The Difference Problem of Obtaining the Parameter of a Parabolic Equation

Abstract

The boundary value problem of determining the parameter 𝑝 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.

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 parameter 𝑝: 𝑑𝑣(𝑡)(𝑡)(𝑡)𝑣(0)(1)𝑑𝑡+𝐴𝑣=𝑓+𝑝,0<𝑡<1,=𝜑,𝑣=𝜓 (1.1) 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: exp{−𝑡𝐴}_𝐸→𝐸≤𝑀𝑒^−𝛿𝑡,𝑡𝐴exp{−𝑡𝐴}_𝐸→𝐸≤𝑀,𝑡>0,𝑀>0,𝛿>0. (2.1) From estimate (2.1), it follows that ‖𝑇‖_𝐸→𝐸≤𝑀(𝛿). (2.2) 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 𝜑_𝐶(𝐸)=max_0≤𝑡≤1𝜑(𝑡)_𝐸 (2.3) 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)∈𝐶^𝛽(𝐸)(0<𝛽≤1). Then, for the solution (𝑣(𝑡),𝑝) of problem (1.1) in 𝐶(𝐸)×𝐸, the estimates 𝑝_𝐸≤𝑀‖𝑣(0)‖_𝐸+‖𝑣(1)‖_𝐸+‖𝐴𝑣(1)‖_𝐸+1𝛽𝑓_{𝐶𝛽(𝐸)},‖𝑣‖_𝐶(𝐸)≤𝑀‖𝑣(0)‖_𝐸+‖𝑣(1)‖_𝐸+𝑓_𝐶(𝐸) (2.4) hold, where M does not depend on 𝛽, 𝑣(0), 𝑣(1) and 𝑓(𝑡). Here 𝐶^𝛽(𝐸) is the space obtained by completion of the space of all smooth 𝐸-valued functions 𝜑(𝑡) on [0,1] in the norm 𝜑_{𝐶𝛽(𝐸)}=max_0≤𝑡≤1𝜑(𝑡)_𝐸+sup_{0≤𝑡<𝑡+𝜏≤1}𝜑(𝑡+𝜏)−𝜑(𝑡)_𝐸𝜏^𝛽. (2.5)
With the help of 𝐴, we introduce the fractional space 𝐸_𝛼(𝐸,𝐴), 0<𝛼<1, consisting of all 𝑣∈𝐸 for which the following norms are finite [6, 30]: ‖𝑣‖_𝛼=sup_𝜆>0𝜆^1−𝛼𝐴exp{−𝜆𝐴}𝑣_𝐸+‖𝑣‖_𝐸. (2.6)

We say (𝑣(𝑡),𝑝) is the solution of problem (1.1) in 𝐶₀^𝛽,𝛾(𝐸)×𝐸₁ if the following conditions are satisfied:(i)𝑣^′(𝑡),𝐴𝑣(𝑡)∈𝐶₀^𝛽,𝛾(𝐸), 𝑝∈𝐸₁⊂𝐸,(ii)(𝑣(𝑡),𝑝) satisfies the equation and boundary conditions (1.1).

Here, 𝐶₀^𝛽,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 𝜑_{𝐶0𝛽,𝛾(𝐸)}=max_0≤𝑡≤1𝜑(𝑡)_𝐸+sup_{0≤𝑡<𝑡+𝜏≤1}(𝑡+𝜏)^𝛾𝜑(𝑡+𝜏)−𝜑(𝑡)_𝐸𝜏^𝛽. (2.7)

In the paper [23], the exact estimates in 𝐶₀^𝛽,𝛾(𝐸),(0≤𝛾≤𝛽, 0<𝛽<1) and 𝐶₀^𝛽,𝛾(𝐸_𝛼−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 Theorem 2.1. First, the boundary-value problem on the range {0≤𝑡≤1, 𝑥∈R^𝑛} for the 2𝑚-order multidimensional parabolic equation is considered: 𝜕𝑣(𝑡,𝑥)𝜕𝑡+_|𝑟|=2𝑚𝑎_𝑟(𝑥)𝜕^|𝑟|𝑣(𝑡,𝑥)𝜕𝑥^𝑟1₁⋯𝜕𝑥^𝑟𝑛_𝑛()()(𝑥)𝑣()(𝑥)()(𝑥)+𝜎𝑣𝑡,𝑥=𝑓𝑡,𝑥+𝑝,0<𝑡<1,0,𝑥=𝜑,𝑣1,𝑥=𝜓,𝑥∈R^𝑛,|𝑟|=𝑟₁+⋯+𝑟_𝑛, (2.8) where 𝑎_𝑟(𝑥) and𝑓(𝑡,𝑥) are given as sufficiently smooth functions. Here, 𝜎 is a sufficiently large positive constant.

It is assumed that the symbol 𝐵^𝑥(𝜉)=_|𝑟|=2𝑚𝑎_𝑟(𝑥)𝑖𝜉₁^𝑟1⋯𝑖𝜉_𝑛^𝑟𝑛,𝜉=𝜉₁,…,𝜉_𝑛∈R^𝑛 (2.9) of the differential operator of the form 𝐵^𝑥=_|𝑟|=2𝑚𝑎_𝑟(𝑥)𝜕^|𝑟|𝜕𝑥^𝑟1₁⋯𝜕𝑥^𝑟𝑛_𝑛 (2.10) acting on functions defined on the space R^𝑛 satisfies the inequalities 0<𝑀₁𝜉^2𝑚≤(−1)^𝑚𝐵^𝑥(𝜉)≤𝑀₂𝜉^2𝑚<∞ (2.11) 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=𝐶^𝜇(𝑅^𝑛) of all continuous bounded functions defined on ℝ^𝑛 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: 𝑝_{𝐶𝜇(𝑅𝑛)}≤𝑀𝜑_{𝐶𝜇(𝑅𝑛)}+𝜓_{𝐶2𝑚+𝜇(𝑅𝑛)}+1𝛽𝑓_{𝐶𝛽(𝐶𝜇(𝑅𝑛))},‖𝑣‖_{𝐶(𝐶𝜇(𝑅𝑛))}≤𝑀𝜑_{𝐶𝜇(𝑅𝑛)}+𝜓_{𝐶𝜇(𝑅𝑛)}+𝑓_{𝐶(𝐶𝜇(𝑅𝑛))}, (2.12) where 𝑀 is independent of 𝜑(𝑥), 𝜓(𝑥), and 𝑓(𝑡,𝑥).

The proof of Theorem 2.2 is based on the abstract Theorem 2.1 and on the strongly positivity of the operator 𝐴=𝐵^𝑥+𝜎𝐼 defined by formula (2.10) (see, [31–33]).

Second, let Ω be the unit open cube in the 𝑛-dimensional Euclidean space ℝ^𝑛(0<𝑥_𝑘<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 𝜕𝑣(𝑡,𝑥)−𝜕𝑡^𝑛_𝑟=1𝛼_𝑟(𝑥)𝜕²𝑣(𝑡,𝑥)𝜕𝑥²_𝑟()()(𝑥),𝑥+𝜎𝑣𝑡,𝑥=𝑓𝑡,𝑥+𝑝𝑥=₁,…,𝑥_𝑛𝑣()(𝑥)()(𝑥)∈Ω,0<𝑡<1,0,𝑥=𝜑,𝑣1,𝑥=𝜓,𝑥∈𝑣()Ω,𝑡,𝑥=0,𝑥∈𝑆, (2.13) where 𝛼_𝑟(𝑥)(𝑥∈Ω), 𝜑(𝑥), 𝜓(𝑥)(Ω), and 𝑓(𝑡,𝑥)(𝑡∈(0,1),𝑥∈Ω) are given smooth functions and 𝛼_𝑟(𝑥)≥𝑎>0.Here, 𝜎 is a sufficiently large positive constant.

We introduce the Banach spaces 𝐶^𝛽₀₁(Ω)(𝛽=(𝛽₁,…,𝛽_𝑛), 0<𝑥_𝑘<1, 𝑘=1,…,𝑛) of all continuous functions satisfying a Hölder condition with the indicator 𝛽=(𝛽₁,...,𝛽_𝑛), 𝛽_𝑘∈(0,1),1≤𝑘≤𝑛, and with weight 𝑥^𝛽𝑘_𝑘(1−𝑥_𝑘−ℎ_𝑘)^𝛽𝑘, 0≤𝑥_𝑘<𝑥_𝑘+ℎ_𝑘≤1, 1≤𝑘≤𝑛 which is equipped with the norm 𝑓_{𝐶𝛽01(Ω)}=𝑓_𝐶(Ω)+sup_{0≤𝑥𝑘<𝑥𝑘+ℎ𝑘≤1,1≤𝑘≤𝑛}𝑓𝑥₁,…,𝑥_𝑛−𝑓𝑥₁+ℎ₁,…,𝑥_𝑛+ℎ_𝑛×^𝑛_𝑘=1ℎ^−𝛽𝑘_𝑘𝑥^𝛽𝑘_𝑘1−𝑥_𝑘−ℎ_𝑘^𝛽𝑘, (2.14)

where 𝐶(Ω) is the space of the all continuous functions defined on Ω, equipped with the norm 𝑓_𝐶(Ω)=max_𝑥∈Ω𝑓(𝑥). (2.15) It is known that the differential expression [34] 𝐴𝑣(𝑥)=−^𝑛_𝑟=1𝛼_𝑟(𝑥)𝜕²𝑣(𝑥)𝜕𝑥²_𝑟+𝜎𝑣(𝑥) (2.16) defines a positive operator 𝐴 acting on 𝐶^𝛽₀₁(Ω) with the domain 𝐷(𝐴)={𝑣(𝑥),𝜕²𝑣(𝑥)/𝜕𝑥²_𝑟∈𝐶^𝛽₀₁(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: 𝑝_{𝐶𝜇(𝑅𝑛)}≤𝑀𝜑_𝐶^𝜇_01(Ω)+𝜓_𝐶^24𝜇_01(Ω)+1𝛽𝑓_{𝐶𝛽(𝐶}^𝜇_01(Ω)),‖𝑣‖_𝐶(𝐶^𝜇_01(Ω))≤𝑀𝜑_𝐶^𝜇_01(Ω)+𝜓_𝐶^𝜇_01(Ω))+𝑓_𝐶(𝐶^𝜇_01(Ω))), (2.17) 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 𝜕𝑣(𝑡,𝑥)(𝑥)𝜕𝜕𝑡−𝑎²𝑣(𝑡,𝑥)𝜕𝑥²()()(𝑥)𝑣()(𝑥)()(𝑥)𝑣()()+𝜎𝑣𝑡,𝑥=𝑓𝑡,𝑥+𝑝,0<𝑡<1,0<𝑥<1,0,𝑥=𝜑,𝑣1,𝑥=𝜓,0≤𝑥≤1,𝑡,0=𝑣𝑡,1,𝑣_𝑥(𝑡,0)=𝑣_𝑥(𝑡,1),0≤𝑡≤1, (2.18) 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 spaces 𝐶^𝛽[0,1](0<𝛽<1) of all continuous functions 𝜑(𝑥) satisfying a Hölder condition for which the following norms are finite, 𝜑_{𝐶𝛽[0,1]}=𝜑_𝐶[0,1]+sup_{0≤𝑥<𝑥+𝜏≤1}𝜑(𝑥+𝜏)−𝜑(𝑥)𝜏^𝛽, (2.19) where 𝐶[0,1] is the space of the all continuous functions 𝜑(𝑥) defined on [0,1] with the usual norm 𝜑_C[0,1]=max_0≤𝑥≤1𝜑(𝑥). (2.20) It is known that the differential expression [30] 𝐴𝑣=−𝑎(𝑥)𝑣^′′(𝑥)+𝜎𝑣(𝑥) (2.21) defines a positive operator 𝐴 acting in 𝐶^𝛽[0,1] with the domain 𝐷(𝐴)=𝑣(𝑥),𝑣^′′(𝑥)∈𝐶^𝛽[0,1],𝑣(0)=𝑣(1),𝑣_𝑥(0)=𝑣_𝑥(1). (2.22)

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: 𝑝_{𝐶𝜇[0,1]}≤𝑀𝜑_{𝐶𝜇[0,1]}+𝜓_{𝐶2+𝜇[0,1]}+1𝛽𝑓_{𝐶𝛽(𝐶𝜇[0,1])},‖𝑣‖_{𝐶(𝐶𝜇[0,1])}≤𝑀𝜑_{𝐶𝜇[0,1]}+𝜓_{𝐶𝜇[0,1]}+𝑓_{𝐶(𝐶𝜇[0,1])}, (2.23) 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 𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1+𝐴𝑢_𝑘=𝜑_𝑘+𝑝,𝜑_𝑘=𝑓𝑡_𝑘,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁,𝑁𝜏=1,₀=𝜑,𝑢_𝑁=𝜓 (3.1) for approximately solving problem (1.1).

Let [0,1]_𝜏={𝑡_𝑘=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 𝜑^𝜏={𝜑_𝑘}^𝑁₁ with values in the Banach space 𝐸.

Let 𝐶_𝜏(𝐸)=𝐶([0,1]_𝜏,𝐸) be the Banach space of bounded grid functions with the norm 𝜑^𝜏_{𝐶𝜏(𝐸)}=max_{1≤𝑘≤𝑁}𝜑_𝑘𝐸. (3.2) For 𝛼∈[0,1], let 𝐶^𝛼(𝐸)=𝐶^𝛼([0,1]_𝜏,𝐸) be the Hölder space with the following norm: 𝜑^𝜏_{𝐶𝛼(𝐸)}=𝜑^𝜏_{𝐶𝜏(𝐸)}+max_{1≤𝑘<𝑘+𝑟≤𝑁}𝜑_𝑘+𝑟−𝜑_𝑘𝐸(𝑟𝜏)^𝛼. (3.3)

Let us start with some lemmas we need in the following.

Lemma 3.1 (see [31]). The following estimates hold: 𝑅^𝑘_𝐸→𝐸≤1(1+𝛿𝜏)^𝑘,𝑘≥1,𝜏AR^𝑘_𝐸→𝐸≤1𝑘,𝑘≥1, (3.4) 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 operator 𝐼−𝑅^𝑁 has an inverse 𝑇_𝜏=(𝐼−𝑅^𝑁)⁻¹and the following estimate is satisfied: 𝑇_{𝜏𝐸→𝐸}≤𝑀(𝛿). (3.5)

Let us now obtain the formula for the solution of problem (3.1). It is clear that the first order of accuracy difference scheme 𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1+𝐴𝑢_𝑘=𝑝+𝜑_𝑘,𝜑_𝑘=𝑓𝑡_𝑘,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁,𝑁𝜏=1,₀=𝜑, (3.6) has a solution and the following formula holds: 𝑢_𝑘=𝑅^𝑘𝜑+^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝑝+𝜑_𝑗𝜏,1≤𝑘≤𝑁. (3.7)

Applying formula (3.7) and the boundary condition 𝑢_𝑁=𝜓, (3.8) we can write 𝜓=𝑅^𝑁𝜑+^𝑁_𝑗=1𝑅^{𝑁−𝑗+1}𝜑_𝑗𝜏+^𝑁_𝑗=1𝑅^{𝑁−𝑗+1}𝜏𝑝. (3.9) Since ^𝑁_𝑗=1𝑅^{𝑁−𝑗+1}𝜏=𝐴⁻¹(𝐼−𝑅)^𝑁_𝑗=1𝑅^𝑁−𝑗=𝐴⁻¹𝐼−𝑅^𝑁, (3.10) we have that 𝜓=𝑅^𝑁𝜑+^𝑁_𝑗=1𝑅^{𝑁−𝑗+1}𝜑_𝑗𝜏+𝐴⁻¹𝐼−𝑅^N𝑝. (3.11) Using Lemma 3.2, we get 𝑝=𝑇_𝜏𝐴𝜓−𝐴𝑅^𝑁𝜑−^𝑁_𝑗=1𝐴𝑅^{𝑁−𝑗+1}𝜑_𝑗𝜏. (3.12) Using formulas (3.7) and (3.12), we get 𝑢_𝑘=𝑅^𝑘𝜑+^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝜑_𝑗𝜏+^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝜏𝑇_𝜏𝐴𝜓−𝐴𝑅^𝑁𝜑−^𝑁_𝑗=1𝐴𝑅^{𝑁−𝑗+1}𝜑_𝑗𝜏,1≤𝑘≤𝑁. (3.13) Since ^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝜏=𝐴⁻¹(𝐼−𝑅)^𝑘_𝑗=1𝑅^𝑘−𝑗=𝐴⁻¹𝐼−𝑅^𝑘, (3.14) we have that 𝑢_𝑘=𝑅^𝑘𝜑+^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝜑_𝑗𝜏+𝐼−𝑅^𝑘𝑇_𝜏𝜓−𝑅^𝑁𝜑−^𝑁_𝑗=1𝑅^{𝑁−𝑗+1}𝜑_𝑗𝜏,1≤𝑘≤𝑁. (3.15) 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 ({𝑢_𝑘}^𝑁_𝑘=1,𝑝) of problem (3.1) in 𝐶_𝜏(𝐸)×𝐸, the stability estimates 𝑝_𝐸≤𝑀𝜑_𝐸+𝐴𝜓_𝐸+1𝛽𝜑_𝑘^𝑁_{𝑘=1𝐶𝛽𝜏(𝐸)},𝑢_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)}≤𝑀𝜑_𝐸+𝜓_𝐸+𝜑_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)} (3.16)(3.17) hold, where 𝑀 is independent of 𝜏, 𝜑, 𝜓, and {𝜑_𝑘}^𝑁_𝑘=1.

Proof. From formulas (3.7) and (3.12), it follows that 𝑝=𝑇_𝜏𝐴𝜓−𝐴𝑅^𝑁𝜑−^𝑁−1_𝑗=1𝐴𝑅^{𝑁−𝑗+1}𝜑_𝑗−𝜑_𝑁𝜏−𝐼−𝑅^𝑁𝜑_𝑁. (3.18) Using this formula, the triangle inequality, and estimates (3.4), we obtain 𝑝_𝐸≤𝑇_{𝜏𝐸→𝐸}𝐴𝜓_𝐻+𝐴𝑅^𝑁_𝐸→𝐸𝜑_𝐸+^𝑁−1_𝑗=1𝐴𝑅^{𝑁−𝑗+1}_𝐸→𝐸𝜑_𝑗−𝜑_𝑁𝐸𝜏+1+𝑅^𝑁_𝐸→𝐸𝜑_𝑁𝐸≤𝑀𝜑_𝐸+𝐴𝜓_𝐸+1𝛽𝜑_𝑘^𝑁_{𝑘=1𝐶𝛽𝜏(𝐸)}. (3.19)

The estimate (3.16) is proved. Using formula (3.15), the triangle inequality, and estimates (3.4), we obtain 𝑢_𝑘𝐸≤𝑅^𝑘_𝐸→𝐸𝜑_𝐸+^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}_𝐸→𝐸𝜑_𝑗𝐸𝜏+1+𝑅^𝑘_𝐸→𝐸𝑇_{𝜏𝐸→𝐸}×‖𝜓‖_𝐸+𝑅^𝑁_𝐸→𝐸𝜑_𝐸+^𝑁_𝑗=1𝑅^{𝑁−𝑗+1}_𝐸→𝐸𝜑_𝑗𝐸𝜏≤𝑀𝜑_𝐸+𝜓_𝐸+𝜑_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)} (3.20) for any 𝑘. From that it follows estimate (3.17). Theorem 3.3 is proved.

Theorem 3.4. For the solution ({𝑢_𝑘}^𝑁_𝑘=1,𝑝) of problem (3.1) in 𝐶_𝜏(𝐸)×𝐸, the almost coercive stability estimates 𝑝_𝐸≤𝑀𝜑_𝐸+𝐴𝜓_𝐸1+minln𝜏,ln‖𝐴‖_𝐸→𝐸𝜑_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)},𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1^𝑁_{𝑘=1𝐶𝜏(𝐸)}+𝐴𝑢_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)}≤𝑀𝐴𝜑_𝐸+𝐴𝜓_𝐸1+minln𝜏,ln‖𝐴‖_𝐸→𝐸𝜑_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)} (3.21)(3.22) hold, where 𝑀 does not depend on 𝜏, 𝜑, 𝜓, and {𝜑_𝑘}^𝑁_𝑘=1.

Proof. Using formula (3.12), the triangle inequality and estimates (3.4), we obtain 𝑝_𝐸≤𝑇_{𝜏𝐸→𝐸}𝐴𝜓_𝐻+𝑅^𝑁_𝐸→𝐸𝐴𝜑_𝐸+^𝑁−1_𝑗=1𝐴𝑅^{𝑁−𝑗+1}_𝐸→𝐸𝜑_𝑗𝐸𝜏≤𝑀𝐴𝜑_𝐸+𝐴𝜓_𝐸+^𝑁−1_𝑗=1𝐴𝑅^{𝑁−𝑗+1}_𝐸→𝐸𝜏𝜑_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)}. (3.23) Since [31] ^𝑁−1_𝑗=1𝐴𝑅^{𝑁−𝑗+1}_𝐸→𝐸1𝜏≤𝑀minln𝜏,ln‖𝐴‖_𝐸→𝐸, (3.24) we have estimate (3.21). Using formula (3.15), the triangle inequality, and estimates (3.4), (3.24), we obtain 𝐴𝑢_𝑘𝐸≤𝑅^𝑘_𝐸→𝐸𝐴𝜑_𝐸+^𝑘_𝑗=1𝐴𝑅^{𝑘−𝑗+1}_𝐸→𝐸𝜑_𝑗𝐸𝜏+1+𝑅^𝑘_𝐸→𝐸𝑇_{𝜏𝐸→𝐸}×𝐴𝜓_𝐸+𝑅^𝑁_𝐸→𝐸𝐴𝜑_𝐸+^𝑁_𝑗=1𝐴𝑅^{𝑁−𝑗+1}_𝐸→𝐸𝜑_𝑗E𝜏≤𝑀𝐴𝜑_𝐸+𝐴𝜓_𝐸1+minln𝜏,ln‖𝐴‖_𝐸→𝐸𝜑_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)} (3.25) for any 𝑘. Therefore, 𝐴𝑢_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)}≤𝑀𝐴𝜑_𝐸+𝐴𝜓_𝐸1+minln𝜏,ln‖𝐴‖_𝐸→𝐸𝜑_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸)}. (3.26) This estimate, triangle inequality, and (1.1) yield estimate (3.22). Theorem 3.4 is proved.

4. Applications

Now, we consider the applications of Theorems 3.3 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 [0,1]_ℎ=𝑥=𝑥_𝑛∶𝑥_𝑛=𝑛ℎ,0≤𝑛≤𝑀,𝑀ℎ=1. (4.1) Let us introduce the Banach space 𝐶_ℎ=𝐶([0,1]_ℎ) of the grid functions 𝜑^ℎ(𝑥)={𝜑_𝑛}₁^𝑀−1 defined on [0,1]_ℎ, equipped with the norm 𝜑^ℎ_𝐶ℎ=max_{𝑥∈[0,1]ℎ}𝜑^ℎ(𝑥). (4.2) To the differential operator 𝐴 generated by problem (2.18), we assign the difference operator 𝐴^𝑥_ℎ by the formula 𝐴^𝑥_ℎ𝜑^ℎ(𝑥)=−𝑎(𝑥)𝜑_{𝑥𝑥,𝑛}+𝜎𝜑_𝑛1^𝑀−1 (4.3) acting in the space of grid functions 𝜑^ℎ(𝑥)={𝜑_𝑛}₁^𝑀−1 satisfying the conditions 𝜑₀=𝜑_𝑀, 𝜑₁−𝜑₀=𝜑_𝑀−𝜑_𝑀−1. It is wellknown that 𝐴^𝑥_ℎ is a strongly positive operator in 𝐶_ℎ. With the help of 𝐴^𝑥_ℎ, we arrive at the boundary value problem 𝑑𝑢^ℎ(𝑡,𝑥)𝑑𝑡+𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑝^ℎ(𝑥)+𝑓^ℎ(𝑡,𝑥),0<𝑡<1,𝑥∈[0,1]_ℎ,𝑢^ℎ(0,𝑥)=𝜑^ℎ(𝑥),𝑢^ℎ(1,𝑥)=𝜓^ℎ(𝑥),𝑥∈[0,1]_ℎ. (4.4) In the second step, we replace (4.4) with the difference scheme (3.1) 𝑢^ℎ_𝑘(𝑥)−𝑢^ℎ_𝑘−1(𝑥)𝜏+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑝^ℎ(𝑥)+𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁,𝑥∈[0,1]_ℎ,𝑢^ℎ(0,𝑥)=𝜑^ℎ(𝑥),𝑢^ℎ(1,𝑥)=𝜓^ℎ(𝑥),𝑥∈[0,1]_ℎ. (4.5)

Theorem 4.1. The solution pairs ({𝑢^ℎ_𝑘(𝑥)}^𝑁₀,𝑝^ℎ(𝑥)) of problem (4.5) satisfy the stability estimates 𝑝^ℎ_𝐶ℎ≤𝑀₁𝜑^ℎ_𝐶ℎ+𝜓^ℎ_𝐶ℎ+𝐴^𝑥_ℎ𝜓^ℎ_𝐶ℎ+1𝛽𝑓^ℎ_𝑘^𝑁_{1𝐶𝛽𝜏(𝐶ℎ)},𝑢^ℎ_𝑘^𝑁_{1𝐶𝜏(𝐶ℎ)}≤𝑀₂𝜑^ℎ_𝐶ℎ+𝜓^ℎ_𝐶ℎ+𝑓^ℎ_𝑘^𝑁_{1𝐶𝜏(𝐶ℎ)}, (4.6) where 𝑀₁ and 𝑀₂ do not depend on 𝛽,𝜑^ℎ, 𝜓^ℎ, and 𝑓^ℎ_𝑘, 1≤𝑘≤𝑁.
Here, 𝐶^𝛽_𝜏(𝐶_ℎ) is the grid space of grid functions {𝑓^ℎ_𝑘}^𝑁₁ defined on [0,1]_𝜏×[0,1]_ℎ with norm 𝑓^ℎ_𝑘^𝑁_{1𝐶𝛽𝜏(𝐶ℎ)}=𝑓^ℎ_𝑘^𝑁_{1𝐶𝜏(𝐶ℎ)}+sup_{1≤𝑘<𝑘+𝑟≤𝑁}𝑓^ℎ_𝑘+𝑟−𝑓^ℎ_𝑘𝐿2ℎ(𝑟𝜏)^𝛽,𝑓^ℎ_𝑘^𝑁_{1𝐶𝜏(𝐶ℎ)}=max_{1≤𝑘≤𝑁}𝑓^ℎ_𝑘𝐶ℎ. (4.7)

The proof of Theorem 4.1 is based on Theorem 3.3 and the positivity property of the operator 𝐴^𝑥_ℎ defined by formula (4.3).

Theorem 4.2. The solution pairs ({𝑢^ℎ_𝑘(𝑥)}^𝑁₀,𝑝^ℎ(𝑥)) of problem (4.5) satisfy the almost coercive stability estimates 𝑝^ℎ_𝐶ℎ≤𝑀₁𝐴^𝑥_ℎ𝜑^ℎ_𝐶ℎ+𝐴^𝑥_ℎ𝜓^ℎ_𝐶ℎ1+ln𝑓𝜏+ℎ^{ℎ𝑘𝑁1𝐶}_𝜏^(𝐶_ℎ⁾,𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1𝜏^𝑁_{1𝐶𝜏(𝐶ℎ)}+𝐴^𝑥_ℎ𝑢^ℎ_𝑘^𝑁_{1𝐶𝜏(𝐶ℎ)}≤𝑀₂𝐴^𝑥_ℎ𝜑^ℎ_𝐶ℎ+𝐴^𝑥_ℎ𝜓^ℎ_𝐶ℎ1+ln𝑓𝜏+ℎ^{ℎ𝑘𝑁1𝐶}_𝜏^(𝐶_ℎ⁾, (4.8) where 𝑀₁ and 𝑀₂ are independent of 𝜑^ℎ, 𝜓^ℎ, and 𝑓^ℎ_𝑘, 1≤𝑘≤𝑁.

The proof of Theorem 4.2 is based on Theorem 3.4 and the positivity property of the operator 𝐴^𝑥_ℎ defined by formula (4.3) and on the estimate 1minln𝜏,ln𝐴^𝑥_{ℎ𝐶ℎ→𝐶ℎ}1≤𝑀ln.𝜏+ℎ (4.9)

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 parameter 𝑝 of a parabolic equation 𝑣^′(𝑡)+𝐴𝑣(𝑡)=𝑓(𝑡)+𝑝(0≤𝑡≤1),𝑣(0)=𝜑,𝑣(1)=𝜓 (5.1) 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 𝑣^′(𝑡)(𝑡)(𝑡)()(0)(𝜆)𝑣+𝐴𝑣=𝑓+𝑝0≤𝑡≤1,𝑣=𝜑,𝑣=𝜓,0<𝜆≤1,_′(𝑡)−𝐴𝑣(𝑡)=𝑓(𝑡)+𝑝(0≤𝑡≤1),𝑣(1)=𝜑,𝑣(𝜆)=𝜓,0≤𝜆<1 (5.2) in an arbitrary Banach space with positive operator 𝐴 and an unknown parameter 𝑝 hold.

Acknowledgment

The authors are grateful to Professor Allaberen Ashyralyev (Fatih University,Turkey) for his comments and suggestions to improve the quality of the paper.