# The Difference Problem of Obtaining the Parameter of a Parabolic Equation

Authors: Charyyar Ashyralyyev; Oznur Demirdag
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The boundary value problem of determining the parameterp of a parabolic equation υ′(t)+Aυ(t)=f(t)+p(0≤t≤1),υ(0)=φ,υ(1)=ψ in an arbitrary Banach space E with the strongly positive operator A is considered. The first order of accuracy stable difference scheme for the approximate solution of this problem
is investigated. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained.


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## Body

## 1. Introduction

The differential equations with parameters play a very important role in many branches of science and engineering. Some examples were given in temperature overspecification by Dehghan [1], chemistry (chromatography) by Kimura and Suzuki [2], physics (optical tomography) by Gryazin et al. [3].The differential equations with parameters have been studied extensively by many researchers (see, e.g., [4–20] and the references therein). However, such problems were not well investigated in general.As a result, considerable efforts have been expanded in formulating numerical solution methods that are both accurate and efficient. Methods of numerical solutions of parabolic problems with parameters have been studied by researchers (see, e.g., [21–29] and the references therein).It is known that various boundary value problems for parabolic equations with parameter can be reduced to the boundary value problem for the differential equation with parameterp:
(1.1)dv(t)dt+Av(t)=f(t)+p,0<t<1,v(0)=φ,v(1)=ψ
in an arbitrary Banach space E with the strongly positive operator A. In the present work, the first order of accuracy difference scheme for the approximate solution of boundary value problem (1.1) is studied. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained.


## 2. The Boundary Value Problem for Parabolic Equations

Throughout this work,E is a Banach space, -A is the generator of the analytic semigroup exp{-tA}(t≥0) with exponentially decreasing norm, when t→+∞, that is, the following estimates hold:
(2.1)‖exp⁡{-tA}‖E→E≤Me-δt,t‖Aexp⁡{-tA}‖E→E≤M,t>0,M>0,δ>0.
From estimate (2.1), it follows that
(2.2)‖T‖E→E≤M(δ).
Here, T=(I-exp{-A})-1.Abstract problem (1.1) was investigated in the paper [4] by applying estimates (2.1) and (2.2). The solvability of problem (1.1) in the space C(E) of the continuous E-valued functions φ(t) defined on [0,1] equipped with the norm
(2.3)‖φ‖C(E)=max⁡0≤t≤1‖φ(t)‖E
was studied under the necessary and sufficient conditions for the operator A. The solution depends continuously on the initial and boundary data.  More pricisely, we have the following result.Theorem 2.1.
Assume that-A is the generator of the analytic semigroup exp{-tA}(t≥0) and all points 2πik, k∈Z, k≠0 do not belong to the spectrum σ(A). Let v(0)∈E, v(1)∈D(A), and f(t)∈Cβ(E)(0<β≤1). Then, for the solution (v(t),p) of problem (1.1) in C(E)×E, the estimates
(2.4)‖p‖E≤M[‖v(0)‖E+‖v(1)‖E+‖Av(1)‖E+1β‖f‖Cβ(E)],‖v‖C(E)≤M[‖v(0)‖E+‖v(1)‖E+‖f‖C(E)]
hold, where M  does not depend on β, v(0), v(1) and f(t). Here Cβ(E) is the space obtained by completion of the space of all smooth E-valued functions φ(t) on [0,1] in the norm
(2.5)‖φ‖Cβ(E)=max⁡0≤t≤1‖φ(t)‖E+sup⁡0≤t<t+τ≤1‖φ(t+τ)-φ(t)‖Eτβ.
With the help ofA, we introduce the fractional space Eα(E,A), 0<α<1, consisting of all v∈E for which the following norms are finite [6, 30]:
(2.6)‖v‖α=sup⁡λ>0‖λ1-αAexp⁡{-λA}v‖E+‖v‖E.We say(v(t),p) is the solution of problem (1.1) in C0β,γ(E)×E1 if the following conditions are satisfied:(i)
v′(t),Av(t)∈C0β,γ(E), p∈E1⊂E,(ii)
(v(t),p) satisfies the equation and boundary conditions (1.1).Here,C0β,γ(E), (0≤γ≤β, 0<β<1) is the Hölder space with weight obtained by completion of the space of all smooth E-valued functions φ(t) on [0,1] in the norm
(2.7)‖φ‖C0β,γ(E)=max⁡0≤t≤1‖φ(t)‖E+sup⁡0≤t<t+τ≤1(t+τ)γ‖φ(t+τ)-φ(t)‖Eτβ.In the paper [23], the exact estimates in C0β,γ(E),(0≤γ≤β, 0<β<1) and C0β,γ(Eα-β)(0≤γ≤β≤α, 0<α<1) Hölder spaces for the solution of problem (1.1) were proved. In applications, exact estimates for the solution of the boundary value problems for parabolic equations were obtained.Now, we consider the application of Theorem2.1. First, the boundary-value problem on the range {0≤t≤1, x∈Rn} for the 2m-order multidimensional parabolic equation is considered:
(2.8)∂v(t,x)∂t+∑|r|=2m‍ar(x)∂|r|v(t,x)∂x1r1⋯∂xnrn+σv(t,x)=f(t,x)+p(x),0<t<1,v(0,x)=φ(x),v(1,x)=ψ(x),x∈Rn,|r|=r1+⋯+rn,
where ar(x) andf(t,x) are given as sufficiently smooth functions. Here, σ is a sufficiently large positive constant.It is assumed that the symbol(2.9)Bx(ξ)=∑|r|=2m‍ar(x)(iξ1)r1⋯(iξn)rn,ξ=(ξ1,…,ξn)∈Rn
of the differential operator of the form
(2.10)Bx=∑|r|=2m‍ar(x)∂|r|∂x1r1⋯∂xnrn
acting on functions defined on the space Rn satisfies the inequalities
(2.11)0<M1|ξ|2m≤(-1)mBx(ξ)≤M2|ξ|2m<∞
for ξ≠0.Problem (2.8) has a unique smooth solution. This allows us to reduce problem (2.8) to problem (1.1) in a Banach space E=Cμ(Rn) of all continuous bounded functions defined on ℝn satisfying a Hölder condition with the indicator μ∈(0,1).Theorem 2.2.
For the solution of boundary problem (2.8), the following estimates are satisfied:
(2.12)‖p‖Cμ(Rn)≤M[‖φ‖Cμ(Rn)+‖ψ‖C2m+μ(Rn)+1β‖f‖Cβ(Cμ(Rn))],‖v‖C(Cμ(Rn))≤M[‖φ‖Cμ(Rn)+‖ψ‖Cμ(Rn)+‖f‖C(Cμ(Rn))],
where M is independent of φ(x), ψ(x), and f(t,x).The proof of Theorem2.2 is based on the abstract Theorem 2.1 and on the strongly positivity of the operator A=Bx+σI defined by formula (2.10) (see, [31–33]).Second, letΩ be the unit open cube in the n-dimensional Euclidean space ℝn(0<xk<1,1≤k≤n) with boundary S,Ω¯=Ω∪S. In [0,1]×Ω, we consider the mixed boundary value problem for the multidimensional parabolic equation
(2.13)∂v(t,x)∂t-∑r=1n‍αr(x)∂2v(t,x)∂xr2+σv(t,x)=f(t,x)+p(x),x=(x1,…,xn)∈Ω,0<t<1,v(0,x)=φ(x),v(1,x)=ψ(x),x∈Ω¯,v(t,x)=0,x∈S,
where αr(x)(x∈Ω), φ(x), ψ(x)(Ω¯), and f(t,x)(t∈(0,1),x∈Ω) are given smooth functions and αr(x)≥a>0.Here, σ is a sufficiently large positive constant.We introduce the Banach spacesC01β(Ω¯)(β=(β1,…,βn), 0<xk<1, k=1,…,n) of all continuous functions satisfying a Hölder condition with the indicator β=(β1,...,βn), βk∈(0,1),1≤k≤n, and with weight xkβk(1-xk-hk)βk, 0≤xk<xk+hk≤1, 1≤k≤n which is equipped with the norm
(2.14)‖f‖C01β(Ω¯)=‖f‖C(Ω¯)+sup⁡0≤xk<xk+hk≤1,1≤k≤n|f(x1,…,xn)-f(x1+h1,…,xn+hn)|×∏k=1n‍hk-βkxkβk(1-xk-hk)βk,whereC(Ω¯) is the space of the all continuous functions defined on Ω¯, equipped with the norm
(2.15)‖f‖C(Ω¯)=max⁡x∈Ω¯|f(x)|.
It is known that the differential expression [34]
(2.16)Av(x)=-∑r=1n‍αr(x)∂2v(x)∂xr2+σv(x)
defines a positive operator A acting on C01β(Ω¯) with the domain D(A)={v(x),∂2v(x)/∂xr2∈C01β(Ω¯),v(x)=0onS}.Therefore, we can replace mixed problem (2.13) by the abstract boundary problem (1.1). Using the results of Theorem 2.1, we can obtain the following theorem on stability.Theorem 2.3.
For the solution of mixed boundary value problem (2.18), the following estimates are valid:
(2.17)‖p‖Cμ(Rn)≤M[‖φ‖C01μ(Ω¯)+‖ψ‖C0124μ(Ω¯)+1β‖f‖Cβ(C01μ(Ω¯))],‖v‖C(C01μ(Ω¯))≤M[‖φ‖C01μ(Ω¯)+‖ψ‖C01μ(Ω¯))+‖f‖C(C01μ(Ω¯)))],
where M does not depend on φ(x), ψ(x), and f(t,x).Third, we consider the mixed boundary value problem for parabolic equation(2.18)∂v(t,x)∂t-a(x)∂2v(t,x)∂x2+σv(t,x)=f(t,x)+p(x),0<t<1,0<x<1,v(0,x)=φ(x),v(1,x)=ψ(x),0≤x≤1,v(t,0)=v(t,1),vx(t,0)=vx(t,1),0≤t≤1,
where a(x), φ(x), ψ(x), and f(t,x) are given sufficiently smooth functions and a(x)≥a>0. Here, σ is a sufficiently large positive constant.We introduce the Banach spacesCβ[0,1](0<β<1) of all continuous functions φ(x) satisfying a Hölder condition for which the following norms are finite,
(2.19)‖φ‖Cβ[0,1]=‖φ‖C[0,1]+sup⁡0≤x<x+τ≤1|φ(x+τ)-φ(x)|τβ,
where C[0,1] is the space of the all continuous functions φ(x) defined on [0,1] with the usual norm
(2.20)‖φ‖C[0,1]=max⁡0≤x≤1|φ(x)|.
It is known that the differential expression [30]
(2.21)Av=-a(x)v′′(x)+σv(x)
defines a positive operator A acting in Cβ[0,1] with the domain
(2.22)D(A)={v(x),v′′(x)∈Cβ[0,1],v(0)=v(1),vx(0)=vx(1)}.Therefore, we can replace the mixed problem (2.18) by the abstract boundary value problem (1.1). Using the result of Theorem 2.1, we can obtain the following theorem on stability.Theorem 2.4.
For the solution of mixed problem (2.18), the following estimates are valid:
(2.23)‖p‖Cμ[0,1]≤M[‖φ‖Cμ[0,1]+‖ψ‖C2+μ[0,1]+1β‖f‖Cβ(Cμ[0,1])],‖v‖C(Cμ[0,1])≤M[‖φ‖Cμ[0,1]+‖ψ‖Cμ[0,1]+‖f‖C(Cμ[0,1])],
where M is independent of φ(x), ψ(x), and f(t,x).


## 3. Rothe Difference Scheme for Parabolic Equations with an Unknown Parameter

In this section, our focus is the well-posedness of the Rothe difference scheme(3.1)τ-1(uk-uk-1)+Auk=φk+p,φk=f(tk),tk=kτ,1≤k≤N,Nτ=1,u0=φ,uN=ψ
for approximately solving problem (1.1).Let[0,1]τ={tk=kτ, k=0,1,…,N, Nτ=1} be the uniform grid space with step size τ>0, where N is a fixed positive integer.Throughout the section,C([0,1]τ,E) denotes the linear space of grid functions φτ={φk}1N with values in the Banach space E.LetCτ(E)=C([0,1]τ,E) be the Banach space of bounded grid functions with the norm
(3.2)‖φτ‖Cτ(E)=max⁡1≤k≤N‖φk‖E.
For α∈[0,1], let Cα(E)=Cα([0,1]τ,E) be the Hölder space with the following norm:
(3.3)‖φτ‖Cα(E)=‖φτ‖Cτ(E)+max⁡1≤k<k+r≤N‖φk+r-φk‖E(rτ)α.Let us start with some lemmas we need in the following.Lemma 3.1 (see [31]).
The following estimates hold:(3.4)‖Rk‖E→E≤1(1+δτ)k,k≥1,‖τARk‖E→E≤1k,k≥1,
for some M,δ>0, which are independent of τ, where τ is a positive small number and R=(I+τA)-1 is the resolvent of A.Lemma 3.2.
The operatorI-RN has an inverse Tτ=(I-RN)-1and the following estimate is satisfied:
(3.5)‖Tτ‖E→E≤M(δ).Let us now obtain the formula for the solution of problem (3.1). It is clear that the first order of accuracy difference scheme
(3.6)τ-1(uk-uk-1)+Auk=p+φk,φk=f(tk),tk=kτ,1≤k≤N,Nτ=1,u0=φ,
has a solution and the following formula holds:
(3.7)uk=Rkφ+∑j=1k‍Rk-j+1(p+φj)τ,1≤k≤N.Applying formula (3.7) and the boundary condition
(3.8)uN=ψ,
we can write
(3.9)ψ=RNφ+∑j=1N‍RN-j+1φjτ+∑j=1N‍RN-j+1τp.
Since
(3.10)∑j=1N‍RN-j+1τ=A-1(I-R)∑j=1N‍RN-j=A-1(I-RN),
we have that
(3.11)ψ=RNφ+∑j=1N‍RN-j+1φjτ+A-1(I-RN)p.
Using Lemma 3.2, we get
(3.12)p=Tτ(Aψ-ARNφ-∑j=1N‍ARN-j+1φjτ).
Using formulas (3.7) and (3.12), we get
(3.13)uk=Rkφ+∑j=1k‍Rk-j+1φjτ+∑j=1k‍Rk-j+1τTτ(Aψ-ARNφ-∑j=1N‍ARN-j+1φjτ),1≤k≤N.
Since
(3.14)∑j=1k‍Rk-j+1τ=A-1(I-R)∑j=1k‍Rk-j=A-1(I-Rk),
we have that
(3.15)uk=Rkφ+∑j=1k‍Rk-j+1φjτ+(I-Rk)Tτ(ψ-RNφ-∑j=1N‍RN-j+1φjτ),1≤k≤N.
Hence, difference equation (3.1) is uniquely solvable, and, for the solution, formulas (3.12) and (3.15) are valid.Theorem 3.3.
For the solution({uk}k=1N,p) of problem (3.1) in Cτ(E)×E, the stability estimates
(3.16)‖p‖E≤M[‖φ‖E+‖Aψ‖E+1β‖{φk}k=1N‖Cτβ(E)],(3.17)‖{uk}k=1N‖Cτ(E)≤M[‖φ‖E+‖ψ‖E+‖{φk}k=1N‖Cτ(E)]
hold, where M is independent of τ, φ, ψ, and {φk}k=1N.Proof.
From formulas (3.7) and (3.12), it follows that
(3.18)p=Tτ(Aψ-ARNφ-∑j=1N-1‍ARN-j+1(φj-φN)τ-(I-RN)φN).
Using this formula, the triangle inequality, and estimates (3.4), we obtain
(3.19)‖p‖E≤‖Tτ‖E→E(∑j=1N-1‍‖Aψ‖H+‖ARN‖E→E‖φ‖E+∑j=1N-1‍‖ARN-j+1‖E→E‖φj-φN‖Eτ+(1+‖RN‖E→E)‖φN‖E)≤M[‖φ‖E+‖Aψ‖E+1β‖{φk}k=1N‖Cτβ(E)].The estimate (3.16) is proved. Using formula (3.15), the triangle inequality, and estimates (3.4), we obtain
(3.20)‖uk‖E≤(‖Rk‖E→E‖φ‖E+∑j=1k‍‖Rk-j+1‖E→E‖φj‖Eτ+(1+‖Rk‖E→E)‖Tτ‖E→E×(∥ψ∥E+‖RN‖E→E‖φ‖E+∑j=1N‍‖RN-j+1‖E→E‖φj‖Eτ))≤M[‖φ‖E+‖ψ‖E+‖{φk}k=1N‖Cτ(E)]
for any k. From that it follows estimate (3.17). Theorem 3.3 is proved.Theorem 3.4.
For the solution({uk}k=1N,p) of problem (3.1) in Cτ(E)×E, the almost coercive stability estimates
(3.21)‖p‖E≤M[‖φ‖E+‖Aψ‖E+min⁡{ln⁡1τ,|ln⁡‖A‖E→E|}‖{φk}k=1N‖Cτ(E)],(3.22)‖{τ-1(uk-uk-1)}k=1N‖Cτ(E)+‖{Auk}k=1N‖Cτ(E)≤M[‖Aφ‖E+‖Aψ‖E+min⁡{ln⁡1τ,|ln⁡‖A‖E→E|}‖{φk}k=1N‖Cτ(E)]
hold, where M  does not depend on τ, φ, ψ, and {φk}k=1N.Proof.
Using formula (3.12), the triangle inequality and estimates (3.4), we obtain
(3.23)‖p‖E≤‖Tτ‖E→E(‖Aψ‖H+‖RN‖E→E‖Aφ‖E+∑j=1N-1‍‖ARN-j+1‖E→E‖φj‖Eτ)≤M[‖Aφ‖E+‖Aψ‖E+∑j=1N-1‍‖ARN-j+1‖E→Eτ‖{φk}k=1N‖Cτ(E)].
Since [31]
(3.24)∑j=1N-1‍‖ARN-j+1‖E→Eτ≤Mmin⁡{ln⁡1τ,|ln⁡‖A‖E→E|},
we have estimate (3.21). Using formula (3.15), the triangle inequality, and estimates (3.4), (3.24), we obtain
(3.25)‖Auk‖E≤(‖Rk‖E→E‖Aφ‖E+∑j=1k‍‖ARk-j+1‖E→E‖φj‖Eτ+(1+‖Rk‖E→E)‖Tτ‖E→E×(‖Aψ‖E+‖RN‖E→E‖Aφ‖E+∑j=1N‍‖ARN-j+1‖E→E‖φj‖Eτ))≤M[‖Aφ‖E+‖Aψ‖E+min⁡{ln⁡1τ,|ln⁡‖A‖E→E|}‖{φk}k=1N‖Cτ(E)]
for any k. Therefore,
(3.26)‖{Auk}k=1N‖Cτ(E)≤M[‖Aφ‖E+‖Aψ‖E+min⁡{ln⁡1τ,|ln⁡‖A‖E→E|}‖{φk}k=1N‖Cτ(E)].
This estimate, triangle inequality, and (1.1) yield estimate (3.22). Theorem 3.4 is proved.


## 4. Applications

Now, we consider the applications of Theorems3.3 and 3.4. The boundary value problem (2.18) for the parabolic differential equation is considered. The discretization of problem (2.18) is carried out in two steps. In the first step, we define the grid space
(4.1)[0,1]h={x=xn:xn=nh,0≤n≤M,Mh=1}.
Let us introduce the Banach space Ch=C([0,1]h) of the grid functions φh(x)={φn}1M-1 defined on [0,1]h, equipped with the norm
(4.2)‖φh‖Ch=max⁡x∈[0,1]h|φh(x)|.
To the differential operator A generated by problem (2.18), we assign the difference operator Ahx by the formula
(4.3)Ahxφh(x)={-(a(x)φx¯)x,n+σφn}1M-1
acting in the space of grid functions φh(x)={φn}1M-1 satisfying the conditions φ0=φM, φ1-φ0=φM-φM-1. It is wellknown that Ahx is a strongly positive operator in Ch. With the help of Ahx, we arrive at the boundary value problem
(4.4)duh(t,x)dt+Ahxuh(t,x)=ph(x)+fh(t,x),0<t<1,x∈[0,1]h,uh(0,x)=φh(x),uh(1,x)=ψh(x),x∈[0,1]h.
In the second step, we replace (4.4) with the difference scheme (3.1)
(4.5)ukh(x)-uk-1h(x)τ+Ahxukh(x)=ph(x)+fkh(x),fkh(x)=fh(tk,x),tk=kτ,1≤k≤N,x∈[0,1]h,uh(0,x)=φh(x),uh(1,x)=ψh(x),x∈[0,1]h.Theorem 4.1.
The solution pairs({ukh(x)}0N,ph(x)) of problem (4.5) satisfy the stability estimates
(4.6)‖ph‖Ch≤M1[‖φh‖Ch+‖ψh‖Ch+‖Ahxψh‖Ch+1β‖{fkh}1N‖Cτβ(Ch)],‖{ukh}1N‖Cτ(Ch)≤M2[‖φh‖Ch+‖ψh‖Ch+‖{fkh}1N‖Cτ(Ch)],
where M1 and M2 do not depend on β,φh, ψh, and fkh, 1≤k≤N.
Here,Cτβ(Ch) is the grid space of grid functions {fkh}1N defined on [0,1]τ×[0,1]h with norm
(4.7)‖{fkh}1N‖Cτβ(Ch)=‖{fkh}1N‖Cτ(Ch)+sup⁡1≤k<k+r≤N‖fk+rh-fkh‖L2h(rτ)β,‖{fkh}1N‖Cτ(Ch)=max⁡1≤k≤N‖fkh‖Ch.The proof of Theorem4.1 is based on Theorem 3.3 and the positivity property of the operator Ahx defined by formula (4.3).Theorem 4.2.
The solution pairs({ukh(x)}0N,ph(x)) of problem (4.5) satisfy the almost coercive stability estimates
(4.8)‖ph‖Ch≤M1[‖Ahxφh‖Ch+‖Ahxψh‖Ch+ln⁡1τ+h‖{fkh}1N‖Cτ(Ch)],‖{ukh-uk-1hτ}1N‖Cτ(Ch)+‖{Ahxukh}1N‖Cτ(Ch)≤M2[‖Ahxφh‖Ch+‖Ahxψh‖Ch+ln⁡1τ+h‖{fkh}1N‖Cτ(Ch)],
where M1 and M2 are independent of φh, ψh, and fkh, 1≤k≤N.The proof of Theorem4.2 is based on Theorem 3.4 and the positivity property of the operator Ahx defined by formula (4.3) and on the estimate
(4.9)min⁡{ln⁡1τ,|ln⁡‖Ahx‖Ch→Ch|}≤Mln⁡1τ+h.Note that, in a similar manner, we can construct the difference schemes of the first order of accuracy with respect to one variable for approximate solutions of boundary value problems (2.8) and (2.13). Abstract theorems given from above permit us to obtain the stability, the almost stability estimates for the solutions of these difference schemes.


## 5. Conclusion

In this work, the first order of accuracy Rothe difference scheme for the approximate solution of the boundary value problem of determining the parameterp of a parabolic equation
(5.1)v′(t)+Av(t)=f(t)+p(0≤t≤1),v(0)=φ,v(1)=ψ
in arbitrary Banach space E with the strongly positive operator A is studied. The well-posedness of the difference scheme is established. Some results in this paper in Hilbert space H with self adjoint positive definite operator A were obtained in the paper [25]. The investigation of this paper in arbitrary Banach space E with the strongly positive operator A permits us to obtain the stability and almost stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained. Of course, such type results for the solution of difference scheme for the following boundary value problems
(5.2)v′(t)+Av(t)=f(t)+p(0≤t≤1),v(0)=φ,v(λ)=ψ,0<λ≤1,v′(t)-Av(t)=f(t)+p(0≤t≤1),v(1)=φ,v(λ)=ψ,0≤λ<1
in an arbitrary Banach space with positive operator A and an unknown parameter p hold.



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