The Difference Problem of Obtaining the Parameter of a Parabolic Equation

The boundary value problem of determining the parameter p of a parabolic equation υ (cid:2) (cid:2) t (cid:3)(cid:4) Aυ (cid:2) t (cid:3) (cid:5) f (cid:2) t (cid:3) (cid:4) p (cid:2) 0 ≤ t ≤ 1 (cid:3) , υ (cid:2) 0 (cid:3) (cid:5) ϕ, υ (cid:2) 1 (cid:3) (cid:5) ψ in an arbitrary Banach space E with the strongly positive operator A is considered. The ﬁrst order of accuracy stable di ﬀ erence scheme for the approximate solution of this problem is investigated. The well-posedness of this di ﬀ erence scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of di ﬀ erence schemes for the approximate solution of di ﬀ erential equations with parameter are obtained. Here, T (cid:5) (cid:2) I − exp A } Abstract problem (cid:2) 1.1 (cid:3) was investigated in the paper (cid:7) 4 (cid:8) by applying estimates (cid:2) 2.1 (cid:3) and (cid:2) 2.2 (cid:3) . The solvability of problem (cid:2) 1.1 (cid:3) in the space C (cid:2) E (cid:3) of the continuous E -valued functions ϕ (cid:2) t (cid:3) deﬁned on (cid:7) 0 , 1 (cid:8) equipped with the norm was studied under the necessary and su ﬃ cient conditions for the operator A . The solution depends continuously on the initial and boundary data. More pricisely, we have the following result. with self adjoint positive deﬁnite operator A were obtained in the . The investigation of this in arbitrary Banach space E with the strongly positive operator A us to obtain the stability and almost stability estimates for the solution of di ﬀ erence schemes for the approximate solution of di ﬀ erential equations with parameter are obtained. Of such type results for the solution of ﬀ erence scheme boundary an hold.


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 dt Av t f t p, 0 < t < 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.

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: From estimate 2.1 , it follows that 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 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 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 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]: We say v t , p is the solution of problem 1.
ii v t , p satisfies the equation and boundary conditions 1.1 .
Here, C β,γ 0 E , 0 ≤ γ ≤ β, 0 < β < 1 is the H ölder space with weight obtained by completion of the space of all smooth E-valued functions ϕ t on 0, 1 in the norm In the paper 23 , the exact estimates in ölder 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: 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 of the differential operator of the form acting on functions defined on the space R n satisfies the inequalities 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 H ölder condition with the indicator μ ∈ 0, 1 .
Theorem 2.2.For the solution of boundary problem 2.8 , the following estimates are satisfied: 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

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

. , n of all continuous functions satisfying a H ölder condition with the indicator
where C Ω is the space of the all continuous functions defined on Ω, equipped with the norm It is known that the differential expression 34 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 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 H ölder condition for which the following norms are finite, where C 0, 1 is the space of the all continuous functions ϕ x defined on 0, 1 with the usual norm 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 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: where M is independent of ϕ x , ψ x , and f t, x .

Rothe Difference Scheme for Parabolic Equations with an Unknown Parameter
In this section, our focus is the well-posedness of the Rothe difference scheme for approximately solving problem 1.1 .
Throughout the section, C 0, 1 τ , E denotes the linear space of grid functions ϕ τ {ϕ k } N 1 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 For α ∈ 0, 1 , let C α E C α 0, 1 τ , E be the H ölder space with the following norm: Let us start with some lemmas we need in the following.

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 −1 and the following estimate is satisfied:

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 has a solution and the following formula holds: Applying formula 3.7 and the boundary condition

3.11
Using Lemma 3.2, we get

3.12
Using formulas 3.7 and 3.12 , we get we have that 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
3.17 hold, where M is independent of τ, ϕ, ψ, and Proof.From formulas 3.7 and 3.12 , it follows that

3.18
Using this formula, the triangle inequality, and estimates 3.4 , we obtain

3.19
The estimate 3.16 is proved.Using formula 3.15 , the triangle inequality, and estimates 3.4 , we obtain for any k.From that it follows estimate 3.17 .Theorem 3.3 is proved.

Theorem 3.4. For the solution {u
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

3.23
Since 31 we have estimate 3.21 .Using formula 3.15 , the triangle inequality, and estimates 3.4 , 3.24 , we obtain for any k.Therefore,

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 } M−1 1 defined on 0, 1 h , equipped with the norm To the differential operator A generated by problem 2.18 , we assign the difference operator A x h by the formula acting in the space of grid functions ϕ h x h is a strongly positive operator in C h .With the help of A x h , we arrive at the boundary value problem

4.4
In the second step, we replace 4.4 with the difference scheme 3.1

4.5
Theorem 4.1.The solution pairs {u h k x } N 0 , p h x of problem 4.5 satisfy the stability estimates where M 1 and M 2 do not depend on β,ϕ h , ψ h , and Here, C β τ C h is the grid space of grid functions {f h k } N 1 defined on 0, 1 τ × 0, 1 h with norm The proof of Theorem where M 1 and M 2 are independent of ϕ h , ψ h , and 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 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.

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 in arbitrary Banach space E with the strongly positive operator A is studied.The wellposedness 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 in an arbitrary Banach space with positive operator A and an unknown parameter p hold.
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.For the solution of mixed boundary value problem 2.18 , the following estimates are valid: 4.1 is based on Theorem 3.3 and the positivity property of the operator A x h defined by formula 4.3 .