Finite Difference Method for the Reverse Parabolic Problem

A ﬁnite di ﬀ erence method for the approximate solution of the reverse multidimensional parabolic di ﬀ erential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the ﬁrst and second orders of accuracy di ﬀ erence schemes are obtained. The theoretical statements are supported by the numerical example. abstract Cauchy problems of order one and boundary value problems of order two. As is well-known regularity of the ﬁrst problems implies regularity of the second ones; they also proved that the converse to hold if the underlying Banach space has the UMD property. A stronger notion of regularity, which is introduced by Sobolevskii, plays an important role in the regularity and Fredholmness of the problem. He also applied the results to nonlocal boundary value problems for and di ﬀ erential equations and their on cylindrical domains. The ﬁrst and second orders of accuracy in t and the second order of accuracy in space variables for the approximate solution of problem (cid:3) 1.8 (cid:4) are presented. Applying the method of papers (cid:5) 23, 24 (cid:6) , the stability, almost coercive stability, and coercive stability estimates for the solution of these di ﬀ erence schemes are obtained. The modiﬁed Gauss elimination method for solving these di ﬀ erence schemes in the case of one-dimensional parabolic partial di ﬀ erential equations is used.


Introduction
In the study of boundary value problems for partial differential equations, the role played by the well-posedness coercivity inequalities is well known see, e.g., 1-3 .Well-posedness of nonlocal boundary value problems for partial differential equations of parabolic type has been studied extensively by many researchers see, e.g., 4-15 and the references therein .
In the paper 4 , Ashyralyev studied the positivity of second-order differential and difference operators with nonlocal condition and the structure of interpolation spaces generated by these operators in a Banach space.Applying this result, he obtained the coercive inequalities for the solutions of the nonlocal boundary value problem for differential and difference equations.
In 5 , Ashyralyev et al. considered the nonlocal boundary value problem regularity and Fredholmness of the problem.He also applied the results to nonlocal boundary value problems for degenerate elliptic and quasielliptic differential equations and their finite or infinite systems on cylindrical domains.It is well known that reverse problems arise in various applications, for example, boundary layer problems in fluid dynamics 16, 17 , plasma physics, and astrophysics in the study of propagation of an electron beam through the solar corona 18 .For further applications of such problems, we refer the reader to 19-22 and the references therein.
In the paper 23 , Ashyralyev et al. considered the multipoint nonlocal boundary value problem for reverse parabolic equations in a Hilbert space H with self-adjoint positive definite operator A. u t is called a solution of problem 1.3 if the following conditions hold: 1 u t is continuously differentiable on the segment 0, 1 .The derivatives at the end points of the segment are understood as the appropriate unilateral derivatives.
2 The element u t belongs to D A for all t ∈ 0, 1 and the function Au t is continuous on the segment 0, 1 .
Throughout the paper, M indicates positive constants which can be different from time to time and we are not interested to make precise.We write M α, β, . . . to stress the fact that the constant depends only on α, β, . . .Under the assumption: Ashyralyev et al. established in 23 the well-posedness of these problems in the space of smooth functions.In applications, they obtained coercivity estimates for the solution of parabolic differential equations.
Moreover, in 24 , Ashyralyev et al. considered the first order of accuracy Rothe difference scheme: for approximately solving problem 1.3 .They established some stability estimates and almost coercivity of the solution for the difference scheme.
In the present paper, multipoint nonlocal boundary value problem for the multidimensional parabolic equation with Dirichlet condition, under the condition 1.6 is considered.Here, a r x , x ∈ Ω , ϕ x x ∈ Ω , and f t, x t ∈ 0, 1 , x ∈ Ω are given smooth functions and a r x ≥ a > 0, and Ω 0, × • • • × 0, is the open cube in the n-dimensional Euclidean space with boundary S, Ω Ω ∪ S. In the Hilbert space H L 2 Ω , we introduce the self-adjoint positive definite operator A defined by Then, problem 1.8 can be written in the abstract form as the nonlocal boundary value problem for reverse parabolic equation 1.3 .
The first and second orders of accuracy in t and the second order of accuracy in space variables for the approximate solution of problem 1.8 are presented.Applying the method of papers 23, 24 , the stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are obtained.The modified Gauss elimination method for solving these difference schemes in the case of one-dimensional parabolic partial differential equations is used.

Difference Schemes: Stability Estimates
We will discretize problem 1.8 in two steps.In the first step, we define the grid spaces Let L 2h L 2 Ω h denote the Banach space of grid functions: defined on Ω h , equipped with the norm

2.3
To the differential operator A generated by problem 1.8 , we assign the second-order approximation difference operator h acting in the space of grid functions u h x , satisfying the condition u h x 0 for all x ∈ S h .Assume that h is bounded operator in L 2h .By using A x h , we arrive at the multipoint nonlocal boundary value problem: for a finite system of ordinary differential equations with a fixed |h|.Note that |h| Therefore, we will try to obtain stability, coercivity stability, and almost coercivity estimates with constants independent of |h|.
In the second step, problem 2.4 is replaced by the first order of accuracy difference scheme and the second order of accuracy difference scheme

2.7
Furthermore, let 0, 1 τ {t k kτ, 1 ≤ k ≤ N, Nτ 1} be the uniform grid space with step size τ > 0, where N is a fixed positive integer.We denote F τ H F 0, 1 τ , H for the linear space of grid functions ϕ τ {ϕ k } N 1 with values in the Hilbert space H.
C α 1 0, 1 τ , H be, respectively, the H ölder space and the weighted H ölder space with the norms

2.8
Here, C τ H C 0, 1 τ , H is the Banach space of bounded grid functions with norm: Theorem 2.1.Let τ and |h| be sufficiently small positive numbers.Then, for the solutions of difference schemes 2.5 and 2.6 , the following stability estimate holds: where M δ, θ p is independent of τ, h, ϕ h x , and f h k x and k 1, . . ., N.
Proof.The proof of Theorem 2.1 is based on the formulas for the solution of difference scheme 2.5 and for the solution of difference scheme 2.6 Here,

2.15
By the spectral representation of self-adjoint positive definite operator and the triangle inequality, we have 2.16 Similarly, we have 2.17 Estimates 2.16 and 2.17 conclude the proof of Theorem 2.1.
Theorem 2.2.Let τ and |h| be sufficiently small positive numbers.Then, for the solutions of difference problem 2.5 and 2.6 , the following almost coercivity inequality is valid, where M δ, θ p does not depend on τ, h, ϕ h x ,f h k x , k 1, . . ., N.

2.19
Since we have that From that, inequality 2.19 , and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L 2h it follows inequality 2.18 .Theorem 2.2 is proved.

2.22
the following coercivity inequality holds: where M does not depend on h and ω h .
Theorem 2.4.Let τ and |h| be sufficiently small positive numbers.Then, the solutions of difference problem 2.5 and 2.6 satisfy the following coercivity stability estimate: where M δ, θ p , α is independent of τ, h, f h k x , and ϕ h x , k 1, . . ., N.

Theorem 2.5. Let
Then, for solutions of problem 2.5 and 2.6 , the following coercive stability estimate holds: where M δ, θ p , α does not depend on τ, h, f h k x , and ϕ h x , k 1, . . ., N.
The proofs of Theorems 2.4-2.5 are based on the formulas: the self-adjoint positive definiteness of the operator A x h in L 2h , estimates 2.16 and 2.17 , the triangle inequality, and assumption 1.6 .

Numerical Results
For the numerical result, we consider the nonlocal boundary value problem: for the reverse parabolic equation.It is easy to see that u t, x t 2 1 − t 2 sin x is the exact solution of 3.1 .
For the approximate solution of nonlocal boundary value problem 3.1 , consider the set 0, 1 τ × 0, π h of a family of grid points depending on the small parameters τ and h

3.2
Applying 2.5 , we get the first order of accuracy in t and the second order of accuracy in x for the approximate solutions of the nonlocal boundary value problem 3.1 .Note that for difference scheme 3.3 , we have that where

3.5
It is easy to see that C x h C x h * and C x h ≥ δI h , and where I h is the identity operator.So, Theorems 2.1, 2.2, 2.4, and 2.5 are compatible for the solution of 3.3 .We can write 3.3 as in the matrix form

3.7
Here, ϕ n is an here and in the future I is the N 1 × N 1 identity matrix, , s n − 1, n, n 1.

3.9
Samarskii and Nikolaev studied this type of system in 27 for difference equations.We seek the solution of 3.7 by the formula 1, 3.10 Abstract and Applied Analysis 13 where u M 0, α n n 1, . . ., M − 1 are N 1 × N 1 square matrices and β n n 1, . . ., M − 1 are N 1 × 1 column matrices.For the solution of difference equation 3.7 we need to use the following formulas for α n , β n : where α 1 is the N 1 × N 1 zero matrix and β 1 is the N 1 × 1 zero column vector.Second, we consider again the nonlocal boundary value problem 3.1 .Applying 2.6 and formulas: 3.12 we get the second order of accuracy in t and x for the approximate solutions of the nonlocal boundary value problem 3.1 .We can rewrite this system in the following matrix form: 3.17 For the solution of the last matrix equation, we use the modified variant of Gauss elimination method.We seek a solution of the matrix equation of the matrix equation in the following form:

3.19
Now, let us give the results of the numerical analysis.In order to get the solution, we used MATLAB programs.The numerical solutions are recorded for different values of N M and u k n represents the numerical solutions of these difference schemes at t k , x n .For their comparison, the errors are computed by

3.20
Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes.Table 1 is constructed for N M 20, 40, and 60, respectively.Hence, the second order of accuracy difference scheme is more accurate compared with the first order of accuracy difference scheme.  1 is the error analysis between the exact solution and solutions derived by difference schemes.

Table 1 :
Error analysis.Table