# A Note on the Right-Hand Side Identification Problem Arising in Biofluid Mechanics

Authors: Abdullah Said Erdogan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The inverse problem of reconstructing the right-hand side (RHS) of a mixed problem for one-dimensional diffusion equation with variable space operator is considered. The well-posedness of this problem in Hölder spaces is established.


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## Body

## 1. Introduction

It is known that many applied problems in fluid mechanics, other areas of physics, and mathematical biology were formulated as the mathematical model of partial differential equations of the variable types [1–3]. A model for transport across microvessel endothelium was developed to determine the forces and bending moments acting on the structure of the flow over endothelial cells (ECs) [4]. Computational blood flow analysis through glycocalyx on the EC is performed as a direct problem previously under smooth and nonsmooth initial conditions (see [5–7]). But it is known that, due to the lack of some data and/or coefficients, many real-life problems are modeled as inverse problems [8–11].In this paper, the well-posedness of the inverse problem of reconstructing the right side of a parabolic equation arisen in computational blood flow analysis is investigated. The importance of well-posedness has been widely recognized by the researchers in the field of partial differential equations [12–16]. Moreover, the well-posedness of the RHS identification problems for a parabolic equation where the unknown function p is in space variable and in time variable is well investigated [17–27]. As it is known, well-posedness in the sense of Hadamard means that there is existence and uniqueness of the solution and the solution is stable. In this study, we deal with the stability analysis of the inverse problem of reconstructing the right-hand side. The existence of a solution for two-phase flow in porous media has been studied previously (for instance, see [28]).


### 1.1. Problem Formulation

Blood flow over the EC inside the arteries is modeled in two regions (see [6]). Core region (0<x<l) flow is defined through the center of capillary and porous region (l<x<L) flow is through the glycocalyx. RHS function includes the pressure difference along the microchannels under unsteady fluid flow conditions. When the pressure difference is an unknown function of t, we reach a new model, and, by overdetermined (additional) conditions derived from an observation point, the solution of this problem can be obtained. The model can be considered as the mixed problem for one-dimensional diffusion equation with variable space operator:
(1.1)∂u(t,x)∂t=a(x)∂2u(t,x)∂x2+p(t)q(x)+f(t,x),x∈(0,l),t∈(0,T],∂u(t,x)∂t=a(x)∂2u(t,x)∂x2+b(t,x)u(t,x)+p(t)q(x)+g(t,x),x∈(l,L),t∈(0,T],u(0,x)=φ(x),x∈[0,L],ux(t,0)=0,u(t,L)=0,t∈[0,T],u(t,l+)=u(t,l-),ux(t,l+)=ux(t,l-),t∈[0,T],u(t,x*)=ρ(t),0≤x*≤l,0≤t≤T.
Here, u(t,x) and p(t) are unknown functions, a(x),   b(t,x),f(t,x),g(t,x),ρ(t), and φ(x) are given sufficiently smooth functions, and a(x)⩾a>0. Also, q(x) is a sufficiently smooth function assuming that q′(0)=q(L)=0 and q(x*)≠0.


## 1.1. Problem Formulation

Blood flow over the EC inside the arteries is modeled in two regions (see [6]). Core region (0<x<l) flow is defined through the center of capillary and porous region (l<x<L) flow is through the glycocalyx. RHS function includes the pressure difference along the microchannels under unsteady fluid flow conditions. When the pressure difference is an unknown function of t, we reach a new model, and, by overdetermined (additional) conditions derived from an observation point, the solution of this problem can be obtained. The model can be considered as the mixed problem for one-dimensional diffusion equation with variable space operator:
(1.1)∂u(t,x)∂t=a(x)∂2u(t,x)∂x2+p(t)q(x)+f(t,x),x∈(0,l),t∈(0,T],∂u(t,x)∂t=a(x)∂2u(t,x)∂x2+b(t,x)u(t,x)+p(t)q(x)+g(t,x),x∈(l,L),t∈(0,T],u(0,x)=φ(x),x∈[0,L],ux(t,0)=0,u(t,L)=0,t∈[0,T],u(t,l+)=u(t,l-),ux(t,l+)=ux(t,l-),t∈[0,T],u(t,x*)=ρ(t),0≤x*≤l,0≤t≤T.
Here, u(t,x) and p(t) are unknown functions, a(x),   b(t,x),f(t,x),g(t,x),ρ(t), and φ(x) are given sufficiently smooth functions, and a(x)⩾a>0. Also, q(x) is a sufficiently smooth function assuming that q′(0)=q(L)=0 and q(x*)≠0.


## 2. Main Results

### 2.1. Differential Case

To formulate our results, we introduce the Banach spaceC°α[0,L],   α∈(0,1), of all continuous functions ϕ(x) defined on [0,L] with ϕ′(0)=ϕ(L)=0 satisfying a Hölder condition for which the following norm is finite:
(2.1)‖ϕ‖C°α[0,L]=max⁡0≤x≤L|ϕ(x)|+sup⁡0≤x<x+h≤L|ϕ(x+h)-ϕ(x)|hα.In a Banach spaceE, with the help of a positive operator A we introduce the fractional spaces Eα,0<α<1, consisting of all v∈E for which the following norm is finite:
(2.2)‖v‖Eα=‖v‖E+sup⁡λ>0λ1-α‖Aexp⁡{-λA}v‖E.
Positive constants will be indicated by M which can be differ in time. On the other hand Mi(α,β,…) is used to focus on the fact that the constant depends only on α,β,…, and the subindex i is used to indicate a different constant.Theorem 2.1.
Letφ∈C°2α+2[0,L],  F1∈C([0,T],C°2α[0,L]), and ρ′∈C[0,T]. Then for the solution of problem (1.1), the following coercive stability estimates
(2.3)‖ut‖C([0,T],C°2α[0,L])+‖u‖C([0,T],C°2α+2[0,L])≤M(x*,q)‖ρ′‖C[0,T]+M(a,δ,σ,α,x*,q,T)×(‖φ‖C°2α+2[0,L]+‖F1‖C([0,T],C°2α[0,L])+‖ρ‖C[0,T]),‖p‖C[0,T]≤M(x*,q)‖ρ′‖C[0,T]+M(a,δ,σ,α,x*,q,T)[‖φ‖C°2α+2[0,L]+‖F1‖C([0,T],C°2α[0,L])+‖ρ‖C[0,T]]
hold.Proof.
Let us search for the solution of inverse problem (1.1) in the following form (see [23]):
(2.4)u(t,x)=η(t)q(x)+w(t,x),
where
(2.5)η(t)=∫0t‍p(s)ds.
Taking derivatives from (2.4) with respect to t and x, we get
(2.6)∂u(t,x)∂t=p(t)q(x)+∂w(t,x)∂t,∂2u(t,x)∂x2=η(t)d2q(x)dx2+∂2w(t,x)∂x2.
Moreover, substituting x by x* in (2.4), we obtain
(2.7)u(t,x*)=η(t)q(x*)+w(t,x*)=ρ(t),η(t)=ρ(t)-w(t,x*)q(x*).
Differentiating both sides of (2.7) with respect to t, we get
(2.8)p(t)=ρ′(t)-wt(t,x*)q(x*).
From identity (2.8) and the triangle inequality, it follows that
(2.9)|p(t)|=|ρ′(t)-wt(t,x*)q(x*)|≤M(x*,q)(|ρ′(t)|+|wt(t,x*)|)≤M(x*,q)(max⁡0≤t≤T⁡|ρ′(t)|+max⁡0≤t≤T⁡max⁡0≤x≤L|wt(t,x)|)≤M(x*,q)(max⁡0≤t≤T⁡|ρ′(t)|+max⁡0≤t≤T⁡‖wt(t)‖C°2α[0,L]),
for any t,t∈[0,T]. Using problem (1.1) and (2.4)–(2.7), one can show that w(t,x) is the solution of the following problem:
(2.10)∂w(t,x)∂t=a(x)∂2w(t,x)∂x2+a(x)ρ(t)-w(t,x*)q(x*)d2q(x)dx2+f(t,x),x∈(0,l),t∈(0,T],∂w(t,x)∂t=a(x)∂2w(t,x)∂x2+b(t,x)w(t,x)+ρ(t)-w(t,x*)q(x*)(a(x)d2q(x)dx2+b(t,x)q(x))+g(t,x),x∈(l,L),t∈(0,T],w(0,x)=φ(x),x∈[0,L],wx(t,0)=0,w(t,L)=0,t∈[0,T],w(t,l+)=w(t,l-),wx(t,l+)=wx(t,l-),t∈[0,T],
under the same assumptions on q(x). Estimate (2.9) and the following theorem conclude the proof of Theorem 2.1.Theorem 2.2.
For the solution of problem (2.10), the following coercive stability estimate
(2.11)‖wt‖C°2α[0,L]≤M(a,δ,σ,α,x*,q,T)×(‖φ‖C°2α+2[0,L]+‖F1‖C([0,T],C°2α[0,L])+‖ρ‖C[0,T])
holds.Proof.
Let us rewrite problem (2.10) in the abstract form as an initial-value problem:
(2.12)wt+Aw+Bw=(aq′′-σq)ρ(t)-w(t,x*)q*+F1(t)+F2(t),0<t≤T,w(0)=φ
in the Banach space E=C°[0,L]. Here, the positive operator A is defined by
(2.13)Au=-a(x)∂2u(t,x)∂x2+σu,
with
(2.14)D(A)={u(x):u,u′,u′′∈C[0,L],ux(0)=u(L)=0},
and for every fixed t∈[0,T], the differential operator B is given by the formula
(2.15)B(t)u={-σun,0≤x<l,-(σ-b(t))u,b(t)=b(t,x),l<x≤L.
Here, σ is a positive constant. The right-hand side functions are defined by
(2.16)F1(t)={f(t),0≤x<l,0,g(t)+ρ(t)q*b(t)q,l<x≤L,F2(t)={0,0≤x<l,0,-w(t,x*)q*b(t)q,l<x≤L,
where f(t)=f(t,x),g(t)=g(t,x),b(t)=b(t,x) are known, and w(t)=w(t,x) is unknown abstract functions defined on [0,T] with values in E=C°[0,L],    w(t,x*) is unknown scalar function defined on [0,T],   q=q(x),   q′′=q′′(x),   φ=φ(x), and  a=a(x) are elements of E=C°[0,L], and q*=q(x*) is a number.
It is known that operator-A generates an analytic semigroup exp{-tA}(t>0) and the following estimate holds:
(2.17)‖Aαexp⁡{-tA}‖E→E≤Me-δtt-α,0≤α≤1,
where t,δ,M>0 [29].
By the Cauchy formula, the solution can be written as(2.18)w(t)=e-tAφ-∫0te-(t-s)Aaq′′-σqq*w(s,x*)ds+∫0te-(t-s)Aρ(s)(aq′′-σq)q*ds+∫0te-(t-s)AF1(s)ds+∫0te-(t-s)AF2(s)ds-∫0te-(t-s)AB(s)w(s)ds.
Then, the following presentation of the solution of abstract problem (2.12) exists:
(2.19)Aw(t)=Ae-tAφ-∫0tAe-(t-s)Aaq′′-σqq*w(s,x*)ds+∫0tAe-(t-s)Aρ(s)(aq′′-σq)q*ds+∫0tAe-(t-s)AF1(s)ds+∫0tAe-(t-s)AF2(s)ds+∫0tAe-(t-s)AB(s)w(s)ds=∑k=16‍Gk(t).
Here,
(2.20)G1(t)=Ae-tAφ,G2(t)=-∫0tAe-(t-s)Aaq′′-σqq*‍w(s,x*)ds,G3(t)=-∫0tAe-(t-s)Aρ(s)(aq′′-σq)q*ds,G4(t)=∫0tAe-(t-s)AF1(s)ds,G5(t)=∫0tAe-(t-s)AF2(s)ds,G6(t)=∫0tAe-(t-s)AB(s)w(s)ds.
From the fact that the operators R,exp{-λA} and A commute, it follows that [29]
(2.21)‖R‖Eα→Eα≤‖R‖E→E.
Now, we estimate Gk(t) for k=1,2,…,5 separately. Applying the definition of norm of the spaces Eα and estimate (2.21), we get
(2.22)‖G1(t)‖Eα=‖Ae-tAφ‖Eα≤‖e-tA‖Eα→Eα‖Aφ‖Eα≤‖e-tA‖E→E‖Aφ‖Eα.
Then, using estimate (2.17) for α=0, we reach to
(2.23)‖G1(t)‖Eα≤M1‖Aφ‖Eα,
for any t,   t∈[0,T].
Let us estimateG2(t):
(2.24)‖G2(t)‖Eα=‖∫0tAe-(t-s)A‍aq′′-σqq*w(s,x*)ds‖≤∫0t‍‖Ae-(t-s)Aaq′′-σqq*‖Eα|w(s,x*)|ds.
By the definition of norm of the spaces Eα, we have that
(2.25)∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eαds=∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eds+sup⁡λ>0∫0t‖λ1-αAe-λAAe-(t-s)Aaq′′-σqq*‖Eds.
Let us estimate the first term. From the definition of norm of the spaces Eα it follows that
(2.26)∫0t‖Ae-(t-s)Aaq′′-σqq*‖E‍ds=∫0t(t-s)α-1‖(t-s)1-αAe-(t-s)Aaq′′-σqq*‖Eds≤∫0t(t-s)α-1ds‖aq′′-σqq*‖Eα≤Tαα‖aq′′-σqq*‖Eα=M2(a,σ,α,x*,q,T).
Using estimate (2.17), we obtain
(2.27)∫0t‖λ1-αAe-λAAe-(t-s)Aaq′′-σqq*‖Eds≤∫0t22-αλ1-α(λ+t-s)2-αds‖λ+t-s2Ae-((λ+t-s)/2)A‖E→E×‖(λ+t-s2)1-αAe-((λ+t-s)/2)Aaq′′-σqq*‖E≤M3(α)‖aq′′-σqq*‖Eα∫0tλ1-α(λ+t-s)2-αds≤M3(α)‖aq′′-σqq*‖Eα(λ1-α(1-α)(λ+t)1-α),
for any λ>0. From that it follows
(2.28)sup⁡λ>0∫0t‖λ1-αAe-λAAe-(t-s)Aaq′′-σqq*‖Eds≤M3(α)‖aq′′-σqq*‖Eα1(1-α)=M4(a,σ,α,x*,q).
Then, we get
(2.29)∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eαds≤M5(a,σ,α,x*,q,T)(2.30)‖G2(t)‖Eα≤M6(a,σ,α,x*,q,T)∫0t|w(s,x*)|ds.
Using definitions of norm of spaces E and Eα and estimate (2.21), we obtain that
(2.31)|w(s,x*)|≤‖w‖E≤‖w‖Eα=‖A-1Aw‖Eα≤‖A-1‖E→E‖Aw‖Eα≤M‖Aw‖Eα,(2.32)‖G2(t)‖Eα≤M7(a,σ,α,x*,q,T)‖Aw‖Eα,
for any t∈[0,T].
From estimate (2.29), the estimate of G3(t) is as follows:
(2.33)‖G3(t)‖Eα=‖∫0tAe-(t-s)Aρ(s)aq′′-σqq*ds‖Eα≤∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eαds‖ρ‖C[0,T]≤M8(a,σ,α,x*,q,T)‖ρ‖C[0,T].
Now, let us estimate G4(t). By the definition of the norm of the spaces Eα, we get
(2.34)‖G4(t)‖Eα=‖∫0tAe-(t-s)AF1(s)ds‖Eα=‖∫0tAe-(t-s)AF1(s)ds‖E+sup⁡λ>0λ1-α‖Ae-λA∫0tAe-(t-s)AF1(s)ds‖E.
Equation (2.2) yields that
(2.35)‖∫0tAe-(t-s)AF1(s)ds‖E=∫0t(t-s)α-1‖(t-s)1-αAe-(t-s)AF1(s)‖Eds≤∫0t(t-s)α-1ds‖F1‖C(Eα)=tαα‖F1‖C(Eα)≤M9(α,T)‖F1‖C(Eα).
Now, we consider the second term. Using (2.2), we get
(2.36)λ1-α‖Ae-λA∫0tAe-(t-s)AF1(s)ds‖E≤λ1-α∫0t(t-s+λ2)α-1(t-s+λ2)-1×‖t-s+λ2Ae-((t-s+λ)/2)A‖E→E×‖(t-s+λ2)1-αAe-((t-s+λ)/2)AF1(s)‖Eds≤M10λ1-α∫0t(t-s+λ2)α-2‖F1‖Eαds≤M11λ1-α∫0t(t-s+λ2)α-2ds‖F1‖C(Eα),
for any λ>0. Then,
(2.37)sup⁡λ>0λ1-α‖Ae-λA∫0tAe-(t-s)AF1(s)ds‖E≤M921-α1-α‖F1‖C(Eα)=M11(α)‖F1‖C(Eα).
Combining estimates (2.35) and (2.37), we obtain
(2.38)‖G4(t)‖Eα≤M12(α,T)‖F1‖C(Eα).
The estimate of G5(t) is as follows. Since operators A and e-tA commute, we can write that
(2.39)‖G5(t)‖Eα≤‖∫0tAe-(t-s)AF2(s)ds‖Eα≤M13(α,T)‖Aw‖Eα.
Let us estimate G6(t):(2.40)‖G6(t)‖Eα=‖∫0tAe-(t-s)AB(s)w(s)ds‖Eα=‖∫0tAe-(t-s)AB(s)A-1Aw(s)ds‖Eα≤∫0t‖Ae-(t-s)AB(s)A-1‖Eα→Eα‖Aw(s)‖Eαds.
Since
(2.41)‖e-tA‖Eα→Eα≤‖e-tA‖E→E≤Me-δt,‖AB(s)A-1‖Eα→Eα≤M,
we get
(2.42)‖G6(t)‖Eα≤M14∫0t‖Aw(s)‖Eαds.
Finally combining estimates (2.23), (2.32), (2.33), (2.38), (2.39), and (2.42), we get
(2.43)‖Aw‖Eα≤M1‖Aφ‖Eα+M8(a,σ,α,x*,q,T)‖ρ‖C[0,T]+M12(α,T)‖F1‖C(Eα)+M15∫0t‖Aw(s)‖Eαds,
where M15=M7+M13+M14.
Using Gronwall's inequality, we can write(2.44)‖Aw‖Eα≤eM15T[M1‖Aφ‖Eα+M8(a,σ,α,x*,q,T)‖ρ‖C[0,T]+M12(α,T)‖F1‖C(Eα)].
Applying the formulas
(2.45)w(t,x*)=w(0,x*)+∫0twz(z,x*)dz=φ(x*)+∫0twz(z,x*)dz,|φ(x*)|≤max⁡0≤x≤L|φ(x)|=‖φ‖E≤‖φ‖Eα≤‖A-1‖Eα→Eα‖Aφ‖Eα≤M‖Aφ‖Eα
and the triangle inequality, we can write
(2.46)‖(aq′′-σq)q*(ρ(t)-w(t,x*))‖Eα≤‖(aq′′-σq)q*‖Eα(‖ρ‖C[0,T]+M13‖Aφ‖Eα+∫0t‖wz‖Eαdz).
Using boundedness of B, problem (2.12), and estimate (2.46), we have
(2.47)‖wt‖Eα≤‖Aw‖Eα+‖Bw‖Eα+‖F1‖C(Eα)+‖(aq′′-σq)q*‖Eα(‖ρ‖C[0,T]+M13‖Aφ‖Eα+∫0t‖wz‖Eαdz).
So, Gronwall's inequality and the following theorem finish the proof of Theorem 2.3.Theorem 2.3 (see [29]).
For0<α<1/2, the spaces Eα(C[0,L],A) and C2α[0,L] coincide and their norms are equivalent.


### 2.2. Difference Case

For the approximate solution of problem (1.1), the Rothe difference scheme
(2.48)unk-unk-1τ=a(xn)un+1k-2unk+un-1kh2+pkqn+f(tk,xn),1≤k≤N,1≤n≤Ml-1,Mlh=l,Nτ=T,unk-unk-1τ=a(xn)un+1k-2unk+un-1kh2+b(tk,xn)unk+pkqn+g(tk,xn),1≤k≤N,Ml+1≤n≤M,Mh=L,Nτ=T,un0=φ(xn),0≤n≤M,u1k-u0k=uMk=0,0≤k≤N,uMl+1k-uMlk=uMlk-uMl-1k,0≤k≤N,u⌊x*/h⌋k=usk=ρ(tk),0≤k≤N,0≤s≤M,
where pk=p(tk),qn=q(xn),xn=nh, and tk=kτ is constructed. Here, qs≠0 and q1-q0=qM=0 are assumed. ⌊x⌋ represents the floor function of x.With the help of a positive operatorA, we introduce the fractional spaces Eα′,0<α<1, consisting of all v∈E for which the following norm is finite:
(2.49)‖v‖Eα′=‖v‖E+sup⁡λ>0λα‖A(λ+A)-1v‖E.
To formulate our results, we introduce the Banach space Ch°α=C°α[0,L]h,  α∈(0,1), of all grid functions ϕh={ϕn}n=1M-1 defined on
(2.50)[0,L]h={xn=nh,0≤n≤M,Mh=L},
with ϕ1-ϕ0=ϕM=0 equipped with the norm
(2.51)‖ϕh‖C°hα=‖ϕh‖Ch+sup⁡1≤n<n+r≤M|ϕn+r-ϕn|(rh)-α,‖ϕh‖Ch=max⁡1≤n≤M|ϕn|.
Moreover, Cτ(E)=C([0,T]τ,E) is the Banach space of all grid functions ϕτ={ϕ(tk)}k=1N-1 defined on [0,T]τ={tk=kτ,0≤k≤N,Nh=T} with values in E equipped with the norm
(2.52)‖ϕτ‖Cτ(E)=max⁡1≤k≤N‖ϕ(tk)‖E.
Then, the following theorem on well-posedness of problem (2.48) is established.Theorem 2.4.
For the solution of problem (2.48), the following coercive stability estimates
(2.53)‖{ukh-uk-1hτ}k=1N‖Cτ(C°h2α)+‖{Dh2ukh}k=1N‖Cτ(C°h2α)≤M(q,s)‖{ρ(tk)-ρ(tk-1)τ}k=1N‖C[0,T]τ+M(a~,ϕ,α,T)(‖Dh2φh‖C°h2α+‖{F1h(tk)}k=1N‖Cτ(C°h2α)+‖ρτ‖C[0,T]τ),‖pτ‖C[0,T]τ≤M(q,s)‖{ρ(tk)-ρ(tk-1)τ}k=1N‖C[0,T]τ+M(a~,ϕ,α,T)[‖Dh2φh‖C°h2α+‖{F1h(tk)}k=1N‖Cτ(C°h2α)+‖ρτ‖C[0,T]τ]
hold. Here,
(2.54)F1h(tk)={f(tk,xn)0b(tk,xn)ρ(tk)qsqn+g(tk,xn)}n=1M-1,φh={φ(xn)}n=1M-1,ρτ={ρ(tk)}k=0N,Dh2uh={un+1-2un+un-1h2}n=1M-1a~=1qs(aDh2qh-σqh).Proof.
The solution of problem (2.48) is searched in the following form:
(2.55)unk=ηkqn+wnk,
where
(2.56)ηk=∑i=1k‍piτ,1≤k≤N,η0=0.
Difference derivatives of (2.55) can be written as
(2.57)unk-unk-1τ=ηk-ηk-1τqn+wnk-wnk-1τ=pkqn+wnk-wnk-1τ,un+1k-2unk+un-1kh2=ηkqn+1-2qn+qn-1h2+wn+1k-2wnk+wn-1kh2,
for any n,1≤n≤M-1. At the interior grid point s=⌊x*/h⌋, we have that
(2.58)usk=ηkqs+wsk=ρ(tk),ηk=ρ(tk)-wskqs.
Taking the difference derivative of the last equality and using the triangle inequality, we obtain
(2.59)pk=1qs(ρ(tk)-ρ(tk-1)τ-wsk-wsk-1τ),(2.60)|pk|≤M(q,s)(|ρ(tk)-ρ(tk-1)τ|+|wsk-wsk-1τ|)≤M(q,s)(max⁡1≤k≤N|ρ(tk)-ρ(tk-1)τ|+max⁡1≤k≤Nmax⁡0≤s≤M|wsk-wsk-1τ|)≤M(q,s)(max⁡1≤k≤N|ρ(tk)-ρ(tk-1)τ|+max⁡1≤k≤N‖wkh-wk-1hτ‖C°h2α),
for any k,1≤k≤N.
In estimate (2.60), {wkh}k=0N is the solution of the following difference scheme:
(2.61)wnk-wnk-1τ=a(xn)wn+1k-2wnk+wn-1kh2+a(xn)ρ(tk)-wskqsqn+1-2qn+qn-1h2+f(tk,xn),1≤k≤N,1≤n≤Ml-1,Mlh=l,Nτ=T,wnk-wnk-1τ=a(xn)wn+1k-2wnk+wn-1kh2+a(xn)ρ(tk)-wskqsqn+1-2qn+qn-1h2b(tk,xn)wnk+b(tk,xn)ρ(tk)-wskqsqn+g(tk,xn),1≤k≤N,Ml+1≤n≤M-1,Mh=L,Nτ=T,wn0=φ(xn),0≤n≤M,w1k-w0k=wMk=0,0≤k≤N,wMl+1k-wMlk=wMlk-wMl-1k,0≤k≤N,
where xn=nh,tk=kτ. Therefore, estimate (2.60) and the following theorem finish the proof of Theorem 2.5.Theorem 2.5.
For the solution of problem (2.61), the following coercive stability estimate
(2.62)‖{wkh-wk-1hτ}k=1N‖Cτ(C°h2α)≤M(a~,ϕ,α,T)×(‖φh‖C°h2α+‖{F1h(tk)}k=1N‖Cτ(C°h2α)+‖ρτ‖C[0,T]τ)
holds.Proof.
We can rewrite difference scheme (2.61) in the abstract form:
(2.63)wkh-wk-1hτ+Ahxwkh+Bhxwkh=(aqn+1-2qn+qn-1h2-σq)ρ(tk)-wskqs+F1h(tk)+F2h(tk),tk=kτ,1≤k≤N,Nτ=T,w0h=φh,
in a Banach space E=C°[0,l]h with the positive operator Ahx defined by
(2.64)Ahxuh={-a(xn)un+1-2un+un-1h2+σu}n=1M-1,
acting on grid functions uh such that it satisfies the condition
(2.65)u1-u0=uM=0.
For every fixed t∈[0,T], the difference operators Bhx(t) are given by the formula
(2.66)Bhx(t)uh={-σun,1≤n≤Ml,-(σ-bn(t))un,bn(t)=b(t,xn),xn=nh,Ml+1≤n≤M-1,}n=1M-1,
where σ is a positive constant and the right-hand side functions are
(2.67)F1h(tk)={f(tk),1≤n≤Ml,b(tk)qnρ(tk)qs+g(tk),Ml+1≤n≤M-1,}n=1M-1,F2h(tk)={0,1≤n≤Ml,-b(tk)qnwskqs,Ml+1≤n≤M-1,}n=1M-1.
Let us denote R=(I+τAhx)-1. In problem (2.63), we have that
(2.68)wkh=Rwk-1h+Rτ(aqn+1-2qn+qn-1h2ρ(tk)-wskqs-Bhx(t)wkh+F1h(tk)+F2h(tk)),
for all k,1≤k≤N. By recurrence relations, we get
(2.69)wkh=Rkφh+∑m=1k‍Rk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm)-∑m=1k‍Rk-m+1τqsaqn+1-2qn+qn-1h2wsm-∑m=1k‍Rk-m+1Bhx(t)τwsm+∑m=1k‍Rk-m+1τF1h(tm)+∑m=1k‍Rk-m+1τF2h(tm).
Then, the following presentation of the solution of problem (2.63)
(2.70)Ahxwkh=AhxRkφh+∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm)-∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2wsm-∑m=1k‍AhxRk-m+1Bhx(t)τwsm+∑m=1k‍AhxRk-m+1τF1h(tm)+∑m=1k‍AhxRk-m+1τF2h(tm)=∑k=16‍Jk
is obtained. Here,
(2.71)J1k=AhxRkφh,J2k=∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm),J3k=-∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2wsm,J4k=-∑m=1k‍AhxRk-m+1Bhx(t)τwsm,J5k=∑m=1k‍AhxRk-m+1τF1h(tm),J6k=∑m=1k‍AhxRk-m+1τF2h(tm).
Now, let us estimateJrk for r=1,2,…,6 separately. We start with J1k. Applying the definition of norm of the spaces Eα′, we get
(2.72)‖J1k‖Eα′=‖RkAhxφh‖Eα′≤‖Rk‖Eα′→Eα′‖Ahxφh‖Eα′≤‖Rk‖E→E‖Ahxφh‖Eα′.
Using estimate(2.73)‖Rk‖E→E≤M,
we get
(2.74)‖J1k‖Eα′≤M1‖Ahxφh‖Eα′,
for any k,   1≤k≤N.
Let us estimateJ2k:(2.75)‖J2k‖Eα′=‖∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm)‖Eα′≤max⁡1≤m≤Nρ(tm)∑m=1k‍‖AhxRk-m+1τa~‖Eα′,
where
(2.76)a~=aqn+1-2qn+qn-1qsh2.
From the definition of norm of the spaces Eα′, it follows that
(2.77)‖∑m=1k‍AhxRk-m+1τa~‖Eα′≤∑m=1k‍‖AhxRk-m+1τa~‖E+sup⁡λ>0∑m=1k‍‖λαAhx(λ+Ahx)-1AhxRk-m+1τa~‖E.
Let us estimate each term separately. We divide first term into two parts:
(2.78)∑m=1k‍‖AhxRk-m+1τa~‖E=∑m=1k-1‍‖AhxRk-m+1τa~‖E+‖AhxRτa~‖E.
In the first part, by the definition of norm of the spaces Eα′ and the identity (see [29])
(2.79)(I+τA)-k=1(k-1)!∫0∞tk-1e-texp⁡{-τtA}dt,k≥2,
we deduce that
(2.80)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤∑m=1k-1‍τ(k-m)!∫0∞tk-m(τt)1-αe-t‖(τt)1-αAhxe-τtAa~‖Edt≤‖a~‖Eα′∑m=1k-1‍τ(k-m)!∫0∞‍tk-m(τt)1-αe-tdt=‖a~‖Eα′∑m=1k-1‍τα(k-m)!∫0∞‍tk-m-1+αe-tdt=‖a~‖Eα′∑m=1k-1‍τα(k-m)!∫0∞t(k-m-1)α+αe-αtt(k-m-1)(1-α)e-(1-α)tdt.
The Hölder inequality with p=1/α,q=1/(1-α) and the definition of the gamma function yield that
(2.81)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤‖a~‖Eα′∑m=1k-1‍τα(k-m)!(∫0∞(t(k-m-1)α+αe-αt)1/αdt)α×(∫0∞(t(k-m-1)(1-α)e-(1-α)t)1/(1-α)dt)1-α=‖a~‖Eα′∑m=1k-1‍τα(k-m)!(∫0∞tk-me-tdt)α(∫0∞tk-m-1e-tdt)1-α=‖a~‖Eα′∑m=1k-1‍τα(k-m)!(Γ(k-m+1))α(Γ(k-m))1-α.
By the fact that Γ(n)=(n-1)! and Γ(n)=(n-1)Γ(n-1), we get
(2.82)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤‖a~‖Eα′∑m=1k-1‍τα(k-m)!(k-m)αΓ(k-m)=‖a~‖Eα′∑m=1k-1‍τα(k-m)1-α=‖a~‖Eα′∑m=1k-1‍τ((k-m)τ)1-α≤M2‖a~‖Eα′∫0kτ‍1(kτ-s)1-αds=M2‖a~‖Eα′([-(kτ-s)αα]0kτ).
So, we have that
(2.83)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤M2‖a~‖Eα′(kτ)αα≤M3(α,T)‖a~‖Eα′.
In the second part, we have that
(2.84)‖AhxRτa~‖E≤‖AhxRτ‖E→E‖a~‖E≤M4‖a~‖Eα′.
Combining estimates (2.83) and (2.84), we obtain
(2.85)∑m=1k‍‖AhxRk-m+1τa~‖E≤M5(α,T)‖a~‖Eα′.
Let us estimate the second term. From the Cauchy-Riesz formula (see [29])
(2.86)f(A)=12πi∫Γf(z)(z-A)-1dz,
it follows that
(2.87)∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~=12πi∫S1∪S2∑m=1k‍z(1+z)k-m+1λαλ+zτ-1Ahx(z-τAhx)-1a~dz=12πi∫S1∪S2∑m=1k‍(z/τ)-α(1+z)k-m+1λαλτ+z(zτ)αAhx(zτ-Ahx)-1a~dz.
Since z=ρe±iϕ, with |ϕ|≤π/2, the estimate (see [29])
(2.88)‖(λ-A)-1‖E→E≤M(ϕ)1+|λ|
yields
(2.89)‖(zτ)αAhx(zτ-Ahx)-1a~‖E≤M6‖(ρτ)αAhx(ρτ+Ahx)-1a~‖E,1|λτ+z|≤M6λτ+ρ.
Hence,
(2.90)‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M6∫0∞∑m=1k‍ρ1-α[1+2ρcos⁡ϕ+ρ2](k-m+1)/2(λτ)αdρλτ+ρ‖a~‖Eα′.
Summing the geometric progression, we get
(2.91)‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M6∫0∞∑m=1k‍ρ1-α[1+2ρcos⁡ϕ+ρ2]1/2×(1-1[1+2ρcos⁡ϕ+ρ2]1/2)-1(λτ)αdρλτ+ρ‖a~‖Eα′≤M6∫0∞‍(λτ)αϰ(ρ)dρ(λτ+ρ)ρα‖a~‖Eα′.
Since the function
(2.92)ϰ(ρ)=ρ[1+2ρcos⁡ϕ+ρ2]1/2-1=1+[1+2ρcos⁡ϕ+ρ2]1/22cos⁡ϕ+ρ
does not increase for ρ≥0, we have ϰ(0)=1/cosϕ≥ϰ(ρ) for all ρ>0. Consequently,
(2.93)‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M6cos⁡ϕ∫0∞(λτ)αdρ(λτ+ρ)ρα‖a~‖Eα′
for any λ>0. Hence,
(2.94)sup⁡λ>0‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M7(ϕ,α)‖a~‖Eα′.
Then, using estimates (2.85) and (2.94), we get
(2.95)‖∑m=1k‍(Rk-m+1-Rk-m)a~‖Eα′≤M8(ϕ,α,T)‖a~‖Eα′,(2.96)‖J2k‖Eα′≤max⁡1≤m≤Nρ(tm)M8(ϕ,α,T)‖a~‖Eα′.
Now, let us estimate J3k:(2.97)‖J3k‖Eα′=‖-∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2wsm‖Eα′≤∑m=1k‍‖AhxRk-m+1τqs(aqn+1-2qn+qn-1h2-σq)‖Eα′|wsm|.
Since
(2.98)|wsm|≤max⁡0≤s≤M|wsm|=‖wh‖E≤‖wh‖Eα′≤‖(Ahx)-1‖Eα′→Eα′‖Ahxwh‖Eα′≤M‖Ahxwh‖Eα′,
and using estimate (2.95), we obtain
(2.99)‖J3‖Eα′≤M9(ϕ,α,T,τ)‖a~‖Eα′∑m=1k‍‖Ahxwh‖Eα′τ.J4k can be estimated as follows:
(2.100)‖J4k‖Eα′=‖-∑m=1k‍AhxRk-m+1Bhx(t)τwsm‖Eα′=‖∑m=1k‍AhxRk-m+1Bhx(t)(Ahx)-1Ahxτwsm‖Eα′≤∑m=1k‍‖AhxRk-m+1Bhx(t)(Ahx)-1‖Eα′→Eα′‖Ahxwsm‖Eα′τ.
From
(2.101)‖Rk‖Eα′→Eα′≤‖Rk‖E→E≤M,‖AhxBhx(t)(Ahx)-1‖Eα′→Eα′≤M
it follows that
(2.102)‖J4k‖Eα′≤M10∑m=1k‍‖Ahxwsm‖Eα′τ.
The estimations of J5k and J6k are as follows. By the definition of the norm of the spaces Eα′ and (2.95), we get
(2.103)‖J5k‖Eα′≤‖∑m=1kAhxRk-m+1τF1h(tm)‖Eα′+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′),(2.104)‖J6‖Eα′≤M12(ϕ,α)∑m=1k‍‖Ahxwh‖Eα′τ.
Combining estimates (2.74), (2.96), (2.99), and (2.102)–(2.104), we get
(2.105)‖Ahxwkh‖Eα′≤M1‖Ahxφh‖Eα′+max⁡1≤m≤Nρ(tm)M8(ϕ,α,T)‖a~‖Eα′+(M9(ϕ,α,T,τ)‖a~‖Eα′+M10+M12(ϕ,α))∑m=1k‍‖Ahxwkh‖Eατ+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′).
Using the discrete analogue of Gronwall's inequality, we get
(2.106)‖Ahxwkh‖Eα′≤eM13(a~,ϕ,α,T,τ)×[M1‖Ahxφh‖Eα′+M14(a~,ϕ,α,T)‖ρτ‖C[0,T]τ+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′)].
It follows from (2.63) and the triangle inequality that
(2.107)‖wkh-wk-1hτ‖Eα′≤eM12(a~,ϕ,α,T)×[M1‖Ahxφh‖Eα′+M13(a~,ϕ,α,T)‖ρτ‖C[0,T]τ+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′)],
for every k,   1≤k≤N. Then, we have that
(2.108)‖{wkh-wk-1hτ}k=1N‖Cτ(Eα′)≤M14(a~,ϕ,α,T)×(‖Ahxφh‖Eα′+‖{F1h(tk)}k=1N‖Cτ(Eα′)+‖ρτ‖C[0,T]τ).
The following theorem finishes the proof of Theorem2.6.Theorem 2.6 (see [30]).
For0<α<1/2, the spaces Eα′(C[0,L]h,Ahx) and C2α[0,L]h coincide and their norms are equivalent.


## 2.1. Differential Case

To formulate our results, we introduce the Banach spaceC°α[0,L],   α∈(0,1), of all continuous functions ϕ(x) defined on [0,L] with ϕ′(0)=ϕ(L)=0 satisfying a Hölder condition for which the following norm is finite:
(2.1)‖ϕ‖C°α[0,L]=max⁡0≤x≤L|ϕ(x)|+sup⁡0≤x<x+h≤L|ϕ(x+h)-ϕ(x)|hα.In a Banach spaceE, with the help of a positive operator A we introduce the fractional spaces Eα,0<α<1, consisting of all v∈E for which the following norm is finite:
(2.2)‖v‖Eα=‖v‖E+sup⁡λ>0λ1-α‖Aexp⁡{-λA}v‖E.
Positive constants will be indicated by M which can be differ in time. On the other hand Mi(α,β,…) is used to focus on the fact that the constant depends only on α,β,…, and the subindex i is used to indicate a different constant.Theorem 2.1.
Letφ∈C°2α+2[0,L],  F1∈C([0,T],C°2α[0,L]), and ρ′∈C[0,T]. Then for the solution of problem (1.1), the following coercive stability estimates
(2.3)‖ut‖C([0,T],C°2α[0,L])+‖u‖C([0,T],C°2α+2[0,L])≤M(x*,q)‖ρ′‖C[0,T]+M(a,δ,σ,α,x*,q,T)×(‖φ‖C°2α+2[0,L]+‖F1‖C([0,T],C°2α[0,L])+‖ρ‖C[0,T]),‖p‖C[0,T]≤M(x*,q)‖ρ′‖C[0,T]+M(a,δ,σ,α,x*,q,T)[‖φ‖C°2α+2[0,L]+‖F1‖C([0,T],C°2α[0,L])+‖ρ‖C[0,T]]
hold.Proof.
Let us search for the solution of inverse problem (1.1) in the following form (see [23]):
(2.4)u(t,x)=η(t)q(x)+w(t,x),
where
(2.5)η(t)=∫0t‍p(s)ds.
Taking derivatives from (2.4) with respect to t and x, we get
(2.6)∂u(t,x)∂t=p(t)q(x)+∂w(t,x)∂t,∂2u(t,x)∂x2=η(t)d2q(x)dx2+∂2w(t,x)∂x2.
Moreover, substituting x by x* in (2.4), we obtain
(2.7)u(t,x*)=η(t)q(x*)+w(t,x*)=ρ(t),η(t)=ρ(t)-w(t,x*)q(x*).
Differentiating both sides of (2.7) with respect to t, we get
(2.8)p(t)=ρ′(t)-wt(t,x*)q(x*).
From identity (2.8) and the triangle inequality, it follows that
(2.9)|p(t)|=|ρ′(t)-wt(t,x*)q(x*)|≤M(x*,q)(|ρ′(t)|+|wt(t,x*)|)≤M(x*,q)(max⁡0≤t≤T⁡|ρ′(t)|+max⁡0≤t≤T⁡max⁡0≤x≤L|wt(t,x)|)≤M(x*,q)(max⁡0≤t≤T⁡|ρ′(t)|+max⁡0≤t≤T⁡‖wt(t)‖C°2α[0,L]),
for any t,t∈[0,T]. Using problem (1.1) and (2.4)–(2.7), one can show that w(t,x) is the solution of the following problem:
(2.10)∂w(t,x)∂t=a(x)∂2w(t,x)∂x2+a(x)ρ(t)-w(t,x*)q(x*)d2q(x)dx2+f(t,x),x∈(0,l),t∈(0,T],∂w(t,x)∂t=a(x)∂2w(t,x)∂x2+b(t,x)w(t,x)+ρ(t)-w(t,x*)q(x*)(a(x)d2q(x)dx2+b(t,x)q(x))+g(t,x),x∈(l,L),t∈(0,T],w(0,x)=φ(x),x∈[0,L],wx(t,0)=0,w(t,L)=0,t∈[0,T],w(t,l+)=w(t,l-),wx(t,l+)=wx(t,l-),t∈[0,T],
under the same assumptions on q(x). Estimate (2.9) and the following theorem conclude the proof of Theorem 2.1.Theorem 2.2.
For the solution of problem (2.10), the following coercive stability estimate
(2.11)‖wt‖C°2α[0,L]≤M(a,δ,σ,α,x*,q,T)×(‖φ‖C°2α+2[0,L]+‖F1‖C([0,T],C°2α[0,L])+‖ρ‖C[0,T])
holds.Proof.
Let us rewrite problem (2.10) in the abstract form as an initial-value problem:
(2.12)wt+Aw+Bw=(aq′′-σq)ρ(t)-w(t,x*)q*+F1(t)+F2(t),0<t≤T,w(0)=φ
in the Banach space E=C°[0,L]. Here, the positive operator A is defined by
(2.13)Au=-a(x)∂2u(t,x)∂x2+σu,
with
(2.14)D(A)={u(x):u,u′,u′′∈C[0,L],ux(0)=u(L)=0},
and for every fixed t∈[0,T], the differential operator B is given by the formula
(2.15)B(t)u={-σun,0≤x<l,-(σ-b(t))u,b(t)=b(t,x),l<x≤L.
Here, σ is a positive constant. The right-hand side functions are defined by
(2.16)F1(t)={f(t),0≤x<l,0,g(t)+ρ(t)q*b(t)q,l<x≤L,F2(t)={0,0≤x<l,0,-w(t,x*)q*b(t)q,l<x≤L,
where f(t)=f(t,x),g(t)=g(t,x),b(t)=b(t,x) are known, and w(t)=w(t,x) is unknown abstract functions defined on [0,T] with values in E=C°[0,L],    w(t,x*) is unknown scalar function defined on [0,T],   q=q(x),   q′′=q′′(x),   φ=φ(x), and  a=a(x) are elements of E=C°[0,L], and q*=q(x*) is a number.
It is known that operator-A generates an analytic semigroup exp{-tA}(t>0) and the following estimate holds:
(2.17)‖Aαexp⁡{-tA}‖E→E≤Me-δtt-α,0≤α≤1,
where t,δ,M>0 [29].
By the Cauchy formula, the solution can be written as(2.18)w(t)=e-tAφ-∫0te-(t-s)Aaq′′-σqq*w(s,x*)ds+∫0te-(t-s)Aρ(s)(aq′′-σq)q*ds+∫0te-(t-s)AF1(s)ds+∫0te-(t-s)AF2(s)ds-∫0te-(t-s)AB(s)w(s)ds.
Then, the following presentation of the solution of abstract problem (2.12) exists:
(2.19)Aw(t)=Ae-tAφ-∫0tAe-(t-s)Aaq′′-σqq*w(s,x*)ds+∫0tAe-(t-s)Aρ(s)(aq′′-σq)q*ds+∫0tAe-(t-s)AF1(s)ds+∫0tAe-(t-s)AF2(s)ds+∫0tAe-(t-s)AB(s)w(s)ds=∑k=16‍Gk(t).
Here,
(2.20)G1(t)=Ae-tAφ,G2(t)=-∫0tAe-(t-s)Aaq′′-σqq*‍w(s,x*)ds,G3(t)=-∫0tAe-(t-s)Aρ(s)(aq′′-σq)q*ds,G4(t)=∫0tAe-(t-s)AF1(s)ds,G5(t)=∫0tAe-(t-s)AF2(s)ds,G6(t)=∫0tAe-(t-s)AB(s)w(s)ds.
From the fact that the operators R,exp{-λA} and A commute, it follows that [29]
(2.21)‖R‖Eα→Eα≤‖R‖E→E.
Now, we estimate Gk(t) for k=1,2,…,5 separately. Applying the definition of norm of the spaces Eα and estimate (2.21), we get
(2.22)‖G1(t)‖Eα=‖Ae-tAφ‖Eα≤‖e-tA‖Eα→Eα‖Aφ‖Eα≤‖e-tA‖E→E‖Aφ‖Eα.
Then, using estimate (2.17) for α=0, we reach to
(2.23)‖G1(t)‖Eα≤M1‖Aφ‖Eα,
for any t,   t∈[0,T].
Let us estimateG2(t):
(2.24)‖G2(t)‖Eα=‖∫0tAe-(t-s)A‍aq′′-σqq*w(s,x*)ds‖≤∫0t‍‖Ae-(t-s)Aaq′′-σqq*‖Eα|w(s,x*)|ds.
By the definition of norm of the spaces Eα, we have that
(2.25)∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eαds=∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eds+sup⁡λ>0∫0t‖λ1-αAe-λAAe-(t-s)Aaq′′-σqq*‖Eds.
Let us estimate the first term. From the definition of norm of the spaces Eα it follows that
(2.26)∫0t‖Ae-(t-s)Aaq′′-σqq*‖E‍ds=∫0t(t-s)α-1‖(t-s)1-αAe-(t-s)Aaq′′-σqq*‖Eds≤∫0t(t-s)α-1ds‖aq′′-σqq*‖Eα≤Tαα‖aq′′-σqq*‖Eα=M2(a,σ,α,x*,q,T).
Using estimate (2.17), we obtain
(2.27)∫0t‖λ1-αAe-λAAe-(t-s)Aaq′′-σqq*‖Eds≤∫0t22-αλ1-α(λ+t-s)2-αds‖λ+t-s2Ae-((λ+t-s)/2)A‖E→E×‖(λ+t-s2)1-αAe-((λ+t-s)/2)Aaq′′-σqq*‖E≤M3(α)‖aq′′-σqq*‖Eα∫0tλ1-α(λ+t-s)2-αds≤M3(α)‖aq′′-σqq*‖Eα(λ1-α(1-α)(λ+t)1-α),
for any λ>0. From that it follows
(2.28)sup⁡λ>0∫0t‖λ1-αAe-λAAe-(t-s)Aaq′′-σqq*‖Eds≤M3(α)‖aq′′-σqq*‖Eα1(1-α)=M4(a,σ,α,x*,q).
Then, we get
(2.29)∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eαds≤M5(a,σ,α,x*,q,T)(2.30)‖G2(t)‖Eα≤M6(a,σ,α,x*,q,T)∫0t|w(s,x*)|ds.
Using definitions of norm of spaces E and Eα and estimate (2.21), we obtain that
(2.31)|w(s,x*)|≤‖w‖E≤‖w‖Eα=‖A-1Aw‖Eα≤‖A-1‖E→E‖Aw‖Eα≤M‖Aw‖Eα,(2.32)‖G2(t)‖Eα≤M7(a,σ,α,x*,q,T)‖Aw‖Eα,
for any t∈[0,T].
From estimate (2.29), the estimate of G3(t) is as follows:
(2.33)‖G3(t)‖Eα=‖∫0tAe-(t-s)Aρ(s)aq′′-σqq*ds‖Eα≤∫0t‖Ae-(t-s)Aaq′′-σqq*‖Eαds‖ρ‖C[0,T]≤M8(a,σ,α,x*,q,T)‖ρ‖C[0,T].
Now, let us estimate G4(t). By the definition of the norm of the spaces Eα, we get
(2.34)‖G4(t)‖Eα=‖∫0tAe-(t-s)AF1(s)ds‖Eα=‖∫0tAe-(t-s)AF1(s)ds‖E+sup⁡λ>0λ1-α‖Ae-λA∫0tAe-(t-s)AF1(s)ds‖E.
Equation (2.2) yields that
(2.35)‖∫0tAe-(t-s)AF1(s)ds‖E=∫0t(t-s)α-1‖(t-s)1-αAe-(t-s)AF1(s)‖Eds≤∫0t(t-s)α-1ds‖F1‖C(Eα)=tαα‖F1‖C(Eα)≤M9(α,T)‖F1‖C(Eα).
Now, we consider the second term. Using (2.2), we get
(2.36)λ1-α‖Ae-λA∫0tAe-(t-s)AF1(s)ds‖E≤λ1-α∫0t(t-s+λ2)α-1(t-s+λ2)-1×‖t-s+λ2Ae-((t-s+λ)/2)A‖E→E×‖(t-s+λ2)1-αAe-((t-s+λ)/2)AF1(s)‖Eds≤M10λ1-α∫0t(t-s+λ2)α-2‖F1‖Eαds≤M11λ1-α∫0t(t-s+λ2)α-2ds‖F1‖C(Eα),
for any λ>0. Then,
(2.37)sup⁡λ>0λ1-α‖Ae-λA∫0tAe-(t-s)AF1(s)ds‖E≤M921-α1-α‖F1‖C(Eα)=M11(α)‖F1‖C(Eα).
Combining estimates (2.35) and (2.37), we obtain
(2.38)‖G4(t)‖Eα≤M12(α,T)‖F1‖C(Eα).
The estimate of G5(t) is as follows. Since operators A and e-tA commute, we can write that
(2.39)‖G5(t)‖Eα≤‖∫0tAe-(t-s)AF2(s)ds‖Eα≤M13(α,T)‖Aw‖Eα.
Let us estimate G6(t):(2.40)‖G6(t)‖Eα=‖∫0tAe-(t-s)AB(s)w(s)ds‖Eα=‖∫0tAe-(t-s)AB(s)A-1Aw(s)ds‖Eα≤∫0t‖Ae-(t-s)AB(s)A-1‖Eα→Eα‖Aw(s)‖Eαds.
Since
(2.41)‖e-tA‖Eα→Eα≤‖e-tA‖E→E≤Me-δt,‖AB(s)A-1‖Eα→Eα≤M,
we get
(2.42)‖G6(t)‖Eα≤M14∫0t‖Aw(s)‖Eαds.
Finally combining estimates (2.23), (2.32), (2.33), (2.38), (2.39), and (2.42), we get
(2.43)‖Aw‖Eα≤M1‖Aφ‖Eα+M8(a,σ,α,x*,q,T)‖ρ‖C[0,T]+M12(α,T)‖F1‖C(Eα)+M15∫0t‖Aw(s)‖Eαds,
where M15=M7+M13+M14.
Using Gronwall's inequality, we can write(2.44)‖Aw‖Eα≤eM15T[M1‖Aφ‖Eα+M8(a,σ,α,x*,q,T)‖ρ‖C[0,T]+M12(α,T)‖F1‖C(Eα)].
Applying the formulas
(2.45)w(t,x*)=w(0,x*)+∫0twz(z,x*)dz=φ(x*)+∫0twz(z,x*)dz,|φ(x*)|≤max⁡0≤x≤L|φ(x)|=‖φ‖E≤‖φ‖Eα≤‖A-1‖Eα→Eα‖Aφ‖Eα≤M‖Aφ‖Eα
and the triangle inequality, we can write
(2.46)‖(aq′′-σq)q*(ρ(t)-w(t,x*))‖Eα≤‖(aq′′-σq)q*‖Eα(‖ρ‖C[0,T]+M13‖Aφ‖Eα+∫0t‖wz‖Eαdz).
Using boundedness of B, problem (2.12), and estimate (2.46), we have
(2.47)‖wt‖Eα≤‖Aw‖Eα+‖Bw‖Eα+‖F1‖C(Eα)+‖(aq′′-σq)q*‖Eα(‖ρ‖C[0,T]+M13‖Aφ‖Eα+∫0t‖wz‖Eαdz).
So, Gronwall's inequality and the following theorem finish the proof of Theorem 2.3.Theorem 2.3 (see [29]).
For0<α<1/2, the spaces Eα(C[0,L],A) and C2α[0,L] coincide and their norms are equivalent.


## 2.2. Difference Case

For the approximate solution of problem (1.1), the Rothe difference scheme
(2.48)unk-unk-1τ=a(xn)un+1k-2unk+un-1kh2+pkqn+f(tk,xn),1≤k≤N,1≤n≤Ml-1,Mlh=l,Nτ=T,unk-unk-1τ=a(xn)un+1k-2unk+un-1kh2+b(tk,xn)unk+pkqn+g(tk,xn),1≤k≤N,Ml+1≤n≤M,Mh=L,Nτ=T,un0=φ(xn),0≤n≤M,u1k-u0k=uMk=0,0≤k≤N,uMl+1k-uMlk=uMlk-uMl-1k,0≤k≤N,u⌊x*/h⌋k=usk=ρ(tk),0≤k≤N,0≤s≤M,
where pk=p(tk),qn=q(xn),xn=nh, and tk=kτ is constructed. Here, qs≠0 and q1-q0=qM=0 are assumed. ⌊x⌋ represents the floor function of x.With the help of a positive operatorA, we introduce the fractional spaces Eα′,0<α<1, consisting of all v∈E for which the following norm is finite:
(2.49)‖v‖Eα′=‖v‖E+sup⁡λ>0λα‖A(λ+A)-1v‖E.
To formulate our results, we introduce the Banach space Ch°α=C°α[0,L]h,  α∈(0,1), of all grid functions ϕh={ϕn}n=1M-1 defined on
(2.50)[0,L]h={xn=nh,0≤n≤M,Mh=L},
with ϕ1-ϕ0=ϕM=0 equipped with the norm
(2.51)‖ϕh‖C°hα=‖ϕh‖Ch+sup⁡1≤n<n+r≤M|ϕn+r-ϕn|(rh)-α,‖ϕh‖Ch=max⁡1≤n≤M|ϕn|.
Moreover, Cτ(E)=C([0,T]τ,E) is the Banach space of all grid functions ϕτ={ϕ(tk)}k=1N-1 defined on [0,T]τ={tk=kτ,0≤k≤N,Nh=T} with values in E equipped with the norm
(2.52)‖ϕτ‖Cτ(E)=max⁡1≤k≤N‖ϕ(tk)‖E.
Then, the following theorem on well-posedness of problem (2.48) is established.Theorem 2.4.
For the solution of problem (2.48), the following coercive stability estimates
(2.53)‖{ukh-uk-1hτ}k=1N‖Cτ(C°h2α)+‖{Dh2ukh}k=1N‖Cτ(C°h2α)≤M(q,s)‖{ρ(tk)-ρ(tk-1)τ}k=1N‖C[0,T]τ+M(a~,ϕ,α,T)(‖Dh2φh‖C°h2α+‖{F1h(tk)}k=1N‖Cτ(C°h2α)+‖ρτ‖C[0,T]τ),‖pτ‖C[0,T]τ≤M(q,s)‖{ρ(tk)-ρ(tk-1)τ}k=1N‖C[0,T]τ+M(a~,ϕ,α,T)[‖Dh2φh‖C°h2α+‖{F1h(tk)}k=1N‖Cτ(C°h2α)+‖ρτ‖C[0,T]τ]
hold. Here,
(2.54)F1h(tk)={f(tk,xn)0b(tk,xn)ρ(tk)qsqn+g(tk,xn)}n=1M-1,φh={φ(xn)}n=1M-1,ρτ={ρ(tk)}k=0N,Dh2uh={un+1-2un+un-1h2}n=1M-1a~=1qs(aDh2qh-σqh).Proof.
The solution of problem (2.48) is searched in the following form:
(2.55)unk=ηkqn+wnk,
where
(2.56)ηk=∑i=1k‍piτ,1≤k≤N,η0=0.
Difference derivatives of (2.55) can be written as
(2.57)unk-unk-1τ=ηk-ηk-1τqn+wnk-wnk-1τ=pkqn+wnk-wnk-1τ,un+1k-2unk+un-1kh2=ηkqn+1-2qn+qn-1h2+wn+1k-2wnk+wn-1kh2,
for any n,1≤n≤M-1. At the interior grid point s=⌊x*/h⌋, we have that
(2.58)usk=ηkqs+wsk=ρ(tk),ηk=ρ(tk)-wskqs.
Taking the difference derivative of the last equality and using the triangle inequality, we obtain
(2.59)pk=1qs(ρ(tk)-ρ(tk-1)τ-wsk-wsk-1τ),(2.60)|pk|≤M(q,s)(|ρ(tk)-ρ(tk-1)τ|+|wsk-wsk-1τ|)≤M(q,s)(max⁡1≤k≤N|ρ(tk)-ρ(tk-1)τ|+max⁡1≤k≤Nmax⁡0≤s≤M|wsk-wsk-1τ|)≤M(q,s)(max⁡1≤k≤N|ρ(tk)-ρ(tk-1)τ|+max⁡1≤k≤N‖wkh-wk-1hτ‖C°h2α),
for any k,1≤k≤N.
In estimate (2.60), {wkh}k=0N is the solution of the following difference scheme:
(2.61)wnk-wnk-1τ=a(xn)wn+1k-2wnk+wn-1kh2+a(xn)ρ(tk)-wskqsqn+1-2qn+qn-1h2+f(tk,xn),1≤k≤N,1≤n≤Ml-1,Mlh=l,Nτ=T,wnk-wnk-1τ=a(xn)wn+1k-2wnk+wn-1kh2+a(xn)ρ(tk)-wskqsqn+1-2qn+qn-1h2b(tk,xn)wnk+b(tk,xn)ρ(tk)-wskqsqn+g(tk,xn),1≤k≤N,Ml+1≤n≤M-1,Mh=L,Nτ=T,wn0=φ(xn),0≤n≤M,w1k-w0k=wMk=0,0≤k≤N,wMl+1k-wMlk=wMlk-wMl-1k,0≤k≤N,
where xn=nh,tk=kτ. Therefore, estimate (2.60) and the following theorem finish the proof of Theorem 2.5.Theorem 2.5.
For the solution of problem (2.61), the following coercive stability estimate
(2.62)‖{wkh-wk-1hτ}k=1N‖Cτ(C°h2α)≤M(a~,ϕ,α,T)×(‖φh‖C°h2α+‖{F1h(tk)}k=1N‖Cτ(C°h2α)+‖ρτ‖C[0,T]τ)
holds.Proof.
We can rewrite difference scheme (2.61) in the abstract form:
(2.63)wkh-wk-1hτ+Ahxwkh+Bhxwkh=(aqn+1-2qn+qn-1h2-σq)ρ(tk)-wskqs+F1h(tk)+F2h(tk),tk=kτ,1≤k≤N,Nτ=T,w0h=φh,
in a Banach space E=C°[0,l]h with the positive operator Ahx defined by
(2.64)Ahxuh={-a(xn)un+1-2un+un-1h2+σu}n=1M-1,
acting on grid functions uh such that it satisfies the condition
(2.65)u1-u0=uM=0.
For every fixed t∈[0,T], the difference operators Bhx(t) are given by the formula
(2.66)Bhx(t)uh={-σun,1≤n≤Ml,-(σ-bn(t))un,bn(t)=b(t,xn),xn=nh,Ml+1≤n≤M-1,}n=1M-1,
where σ is a positive constant and the right-hand side functions are
(2.67)F1h(tk)={f(tk),1≤n≤Ml,b(tk)qnρ(tk)qs+g(tk),Ml+1≤n≤M-1,}n=1M-1,F2h(tk)={0,1≤n≤Ml,-b(tk)qnwskqs,Ml+1≤n≤M-1,}n=1M-1.
Let us denote R=(I+τAhx)-1. In problem (2.63), we have that
(2.68)wkh=Rwk-1h+Rτ(aqn+1-2qn+qn-1h2ρ(tk)-wskqs-Bhx(t)wkh+F1h(tk)+F2h(tk)),
for all k,1≤k≤N. By recurrence relations, we get
(2.69)wkh=Rkφh+∑m=1k‍Rk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm)-∑m=1k‍Rk-m+1τqsaqn+1-2qn+qn-1h2wsm-∑m=1k‍Rk-m+1Bhx(t)τwsm+∑m=1k‍Rk-m+1τF1h(tm)+∑m=1k‍Rk-m+1τF2h(tm).
Then, the following presentation of the solution of problem (2.63)
(2.70)Ahxwkh=AhxRkφh+∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm)-∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2wsm-∑m=1k‍AhxRk-m+1Bhx(t)τwsm+∑m=1k‍AhxRk-m+1τF1h(tm)+∑m=1k‍AhxRk-m+1τF2h(tm)=∑k=16‍Jk
is obtained. Here,
(2.71)J1k=AhxRkφh,J2k=∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm),J3k=-∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2wsm,J4k=-∑m=1k‍AhxRk-m+1Bhx(t)τwsm,J5k=∑m=1k‍AhxRk-m+1τF1h(tm),J6k=∑m=1k‍AhxRk-m+1τF2h(tm).
Now, let us estimateJrk for r=1,2,…,6 separately. We start with J1k. Applying the definition of norm of the spaces Eα′, we get
(2.72)‖J1k‖Eα′=‖RkAhxφh‖Eα′≤‖Rk‖Eα′→Eα′‖Ahxφh‖Eα′≤‖Rk‖E→E‖Ahxφh‖Eα′.
Using estimate(2.73)‖Rk‖E→E≤M,
we get
(2.74)‖J1k‖Eα′≤M1‖Ahxφh‖Eα′,
for any k,   1≤k≤N.
Let us estimateJ2k:(2.75)‖J2k‖Eα′=‖∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2ρ(tm)‖Eα′≤max⁡1≤m≤Nρ(tm)∑m=1k‍‖AhxRk-m+1τa~‖Eα′,
where
(2.76)a~=aqn+1-2qn+qn-1qsh2.
From the definition of norm of the spaces Eα′, it follows that
(2.77)‖∑m=1k‍AhxRk-m+1τa~‖Eα′≤∑m=1k‍‖AhxRk-m+1τa~‖E+sup⁡λ>0∑m=1k‍‖λαAhx(λ+Ahx)-1AhxRk-m+1τa~‖E.
Let us estimate each term separately. We divide first term into two parts:
(2.78)∑m=1k‍‖AhxRk-m+1τa~‖E=∑m=1k-1‍‖AhxRk-m+1τa~‖E+‖AhxRτa~‖E.
In the first part, by the definition of norm of the spaces Eα′ and the identity (see [29])
(2.79)(I+τA)-k=1(k-1)!∫0∞tk-1e-texp⁡{-τtA}dt,k≥2,
we deduce that
(2.80)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤∑m=1k-1‍τ(k-m)!∫0∞tk-m(τt)1-αe-t‖(τt)1-αAhxe-τtAa~‖Edt≤‖a~‖Eα′∑m=1k-1‍τ(k-m)!∫0∞‍tk-m(τt)1-αe-tdt=‖a~‖Eα′∑m=1k-1‍τα(k-m)!∫0∞‍tk-m-1+αe-tdt=‖a~‖Eα′∑m=1k-1‍τα(k-m)!∫0∞t(k-m-1)α+αe-αtt(k-m-1)(1-α)e-(1-α)tdt.
The Hölder inequality with p=1/α,q=1/(1-α) and the definition of the gamma function yield that
(2.81)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤‖a~‖Eα′∑m=1k-1‍τα(k-m)!(∫0∞(t(k-m-1)α+αe-αt)1/αdt)α×(∫0∞(t(k-m-1)(1-α)e-(1-α)t)1/(1-α)dt)1-α=‖a~‖Eα′∑m=1k-1‍τα(k-m)!(∫0∞tk-me-tdt)α(∫0∞tk-m-1e-tdt)1-α=‖a~‖Eα′∑m=1k-1‍τα(k-m)!(Γ(k-m+1))α(Γ(k-m))1-α.
By the fact that Γ(n)=(n-1)! and Γ(n)=(n-1)Γ(n-1), we get
(2.82)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤‖a~‖Eα′∑m=1k-1‍τα(k-m)!(k-m)αΓ(k-m)=‖a~‖Eα′∑m=1k-1‍τα(k-m)1-α=‖a~‖Eα′∑m=1k-1‍τ((k-m)τ)1-α≤M2‖a~‖Eα′∫0kτ‍1(kτ-s)1-αds=M2‖a~‖Eα′([-(kτ-s)αα]0kτ).
So, we have that
(2.83)∑m=1k-1‍‖AhxRk-m+1τa~‖E≤M2‖a~‖Eα′(kτ)αα≤M3(α,T)‖a~‖Eα′.
In the second part, we have that
(2.84)‖AhxRτa~‖E≤‖AhxRτ‖E→E‖a~‖E≤M4‖a~‖Eα′.
Combining estimates (2.83) and (2.84), we obtain
(2.85)∑m=1k‍‖AhxRk-m+1τa~‖E≤M5(α,T)‖a~‖Eα′.
Let us estimate the second term. From the Cauchy-Riesz formula (see [29])
(2.86)f(A)=12πi∫Γf(z)(z-A)-1dz,
it follows that
(2.87)∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~=12πi∫S1∪S2∑m=1k‍z(1+z)k-m+1λαλ+zτ-1Ahx(z-τAhx)-1a~dz=12πi∫S1∪S2∑m=1k‍(z/τ)-α(1+z)k-m+1λαλτ+z(zτ)αAhx(zτ-Ahx)-1a~dz.
Since z=ρe±iϕ, with |ϕ|≤π/2, the estimate (see [29])
(2.88)‖(λ-A)-1‖E→E≤M(ϕ)1+|λ|
yields
(2.89)‖(zτ)αAhx(zτ-Ahx)-1a~‖E≤M6‖(ρτ)αAhx(ρτ+Ahx)-1a~‖E,1|λτ+z|≤M6λτ+ρ.
Hence,
(2.90)‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M6∫0∞∑m=1k‍ρ1-α[1+2ρcos⁡ϕ+ρ2](k-m+1)/2(λτ)αdρλτ+ρ‖a~‖Eα′.
Summing the geometric progression, we get
(2.91)‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M6∫0∞∑m=1k‍ρ1-α[1+2ρcos⁡ϕ+ρ2]1/2×(1-1[1+2ρcos⁡ϕ+ρ2]1/2)-1(λτ)αdρλτ+ρ‖a~‖Eα′≤M6∫0∞‍(λτ)αϰ(ρ)dρ(λτ+ρ)ρα‖a~‖Eα′.
Since the function
(2.92)ϰ(ρ)=ρ[1+2ρcos⁡ϕ+ρ2]1/2-1=1+[1+2ρcos⁡ϕ+ρ2]1/22cos⁡ϕ+ρ
does not increase for ρ≥0, we have ϰ(0)=1/cosϕ≥ϰ(ρ) for all ρ>0. Consequently,
(2.93)‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M6cos⁡ϕ∫0∞(λτ)αdρ(λτ+ρ)ρα‖a~‖Eα′
for any λ>0. Hence,
(2.94)sup⁡λ>0‖∑m=1k‍λαAhx(λ+Ahx)-1Rk-m+1Ahxτa~‖E≤M7(ϕ,α)‖a~‖Eα′.
Then, using estimates (2.85) and (2.94), we get
(2.95)‖∑m=1k‍(Rk-m+1-Rk-m)a~‖Eα′≤M8(ϕ,α,T)‖a~‖Eα′,(2.96)‖J2k‖Eα′≤max⁡1≤m≤Nρ(tm)M8(ϕ,α,T)‖a~‖Eα′.
Now, let us estimate J3k:(2.97)‖J3k‖Eα′=‖-∑m=1k‍AhxRk-m+1τqsaqn+1-2qn+qn-1h2wsm‖Eα′≤∑m=1k‍‖AhxRk-m+1τqs(aqn+1-2qn+qn-1h2-σq)‖Eα′|wsm|.
Since
(2.98)|wsm|≤max⁡0≤s≤M|wsm|=‖wh‖E≤‖wh‖Eα′≤‖(Ahx)-1‖Eα′→Eα′‖Ahxwh‖Eα′≤M‖Ahxwh‖Eα′,
and using estimate (2.95), we obtain
(2.99)‖J3‖Eα′≤M9(ϕ,α,T,τ)‖a~‖Eα′∑m=1k‍‖Ahxwh‖Eα′τ.J4k can be estimated as follows:
(2.100)‖J4k‖Eα′=‖-∑m=1k‍AhxRk-m+1Bhx(t)τwsm‖Eα′=‖∑m=1k‍AhxRk-m+1Bhx(t)(Ahx)-1Ahxτwsm‖Eα′≤∑m=1k‍‖AhxRk-m+1Bhx(t)(Ahx)-1‖Eα′→Eα′‖Ahxwsm‖Eα′τ.
From
(2.101)‖Rk‖Eα′→Eα′≤‖Rk‖E→E≤M,‖AhxBhx(t)(Ahx)-1‖Eα′→Eα′≤M
it follows that
(2.102)‖J4k‖Eα′≤M10∑m=1k‍‖Ahxwsm‖Eα′τ.
The estimations of J5k and J6k are as follows. By the definition of the norm of the spaces Eα′ and (2.95), we get
(2.103)‖J5k‖Eα′≤‖∑m=1kAhxRk-m+1τF1h(tm)‖Eα′+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′),(2.104)‖J6‖Eα′≤M12(ϕ,α)∑m=1k‍‖Ahxwh‖Eα′τ.
Combining estimates (2.74), (2.96), (2.99), and (2.102)–(2.104), we get
(2.105)‖Ahxwkh‖Eα′≤M1‖Ahxφh‖Eα′+max⁡1≤m≤Nρ(tm)M8(ϕ,α,T)‖a~‖Eα′+(M9(ϕ,α,T,τ)‖a~‖Eα′+M10+M12(ϕ,α))∑m=1k‍‖Ahxwkh‖Eατ+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′).
Using the discrete analogue of Gronwall's inequality, we get
(2.106)‖Ahxwkh‖Eα′≤eM13(a~,ϕ,α,T,τ)×[M1‖Ahxφh‖Eα′+M14(a~,ϕ,α,T)‖ρτ‖C[0,T]τ+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′)].
It follows from (2.63) and the triangle inequality that
(2.107)‖wkh-wk-1hτ‖Eα′≤eM12(a~,ϕ,α,T)×[M1‖Ahxφh‖Eα′+M13(a~,ϕ,α,T)‖ρτ‖C[0,T]τ+M11(ϕ,α,T)‖{F1h(tk)}k=1N‖Cτ(Eα′)],
for every k,   1≤k≤N. Then, we have that
(2.108)‖{wkh-wk-1hτ}k=1N‖Cτ(Eα′)≤M14(a~,ϕ,α,T)×(‖Ahxφh‖Eα′+‖{F1h(tk)}k=1N‖Cτ(Eα′)+‖ρτ‖C[0,T]τ).
The following theorem finishes the proof of Theorem2.6.Theorem 2.6 (see [30]).
For0<α<1/2, the spaces Eα′(C[0,L]h,Ahx) and C2α[0,L]h coincide and their norms are equivalent.


## 3. Conclusion

Since artery disease caused by atherosclerosis is one of the most important causes of the death in the world, investigation of the effect of flow over the glycocalyx takes an important place. The flow equations can be formulated as an inverse problem. Here, our aim is to give more detailed understanding of the flow phenomena. Therefore, the well-posedness of the inverse problem of reconstructing the right side of a parabolic equation was investigated. Further, a new computer code regarding the flow analysis for the unknown pressure difference will be written.



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*Source: 548508-2012-07-12.xml*