Definition 
 <ref>, there exists a unique weak solution u(x,t;a) 
 of FDE (<ref>) with the spectral representation 
 (<ref>) and the following regularity estimates:
 
    u_L^2(0,T;H^2(Ω))+ u_L^2([0,T]×Ω)≤ C(F_L^2([0,T]×Ω)
    +T^1-α/2u_0_H^1(Ω)).

Moreover, if the conditions u_0∈ H^2(Ω)∩ H^1_0(Ω) and F∈ C^θ([0,T];L^2(Ω)), 0<θ<1 are added, we have: 
 
    u_C([0,T];H^2(Ω))+ u_C([0,T];L^2(Ω))≤ C(F_C^θ([0,T];L^2(Ω))+u_0_H^2(Ω)).







§ INVERSE PROBLEM–RECONSTRUCTION OF THE DIFFUSION COEFFICIENT A(T)


In this section, we discuss how to recover the coefficient a(t) 
through the output flux data 

    a(t)∂ u/(x_0,t;a)=g(t), x_0∈∂Ω.
 
All cross the inverse problem work, the operator -Ł is assumed to satisfy the condition (<ref>), then the expression ∂ϕ_n/(x_0) makes sense.
We only consider this reconstruction in the space C^+[0,T], which 
can be regarded as the admissible set for a(t).
To this end, we introduce an operator K, which will be shown to have 
a fixed point consisting of the desired coefficient a(t). 



 §.§ Operator K


The operator K is defined as 

    K ψ(t):=g(t)/∂ u/(x_0,t;ψ)
    =g(t)/∑_n=1^∞ u_n(t;ψ)
    ∂ϕ_n/(x_0),
     t∈ [0,T]
 
with domain 

    :={ψ∈ C^+[0,T]:ψ(t)≥ g(t)[∂ u_0/(x_0)
    + [∂ F/(x_0,t)]]^-1, t∈[0,T] }.


To analyze K, we make the following assumptions. 

 u_0, F and g should satisfy the following restrictions:
 
  
  (a) u_0∈ H^3(Ω)∩ H_0^1(Ω) with 
  b_n:=(u_0,ϕ_n)≥ 0, +;
  
  (b) ∃θ∈ (0,1) s.t. F(x,t)∈ C^θ([0,T];H^3(Ω)∩ H_0^1(Ω)) with 
  F_n(t):=(F(·,t),ϕ_n)≥ 0 on [0,T] for each +;
  
  (c) ∃ N∈ℕ^+ s.t. 
  ∂ϕ_N/(x_0)>0, b_N>0 and F_N(t)>0 on [0,T]; 
  
  (d) g∈ C^+[0,T].
 


The next remark shows that the equality in the definition of K is valid. 

Given ψ∈ C^+[0,T] and for each t∈[0,T], by the proofs of Corollaries <ref> and <ref>, we have 

    u^r,+(x,t;ψ)^2_H^3(Ω)   ≤ C (-Ł)^3/2u^r,+^2_L^2(Ω)≤ C ∑_n=1^∞
    |λ_n^3/2u^r,+_n(t;ψ)|^2
       ≤ C ∑_n=1^∞|λ_n^3/2∫_0^t F^+_n(τ) (t-τ)^α-1 
    E_α,α(-λ_n q_ψ(t-τ)^α) dτ|^2
       ≤ C ∑_n=1^∞|λ_n ∫_0^t λ_n^1/2|F^+_n(τ) -F^+_n(t)|(t-τ)^α-1 
    E_α,α(-λ_n q_ψ(t-τ)^α) dτ|^2
          +C ∑_n=1^∞|λ_n^1/2F_n^+(t)(1-E_α,1(-λ_n q_ψ t^α))|^2
       ≤ C ∑_n=1^∞|λ_n ∫_0^t λ_n^1/2|F_n(τ) -F_n(t)|(t-τ)^α-1 
    E_α,α(-λ_n q_ψ(t-τ)^α) dτ|^2
          +C ∑_n=1^∞|λ_n^1/2F_n(t)(1-E_α,1(-λ_n q_ψ t^α))|^2
       ≤ C (-Ł)^1/2 F_C^θ([0,T];L^2(Ω))^2+C(-Ł)^1/2 F(·,t)_L^2(Ω)^2
       ≤ C  F_C^θ([0,T];H^1(Ω))^2+CF(·,t)_H^1(Ω)^2

and 

    u^r,-(x,t;ψ)^2_H^3(Ω)≤ C  F_C^θ([0,T];H^1(Ω))^2+CF(·,t)_H^1(Ω)^2,

which give 
u^r_C([0,T];H^3(Ω))≤ C F_C^θ([0,T];H^1(Ω));

    u^i(x,t;ψ)^2_H^3(Ω)   ≤ C (-Ł)^3/2u^i^2_L^2(Ω)≤ C ∑_n=1^∞λ_n^3/2u_n^i(t;ψ)ϕ_n(x)^2_L^2(Ω)
       ≤ C∑_n=1^∞|λ_n^3/2b_n 
    E_α,1(-λ_nq_ψ t^α)|^2
    ≤ C∑_n=1^∞|λ_n^3/2b_n|^2
       =C(-Ł)^3/2 u_0_L^2(Ω)^2 ≤ C u_0_H^3(Ω)^2,

which gives u^i_C([0,T];H^3(Ω))≤ C u_0_H^3(Ω).
Combining the above two results yields that 

    u_C([0,T];H^3(Ω))≤ C ( F_C^θ([0,T];H^1(Ω))+u_0_H^3(Ω))<∞,

which means for each t∈ [0,T], 
u_H^3(Ω)<∞. Recall that Ω⊂ R^n, n=1,2,3, then the Sobolev Embedding Theorem gives 

    u(x,t;ψ)=∑_n=1^∞ u_n(t;ψ)ϕ_n(x)∈ C^1(Ω) for each t∈ [0,T].
 
Hence, 
∑_n=1^∞ u_n(t;ψ)
∂ϕ_n/(x_0) is well-defined and 

    ∂ u/(x_0,t;ψ)=∑_n=1^∞ u_n(t;ψ)
    ∂ϕ_n/(x_0),   t∈[0,T].



The following two remarks will explain the reasonableness and reason 
for Assumption <ref>.

 For the inverse problem, the right-hand side function F(x,t)
 and the initial condition u_0(x) are input data, which, 
 at least in some circumstance, can be assumed to be controlled. 
 Even though Assumption <ref> 
 (a), (b) and (c) appear restrictive, it is not hard to construct 
 functions that satisfy them. For example, in (a) if u_0=cϕ_k for 
 some c>0, then Assumption <ref> (a) will be 
 satisfied. This will also be true if u_0=∑_k=1^M c_kϕ_k 
 with all c_k>0. Similarly, (b) is satisfied if F(x,t) is also a 
 linear combination of {ϕ_n:+} with positive coefficients. 
 For (c), by the completeness of {ϕ_n:+} in L^2(Ω), there should exist 
 N∈ℕ^+ s.t. ∂ϕ_N/(x_0)>0. Otherwise, for each 
 ψ∈ H^3(Ω)⊂ L^2(Ω), ∂ψ/(x_0)=0 and obviously 
 it is incorrect. Then for this N, we only need to set the 
 coefficients of u_0 and F upon ϕ_N be strictly positive. 
  The output flux data g(t), it is not under our control. 
  However, if there exists a∈ C^+[0,T] s.t. a(t)∂ u/(x_0,t;a)=g(t),  
 Assumption <ref> (a), (b) and Corollary 
 <ref> yield that u_n(t;a)≥ 0; 
  (<ref>) gives
 ∂ϕ_n/(x_0)≥ 0, +;
 Assumption <ref> (c) ensures 
 ∂ϕ_N/(x_0)>0 and 
 u_N(t;a)>0 on [0,T], where the proof can be seen in 
 Lemma <ref>. 
 Consequently, 
    ∂ u/(x_0,t;a)
     =∑_n=1^∞ u_n(t;a)∂ϕ_n/(x_0)
     ≥ u_N(t;a)∂ϕ_N/(x_0)>0, t∈[0,T].
 
 This together with a∈ C^+[0,T] gives that g>0.
 The continuity of g follows from the ones of a and u_n(t;a), +, 
 which are derived from the admissible set C^+[0,T] and 
 Theorem <ref>, respectively.
 Therefore, Assumption <ref> (d) is 
 reasonable and can be attained.



 The well-definedness of the domain  is guaranteed by 
 Assumption <ref> (a), (b), (c) and (d) in the 
 sense that the H^3-regularity of u_0, F and the Sobolev Embedding Theorem support that ∂ u_0/(x_0) and ∂ F/(x_0,t) are well defined, and 
 the dominator of the lower bound of  
 
    ∂ u_0/(x_0)+[∂ F/(x_0,t)]
     =∑_n=1^∞ (b_n+ F_n) ∂ϕ_n/(x_0)
     ≥ (b_N+ F_N) ∂ϕ_N/(x_0)>0

 on [0,T]. Recall that the numerator g>0, so that the lower bound 
 g(t)[∂ u_0/(x_0)
 +[∂ F/(x_0,t)]]^-1>0, which 
 gives that  is a subspace of C^+[0,T]. 
 Also, F(x,t)∈ C^θ([0,T];H^3(Ω)∩ H_0^1(Ω)) yields that 
 F_N(t) is continuous on [0,T], so is 
 (b_N+ F_N) ∂ϕ_N/(x_0). Then ∃ C>0
 s.t. (b_N+ F_N) ∂ϕ_N/(x_0)>C>0, which leads to 
 the dominator 
    ∂ u_0/(x_0)
     +[∂ F/(x_0,t)]>C>0 on [0,T].
 
 The strict positivity of the dominator avoids  degenerating 
 to an empty set. 
 In order to show the well-definedness of K,  
 Assumption <ref> (a), (b) and (c) will 
 be used. Furthermore, Assumption <ref> 
 (a) and (b) are crucial to build the monotonicity of operator K; 
 meanwhile, Assumption <ref> (c) is stated for 
 the uniqueness of fixed points of K.


For the operator K, we have the following lemmas. 

 The operator K is well-defined.


 For each ψ∈, Theorem <ref> ensures that 
  there exists a unique u_n(t;ψ) for +, 
 which implies the existence and uniqueness of Kψ.
 
 Then it is suffice to show the dominator ∑_n=1^∞ u_n(t;ψ)
∂ϕ_n/(x_0)>0 on [0,T].
With (<ref>), Lemma <ref> and Assumption 
<ref> (a) and (b), we have 
u_n(t;ψ)≥ 0 on [0,T], which together with 
∂ϕ_n/(x_0)≥0 gives 
∑_n=1^∞ u_n(t;ψ)∂ϕ_n/(x_0)
≥ u_N(t;ψ)∂ϕ_N/(x_0).
Due to the assumption ∂ϕ_N/(x_0)>0, 
we claim that u_N(t;ψ)>0. 
Assume not, i.e. ∃ t_0∈ [0,T] s.t. u_N(t_0;ψ)≤ 0. 
The result u_N(t;ψ)≥ 0 yields that u_N(t_0;ψ)=0 so that 
u_N(t;ψ) attains its minimum at t=t_0. 
u_N(0;ψ)=b_N>0 implies t_0 0, i.e. t_0∈(0,T]. 
Then Lemma <ref>, u_N(t_0;ψ)=0 and the ODE (<ref>)
show that ^CD_t^αu_N(t_0;ψ)=F_N(t_0)≤ 0, 
which contradicts with Assumption <ref> (c) 
and confirms the claim. Hence, 

    ∑_n=1^∞ u_n(t;ψ)∂ϕ_n/(x_0)
    ≥ u_N(t;ψ)∂ϕ_N/(x_0)>0,
 
which completes the proof.



 K maps  into .


 Given ψ∈. The continuity of Kψ follows from the continuity 
 of u_n(t;ψ) for each + and the continuity of g, which are 
 established by Theorem <ref> and Assumption 
 <ref> (d) respectively.
 
 For each +, (<ref>) ensures u_n(t;ψ) satisfies 
 
    u_n(t;ψ)+λ_n ψ(t) u_n(t;ψ)=F_n(t),
       u_n(0;ψ)=b_n.

Taking  on both sides of the above ODE and using 
 Lemma <ref> yield that 
 
    u_n(t;ψ)+λ_n [ψ(t) u_n(t;ψ)]= F_n+b_n.

 From the proof of Lemma <ref>, we have 
 u_n(t;ψ)≥ 0 on [0,T], which together with 
 λ_n>0, the positivity of ψ and the definition of 
  yields that λ_n [ψ(t) u_n(t;ψ)]≥ 0.
 Since u_n(t;ψ)≥ 0 and 
 λ_n [ψ(t) u_n(t;ψ)]≥ 0, 
 we deduce that 0≤ u_n(t;ψ)≤ F_n+b_n on [0,T].
 Hence, with ∂ϕ_n/(x_0)≥ 0 and 
 the smoothness assumptions u_0∈ H^3(Ω)∩ H_0^1(Ω), 
F∈ C^θ([0,T];H^3(Ω)∩ H_0^1(Ω)) 
stated in Assumption <ref> (a) and (b) respectively,
 the following inequality holds
 
    ∑_n=1^∞ u_n(t;ψ)∂ϕ_n/(x_0)
      ≤∑_n=1^∞ ( F_n+b_n) ∂ϕ_n/(x_0)
      =∂ u_0/(x_0)+[∂ F/(x_0,t)],

which together with g>0 yields that 

    Kψ(t) = g(t)/∑_n=1^∞ u_n(t;ψ)
    ∂ϕ_n/(x_0)≥
    g(t)[∂ u_0/(x_0)
    + [∂ F/(x_0,t)]]^-1>0, t∈[0,T],

where the last inequality follows from Remark <ref>.
The above result and the continuity of Kψ lead to 
Kψ∈, which is the expected result.





 §.§ Monotonicity

In this part, we show the monotonicity of the operator K.

 Given a_1,a_2∈ with a_1≤ a_2, then 
 Ka_1≤ Ka_2 on [0,T].


Pick +, due to (<ref>), u_n(t;a_1) and u_n(t;a_2) 
satisfy 

    u_n(t;a_1)+λ_n a_1(t) u_n(t;a_1)=F_n(t),
       u_n(0;a_1)=b_n;
     u_n(t;a_2)+λ_n a_2(t) u_n(t;a_2)=F_n(t),
       u_n(0;a_2)=b_n,

which together with a_1≤ a_2 and Lemma <ref> yields 

    w+λ_na_1(t)w(t)=λ_nu_n(t;a_2)(a_2(t)-a_1(t))≥ 0,
      w(0)=0,

where w(t)=u_n(t;a_1)-u_n(t;a_2).
Applying Lemma <ref> to the above ODE yields that 
w≥ 0, i.e. u_n(t;a_1)≥ u_n(t;a_2)≥0, which together with 
assumption (<ref>) leads to 

    ∑_n=1^∞ u_n(t;a_1)∂ϕ_n/(x_0)
    ≥∑_n=1^∞ u_n(t;a_2)∂ϕ_n/(x_0) 
    > 0, t∈[0,T].

Therefore, with the condition g>0 stated in 
Assumption <ref> (d), 

    Ka_1(t)=g(t)/∑_n=1^∞ u_n(t;a_1)
    ∂ϕ_n/(x_0)≤g(t)/∑_n=1^∞ u_n(t;a_2)
    ∂ϕ_n/(x_0)=Ka_2(t), t∈[0,T],
 
which completes this proof.





 §.§ Uniqueness

In order to show the uniqueness, we state two lemmas. 

 If a_1,a_2∈ are both fixed points of K with a_1≤ a_2, 
 then a_1≡ a_2.


 Pick a fixed point a(t), then 
 
    a(t)∑_n=1^∞ u_n(t;a)
    ∂ϕ_n/(x_0)
    =∑_n=1^∞ a(t)u_n(t;a)
    ∂ϕ_n/(x_0)=g(t),

which gives 

    ∑_n=1^∞[a(t)u_n(t;a)]
    ∂ϕ_n/(x_0)= g

by taking  on both sides.
Similarly, taking  on the both sides of (<ref>)
and applying Lemma <ref> yield that 

    [a(t)u_n(t;a)]=λ_n^-1 F_n+λ_n^-1b_n
    -λ_n^-1u_n(t;a), +,

which together with (<ref>) generates 

    ∑_n=1^∞λ_n^-1u_n(t;a)
     ∂ϕ_n/(x_0)
     =∑_n=1^∞λ_n^-1( F_n+b_n)
     ∂ϕ_n/(x_0)- g.

In (<ref>), the convergence of the two series in C[0,T] is supported by Assumption <ref>, Remark <ref> and the fact that 0<λ_1≤λ_2≤⋯. 

Given two fixed points a_1,a_2 with a_1≤ a_2, then a_1 and a_2 
should satisfy (<ref>) simultaneously, which gives 

    ∑_n=1^∞λ_n^-1∂ϕ_n/(x_0)(u_n(t;a_1)-u_n(t;a_2))=0.

In the proof of Theorem <ref>, we have shown that  
u_n(t;a_1)≥ u_n(t;a_2)≥0. Also recall that 
λ_n^-1∂ϕ_n/(x_0)≥ 0, +, 
then λ_n^-1∂ϕ_n/(x_0)
(u_n(t;a_1)-u_n(t;a_2))≥ 0 on [0,T] for +.
Hence, (<ref>) implies that 

    λ_n^-1∂ϕ_n/(x_0)
    (u_n(t;a_1)-u_n(t;a_2))= 0, t∈[0,T], +.

Let n=N, λ_N^-1∂ϕ_N/(x_0)>0
gives u_N(t;a_1)≡ u_N(t;a_2) on [0,T].
Set w(t)=u_N(t;a_1)-u_N(t;a_2)=0. Then (<ref>) yields that 

    0= w+λ_Na_1(t)w(t)=λ_Nu_N(t;a_2)(a_2(t)-a_1(t)),

i.e. u_N(t;a_2)(a_2(t)-a_1(t))≡ 0 on [0,T]; while the proof of Lemma <ref> yields that u_N(t;a_2)>0. Hence, we have  
a_1= a_2 on [0,T], which completes the proof.


Before showing uniqueness, we introduce a successive iteration procedure 
which will generate a sequence converging to a fixed point if it exists. Set 

    a_0(t)=g(t)[∂ u_0/(x_0)
    + [∂ F/(x_0,t)]]^-1, a_n+1=Ka_n, .
 Then this iteration reproduces 
a sequence {a_n:} which is contained by 
 due to Lemma <ref>.


 If there exists a fixed point a(t)∈ of operator K, then the 
 sequence {a_n:} will converge to a(t).


 a_0 is the lower bound of  and 
 {a_n:}⊂ yield that 
 a_0≤a_1. Using Theorem <ref>, 
 we have a_1=Ka_0≤ Ka_1=a_2, 
 i.e. a_1≤a_2. The same argument 
 gives a_2=Ka_1≤ Ka_2=
 a_3. Continue this process, we can deduce 
 a_0≤a_1≤a_2≤…, 
 which means {a_n:} is increasing. 
 Since the results that a_0 is the lower bound of  
 and a(t)∈, it holds a_0≤ a. 
 Applying Theorem <ref> 
 to this inequality, we obtain a_1=Ka_0≤ Ka=a, 
 i.e. a_1≤ a.
 This argument generates 
  a_n≤ a, , which means a(t) is an upper bound 
 of {a_n:}.
 
 We have proved {a_n:} is an increasing sequence 
 in  with an upper bound a(t), which leads to {a_n:}
 is convergent in  and the limit is smaller than a(t). 
 Denote the limit of {a_n:} by a. 
 We have a∈, a≤ a and 
 a is a fixed point of K in . 
 Hence, Lemma <ref> yields 
 a=a, which is the desired result. 
 

 Now, we are able to prove the uniqueness of fixed points of K.
 
 There is at most one fixed point of K in .


 Let a_1, a_2∈ be both fixed points of K. 
 Lemma <ref> implies that 
 a_n→ a_1 and a_n→ a_2, which 
 leads to a_1=a_2 and completes this proof.




 §.§ Existence

Assumption <ref> is not sufficient to deduce the 
existence of the fixed points of K since  has 
no upper bound so that an increasing  sequence in  may not be convergent.
In this part, we discuss the existence of fixed points, by providing  
some extra conditions.

Additional assumptions on u_0, F and g:

 
  (a) -Ł u_0∈ H^3(Ω)∩ H_0^1(Ω);
 
  (b) F(x,t)=-Ł u_0(x)· f(t) s.t. f∈ C^θ[0,T], 0<θ<1 and f(t)≥ g(t)
 [∂ u_0/(x_0)]^-1 on [0,T].




Assumption <ref> is set up to make sure that 
F(x,t)=-Ł u_0(x)· f(t)∈ C^θ([0,T];H^3(Ω)∩ H_0^1(Ω)), so that 
F(x,t) also satisfies Assumption <ref>.

Fix u_0 and f, if the measured data g does not satisfy 
Assumption <ref> (b), then we can modify 
u_0 by increasing the value of u_0 in a very small neighborhood 
of the point x_0 so that the value of ∂ u_0/(x_0) 
becomes larger. Meanwhile, since u_0 is changed in a small domain, 
the coefficients {b_n:+} only vary slightly, so do u_n(t;a) and u(x,t;a). 
Hence, ∂ u/(x_0,t;a) and g(t) will not appear 
a significant change that can violate Assumption 
<ref> (b).


Define the subspace ' of  as 

    ':={ψ∈ C^+[0,T]:     g(t)[∂ u_0/(x_0)
    + [∂ F/(x_0,t)]]^-1
       ≤ψ(t)≤ g(t)[∂ u_0/(x_0)]^-1,
     t∈[0,T]}.
 
We have proved the lower bound of ' is positive in Remark 
 <ref> and clearly the upper bound 
 of ' is larger than the lower bound. Consequently, 
 ' is well-defined. 
 
The next lemma concerns the range of K with domain '.

 With Assumptions <ref> and 
 <ref>, K maps ' into '.


 Given ψ∈', we have proved 
 K ψ∈ C^+[0,T] and 
 
    Kψ(t) ≥ g(t)[∂ u_0/(x_0)
    + [∂ F/(x_0,t)]]^-1, t∈[0,T]
 
in the proof of Lemma <ref>, so that 
it is sufficient to show Kψ≤ 
g(t)[∂ u_0/(x_0)]^-1 on [0,T].

For each +, let w_n(t;ψ)=u_n(t;ψ)-b_n, 
(<ref>) yields the following ODE by direct calculation

    w_n(t;ψ)+λ_n ψ(t)w_n(t;ψ)=λ_nb_n(f(t)-ψ(t))≥0,
     w_n(0,ψ)=0,

where λ_nb_n(f(t)-ψ(t))≥ 0 follows from 
the fact ψ(t) ≤ g(t)[∂ u_0/(x_0)]^-1 
and Assumption <ref> (b).
Applying Corollary <ref> to the above ODE gives 
w_n(t;ψ)≥ 0, i.e. u_n(t;ψ)≥ b_n≥ 0 on [0,T]. 
Hence, 

    Kψ(t)=g(t)/∑_n=1^∞ u_n(t;ψ)
    ∂ϕ_n/(x_0)≤g(t)/∑_n=1^∞ b_n∂ϕ_n/(x_0)
    =g(t)[∂ u_0/(x_0)]^-1

and this proof is complete.


The existence conclusion is derived from Lemmas <ref> 
and <ref>.

Suppose Assumptions <ref> and 
<ref> be valid, then there exists a 
fixed point of K in '.


 Lemma <ref> yields the sequence 
 {a_n:} is increasing, while Lemma <ref> 
 gives {a_n:}⊂'. 
 Then {a_n:} is an increasing sequence with an
 upper bound g(t)[∂ u_0/(x_0)]^-1, 
 which implies the convergence of {a_n:}. 
 Denote the limit by a, clearly a 
 is a fixed point of K. Also, the closedness of ' 
 yields that a∈'.  
 Therefore, a is a fixed point of K in ', which 
 confirms the existence.







 §.§ Main theorem for the inverse problem and reconstruction algorithm


Lemma <ref>, Theorems <ref> and 
<ref> allow us to deduce the main theorem for this 
inverse problem.


 Suppose Assumption <ref> holds. 
 
  
  (a) If there exists a fixed point of K in , then it is 
  unique and coincides with the limit of {a_n:}; 
  
  (b) If Assumption <ref> is also valid, 
  then there exists a unique fixed point of K in ', which is the 
  limit of {a_n:}.
 


The following reconstruction algorithm for a(t) is based on 
Theorem <ref>.








§ NUMERICAL RESULTS FOR INVERSE PROBLEM




 §.§ L1 time-stepping of 

The fourth step of Algorithm <ref> includes   
solving the direct problem of FDE (<ref>) numerically. 
To this end, we choose L1 time 
stepping <cit.> to discretize the term 
u(x,t):
 
    u(x,t_N)    = 1/Γ(1-α)∑^N-1_j=0∫^t_j+1_t_j∂ u(x,s)/∂ s 
         (t_N-s)^-α  ds 
       ≈1/Γ(1-α)∑^N-1_j=0u(x,t_j+1)-u(x,t_j)/τ∫_t_j^t_j+1
         (t_N-s)^-αds
       =∑_j=0^N-1b_ju(x,t_N-j)-u(x,t_N-j-1)/τ^α
       =τ^-α [b_0u(x,t_N)-b_N-1u(x,t_0)
         +∑_j=1^N-1(b_j-b_j-1)u(x,t_N-j)] ,

where 

    b_j=((j+1)^1-α-j^1-α)/Γ(2-α), j=0,1,…,N-1.




 §.§ Numerical results for noise free data

In this part, we set Ω=(0,1), x_0=0, T=1, Ł u=u_xx, 
pick u_0(x)=-sinπ x,  F(x,t)=-(t+1)sinπ x 
and consider the following two coefficients:

 
  (a1) smooth coefficient: a(t)=sin5π t+1.3;
 
  (a2) nonsmooth coefficient (“smile” function): 
 
    a(t)   =[0.8sin3π t+1.5]χ_[0,1/3]
     +[-0.5sin(3π t-π)+0.6]χ_(1/3,2/3)
         +[0.8sin(3π t-2π)+1.5]χ_[2/3,1].



In experiment (a1), the exact coefficient we pick is a smooth 
function. Figure <ref> shows the initial guess and the 
first three iterations, while Figure <ref> presents 
the exact and approximate coefficients. From these two figures, we  
observe that {a_n:} converges to a(t) monotonically, 
which illustrates Theorems <ref> and <ref>. 
Moreover, the L^2 error of the approximation in Figure 
<ref> is 
a-a_N_L^2[0,T]=1.04× 10^-6,
which implies us the L^2 error of this approximation may be bounded 
by the stopping criterion number ϵ_0. This guess is confirmed 
by Figure <ref> and can be expressed as 

    a-a_N_L^2[0,T]=O(ϵ_0).
 
Several attempts of experiment (a1) for different α∈(0,1) 
are taken to find the dependence of the convergence rate of Algorithm 
<ref> on the fractional order α, which is shown in Figure 
<ref>. This figure shows the amounts of iterations 
required, i.e. N, corresponding to different α, which imply 
that restricted α∈(0,1), the larger α is, the faster the 
convergence rate of Algorithm <ref> is. This phenomenon 
is explained in <cit.> by a property of the Mittag-Leffler 
function; for α∈(0,1), the larger α is, 
the faster the decay rate of E_α,1(-z) is as z→∞.














The definition of  restricts the coefficient a(t) in the space 
C^+[0,T], however, the results of experiment (a2) indicate that Algorithm 
<ref> still works for nonsmooth a(t), which means the numerical 
restriction on a(t) can possibly be extended from a(t)∈ C^+[0,T] to 
a(t)∈ L^∞ [0,T].  
For discontinuous a(t), Figures <ref> 
and <ref> explain that 
Theorems <ref> and <ref> still hold, 
while Figures <ref> and <ref>   
illustrate the similar conclusions as  
the larger α is, the faster the convergence rate of Algorithm 
<ref> is, and 

    a-a_N_L^2[0,T]=O(ϵ_0).
 











 §.§ Numerical results for noisy data

In this subsection, we will consider data polluted by noise.   
Set g be the exact data and denote the noisy data by g_δ 
with relative noise level δ, i.e. 
(g-g_δ)/g_L^∞[0,T]≤δ. 
Then the perturbed operator K_δ is

    K_δψ(t)
    =g_δ(t)/∑_n=1^∞ u_n(t;ψ)
    ∂ϕ_n/(x_0)
 
with domain

    :={ψ∈ C^+[0,T]:g_δ(t)[∂ u_0/(x_0)
    + [∂ F/(x_0,t)]]^-1≤ψ(t), t∈[0,T] }.

Also, the sequence {a_δ,n:n∈ℕ} can be 
obtained from the iteration 

    a_δ,0=g_δ[∂ u_0/(x_0)
    + [∂ F/(x_0,t)]]^-1, a_δ,n+1=K_δa_δ,n, n∈ℕ.

Since δ is a small positive number and g is a strictly 
positive function, we can assume g_δ is still positive, which 
means Theorem <ref> still holds for K_δ. 
Hence, if there exists a fixed point a_δ∈, 
the sequence {a_δ,n:n∈ℕ} will converge to 
a_δ monotonically and we denote the limit by a_δ.
Algorithm <ref> is still able to be used to recover 
a_δ after a slightly modification-replacing 
g and K by g_δ and K_δ, respectively.

We take the experiments (a1) and (a2) with  
noise level δ>0. Figures <ref> and 
<ref> present 
the exact and approximate coefficients under δ=3% for 
experiments (a1) and (a2) respectively. From figures  
<ref> and <ref>, 
we observe that the smaller |a(t)| is, the better the approximation is. 
This can be explained by δ means the relatively noise level, i.e. 
we pick g_δ=(1+ζδ)g in the codes, where 
ζ follows a uniform distribution on [-1,1].
Figure <ref> illustrates that  

    a-a_δ,N_L^2[0,T]/a_L^2[0,T]=O(δ),
 
showing the domination of the noise level δ in 
relatively L^2 error with the reason that ϵ_0 ≪δ.











 §.§ Numerical results in two dimensional case

In this part, the numerical experiments on a two dimensional domain will be considered. We set α=0.9, ϵ_0=10^-6, Ω=(0,1)^2, x_0=(0,1/2), T=1, Ł u= u, choose u_0(x,y)=-sin[π xy(1-x)(1-y)], F(x,y)=-(t+1)·sin[π xy(1-x)(1-y)], and consider experiments (a1) and (a2). 
Figures <ref> and <ref> confirm the theoretical conclusions in section 4.
 






§ ACKNOWLEDGMENT

The author is indebted to William Rundell for assistance in this work 
and acknowledges partial support from NSF-DMS 1620138. 

abbrv


