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Finite Difference Method for the Reverse Parabolic Problem
A finite difference method for the approximate solution of the reverse multidimensional parabolic differential 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 difference schemes are obtained. The theoretical statements are supported by the numerical example.
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 , 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 , Ashyralyev et al. considered the nonlocal boundary value problem in a Banach space with strongly positive operator . They established the well-posedness of problem (1.1) in Hölder spaces. Moreover, they obtained the exact Schauder's estimates in Hölder norms of solutions of the boundary values problem for -th order multidimensional parabolic equations.
Ashyralyev established in  the well-posedness of the nonlocal boundary-value problem (1.1) in Bochner spaces. He considered the first and second order of accuracy difference schemes for the approximate solutions of problem (1.1). He also established the coercive inequalities for the solutions of these difference schemes. Moreover, in applications, he obtained the almost coercive stability and coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary-value problem for parabolic equation.
Clément and Guerre-Delabriére studied in  maximal regularity (in the -sense) for 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 proofs.
In , Gulin et al. considered the linear heat equation: with Dirichlet condition , and nonlocal boundary conditions , , . They constructed an explicit difference scheme with second order of approximation with respect to the space variables and first order of approximation with respect to . Moreover, using previous results of Ionkin and Morozova for the one-dimensional heat equation with nonlocal boundary conditions, they proved the stability of this scheme with respect to the norm , which is induced by the symmetric and positive-definite matrix .
Liu et al. studied in  a finite difference method for multidimensional nonlinear coupled system of parabolic and hyperbolic equations. By using a variational method, they obtained an a priori estimate. They also proved that the finite difference scheme is uniquely solvable and unconditionally stable. To support the theory, they gave numerical example of two-dimensional problem.
In [11, 12], Martin-Vaquero and Vigo-Aguiar provided algorithms improving the CPU time and accuracy of Crandall's formula. They studied the convergence of the algorithms and compared the efficiency of the methods with well-known numerical examples.
In , Sapagovas applied finite difference approximations to a nonhomogeneous heat equation in one space dimension, subject to nonlocal boundary conditions. He presented a stable difference approximation for the equation and a piecewise constant discretization of the integrals appearing in the boundary conditions. He discussed the stability of the complete problem with respect to two parameters included in the integral terms. He constructed a stability region in the plane of the parameters and gave practical examples with specific choices of the integral conditions. Sapagovas investigated in  the stability of implicit difference schemes for the equation of a thermoelastic rod, which is a parabolic equation subject to integral conditions for the boundaries.
In , Shakhmurov dealt with a nonlocal boundary value problem for a degenerate equation in a Banach space with unbounded operators in . He proved the maximal 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 . For further applications of such problems, we refer the reader to [19–22] and the references therein.
In the paper , Ashyralyev et al. considered the multipoint nonlocal boundary value problem for reverse parabolic equations in a Hilbert space with self-adjoint positive definite operator .
is called a solution of problem (1.3) if the following conditions hold: (1) is continuously differentiable on the segment . The derivatives at the end points of the segment are understood as the appropriate unilateral derivatives.(2)The element belongs to for all and the function is continuous on the segment .(3) satisfies the equation and the nonlocal boundary conditions (1.3).
A solution of problem (1.3) defined in this manner will be from now referred to as a solution of problem (1.3) in the space of all continuous functions defined on with values in equipped with the norm
Throughout the paper, indicates positive constants which can be different from time to time and we are not interested to make precise. We write to stress the fact that the constant depends only on
Under the assumption:
Ashyralyev et al. established in  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 , 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, ,,, and are given smooth functions and , and is the open cube in the -dimensional Euclidean space with boundary ,. In the Hilbert space , we introduce the self-adjoint positive definite operator defined by with domain 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 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.
2. Difference Schemes: Stability Estimates
We will discretize problem (1.8) in two steps. In the first step, we define the grid spaces
We denote that . Let denote the Banach space of grid functions: defined on , equipped with the norm
To the differential operator generated by problem (1.8), we assign the second-order approximation difference operator acting in the space of grid functions , satisfying the condition for all . Assume that is self-adjoint, positive-definite operator in and is bounded operator in .
By using , we arrive at the multipoint nonlocal boundary value problem: for a finite system of ordinary differential equations with a fixed . Note that . Therefore, we will try to obtain stability, coercivity stability, and almost coercivity estimates with constants independent of .
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 where denotes the greatest integer function.
To formulate our results, let and be spaces of the grid functions defined on , equipped with the norms
Furthermore, let be the uniform grid space with step size , where is a fixed positive integer. We denote for the linear space of grid functions with values in the Hilbert space .
For , let and be, respectively, the Hölder space and the weighted Hölder space with the norms Here, is the Banach space of bounded grid functions with norm:
Theorem 2.1. Let and be sufficiently small positive numbers. Then, for the solutions of difference schemes (2.5) and (2.6), the following stability estimate holds: where is independent of ,,, and and .
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)
By the spectral representation of self-adjoint positive definite operator and the triangle inequality, we have
Similarly, we have Estimates (2.16) and (2.17) conclude the proof of Theorem 2.1.
Theorem 2.2. Let and 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 does not depend on ,,.
Proof. Using formulas (2.11)–(2.14), estimates (2.16) and (2.17), the triangle inequality, assumption (1.6), we obtain 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 it follows inequality (2.18). Theorem 2.2 is proved.
Theorem 2.4. Let and be sufficiently small positive numbers. Then, the solutions of difference problem (2.5) and (2.6) satisfy the following coercivity stability estimate: where is independent of ,,, and ,.
3. Numerical Results
For the numerical result, we consider the nonlocal boundary value problem: for the reverse parabolic equation. It is easy to see that is the exact solution of (3.1).
For the approximate solution of nonlocal boundary value problem (3.1), consider the set of a family of grid points depending on the small parameters and
Note that for difference scheme (3.3), we have that where
It is easy to see that and , and where is the identity operator.
We can write (3.3) as in the matrix form Here, is an column matrix, are square matrices, , here and in the future is the identity matrix,
Samarskii and Nikolaev studied this type of system in  for difference equations. We seek the solution of (3.7) by the formula where are square matrices and are column matrices. For the solution of difference equation (3.7) we need to use the following formulas for : where is the zero matrix and is the zero column vector.
Second, we consider again the nonlocal boundary value problem (3.1). Applying (2.6) and formulas: we get the second order of accuracy in and for the approximate solutions of the nonlocal boundary value problem (3.1).
We can rewrite this system in the following matrix form: where is an column matrix, , ,,, are square matrices ,,,, where ,
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: where , are zero matrices, , and
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 and represents the numerical solutions of these difference schemes at . For their comparison, the errors are computed by Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes. Table 1 is constructed for , 40, and 60, respectively. Hence, the second order of accuracy difference scheme is more accurate compared with the first order of accuracy difference scheme.
The authors would like to thank Prof. Dr. Allaberen Ashyralyev (Fatih University, Turkey) on his very helpful comments and suggestions in improving the quality of this work.
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