Well-Posed and Ill-Posed Boundary Value Problems for PDEView this Special Issue
Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces
A first order of accuracy difference scheme for the approximate solution of abstract nonlocal boundary value problem , , , , for differential equations in a Hilbert space with a self-adjoint positive definite operator A is considered. The well-posedness of this difference scheme in Hölder spaces without a weight is established. Moreover, as applications, coercivity estimates in Hölder norms for the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.
Nonlocal boundary value problems for partial differential equations have been applied by various researchers in order to model numerous processes in different fields of applied sciences when they are unable to determine the boundary values of the unknown function (see, e.g., [1–15] and the references therein).
In paper , the well-posedness of abstract nonlocal boundary value problem in Hölder spaces without a weight was established. The coercivity inequalities for solutions of the boundary value problem for elliptic-parabolic equations were obtained.
In the present paper, the first order of accuracy difference scheme for the approximate solution of problem (1.1) is considered. The well-posedness of difference scheme (1.2) in Hölder spaces without a weight is established. As an application, coercivity inequalities for solutions of difference scheme for elliptic-parabolic equations are obtained.
Throughout the paper, denotes a Hilbert space and is a self-adjoint positive definite operator with for some . Then, it is wellknown that is a self-adjoint positive definite operator and . Furthermore, and which are defined on the whole space , are bounded operators, where is the identity operator.
2. Well-Posedness of (1.2)
First of all, let us start with some auxiliary lemmas that are used throughout the paper.
Let be the linear space of mesh functions defined on with values in the Hilbert space . Next, , , , and denote Banach spaces on with norms:
Lemma 2.2. For , the norms of the spaces and are equivalent (see ).
Theorem 2.3. Suppose , , and . Boundary value problem (1.2) is wellposed in Hölder space and the following coercivity inequality holds: where is independent of not only , , and but also of and .
Proof. First of all, let us get the formulae for solution of problem (1.2). By [21, 25],
is the solution of boundary value difference problem:
is the solution of inverse Cauchy problem:
Combining the conditions and formulas (2.6), (2.8), we get formulas
Operator equation follows from formulas (2.10), (2.11), and the condition . As the operator has an inverse it follows that for the solution of operator equation (2.12). Hence, we have formulas (2.10), (2.11), and (2.15) for the solution of difference problem (1.2).
Using formulae (2.10) and (2.15), we can get
Finally, we will get coercivity estimate (2.5). It is based on estimates for the solution of boundary value difference problem (2.7), for the solution of inverse Cauchy difference problem (2.9), and for the solution of problem (1.2). Estimates (2.18) and (2.19) were established in [21, 25], respectively.
Estimates (2.20) are derived from the formulas (2.16) and (2.17) for the solution of problem (1.2), estimates (2.1) and following estimates which were established in . This finalizes the proof of Theorem 2.3.
3. An Application
In this section, an application of the abstract Theorem 2.3 is considered. First, let be the unit open cube in the -dimensional Euclidean space with boundary . In , the mixed boundary value problem for multidimensional mixed equation: is considered. Here, , and are given smooth functions and .
The discretization of problem (3.1) is carried out in two steps. In the first step, the grid sets
are defined. To the differential operator generated by problem (3.1), the difference operator is assigned by formula: acting in the space of grid functions , satisfying the conditions for all . With the help of , we arrive at the nonlocal boundary-value problem for an infinite system of ordinary differential equations.
In the second step, problem (3.4) is replaced by difference scheme (1.2) (see ): To formulate the result, we introduce the Hilbert spaces , and of the grid functions defined on , equipped with the norms:
Theorem 3.1. Let and be sufficiently small numbers. Then, the solutions of difference scheme (3.5) satisfy the following coercivity stability estimate: where is not dependent on , and .
The proof of Theorem 3.1 is based on Theorem 2.3, the symmetry properties of the difference operator defined by formula (3.3), and along with the following theorem on the coercivity inequality for the solution of elliptic difference equation in .
Theorem 3.2. For the solution of elliptic difference problem: the following coercivity inequality holds : Here, depends neither on nor .
The author would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey) for his inspirational contributions and to the anonymous referees whose careful reading of the paper and valuable comments helped to improve it.
M. S. Salakhitdinov, Equations of Mixed-Composite Type, Fan, Tashkent, Uzbekistan, 1974.
T. D. Dzhuraev, Boundary Value Problems for Equations of Mixed and Mixed Composite Types, Fan, Tashkent, Uzbekistan, 1979.
D. G. Gordeziani, On Methods of Resolution of a Class of Nonlocal Boundary Value Problems, Tbilisi University Press, Tbilisi, Georgia, 1981.
V. N. Vragov, Boundary Value Problems for Nonclassical Equations of Mathematical Physics, Textbook for Universities, NGU, Novosibirsk, Russia, 1983.
D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylim, Ashgabat, Turkmenistan, 1995.
A. M. Nakhushev, Equations of Mathematical Biology, Vysshaya Shkola, Moskow, Russia, 1995.
S. N. Glazatov, “Nonlocal boundary value problems for linear and nonlinear equations of variable type,” Sobolev Institute of Mathematics SB RAS, no. 46, p. 26, 1998.View at: Google Scholar
A. Ashyralyev and H. A. Yurtsever, “On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations,” Nonlinear Analysis-Theory, Methods and Applications, vol. 47, no. 5, pp. 3585–3592, 2001.View at: Google Scholar
J. I. Díaz, M. B. Lerena, J. F. Padial, and J. M. Rakotoson, “An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator,” Journal of Differential Equations, vol. 198, no. 2, pp. 321–355, 2004.View at: Publisher Site | Google Scholar | Zentralblatt MATH
J. Martín-Vaquero, A. Queiruga-Dios, and A. H. Encinas, “Numerical algorithms for diffusion-reaction problems with non-classical conditions,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5487–5495, 2012.View at: Google Scholar
A. Ashyralyev and H. Soltanov, “On elliptic-parabolic equations in a Hilbert space,” in Proceedings of the IMM of CS of Turkmenistan, pp. 101–104, Ashgabat, Turkmenistan, 1995.View at: Google Scholar
A. Ashyralyev, “On the well-posedness of the nonlocal boundary value problem for elliptic-parabolic equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 49, pp. 1–16, 2011.View at: Google Scholar
P. E. Sobolevskii, “The theory of semigroups and the stability of difference schemes,” in Operator Theory in Function Spaces, pp. 304–337, Nauka, Novosibirsk, Russia, 1977.View at: Google Scholar
A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1994.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978.
A. Ashyralyev, Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations [Doctor of Sciences Thesis], Kiev, Ukraine, 1992.
P. E. Sobolevskii, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971.View at: Google Scholar
A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004.
P. E. Sobolevskii, On Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.