Volume 2, 2019
Difference & Differential Equations and Applications
|Number of page(s)||7|
|Section||Mathematics - Applied Mathematics|
|Published online||29 May 2019|
To the theory of discrete boundary value problems
Chair of Differential Equations, Belgorod State National Research University, ul. Pobedy, 85, Belgorod 308015, Russia
* Corresponding author: firstname.lastname@example.org
Accepted: 5 April 2019
We consider discrete analogues of pseudo-differential operators and related discrete equations and boundary value problems. Existence and uniqueness results for special elliptic discrete boundary value problem and comparison for discrete and continuous solutions are given for certain smooth data in discrete Sobolev–Slobodetskii spaces.
Mathematics Subject Classification: Primary: 35S15 / Secondary: 65T50
Key words: Digital pseudo-differential operator / Discrete boundary value problem / Periodic factorization / Discrete solution / Error estimate
© O.A. Tarasova & V.B. Vasilyev, Published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Our main goal here is describing a periodic variant of this definition and studying its certain properties related to solvability of corresponding equations in canonical domains of an Euclidean space. In this paper the main result is related to a comparison of discrete and continuous solutions. We try to preserve maximal correspondence for discrete and continuous cases under digitization, it permits to find more appropriate constructions.
This problem is very large and in our opinion it should include the following aspects according to a lot of physical and technical applications of such operators and related equations:
finite and infinite discrete Fourier transform as a natural technique for such equations;
choice of appropriate discrete functional spaces;
studying solvability for infinite discrete equations;
studying solvability of approximating finite discrete equations;
a comparison between continuous and infinite discrete equations;
a comparison between infinite discrete and finite discrete equations.
This is not completed list of questions for studying which we intend to consider. Some results in this direction were obtained for simplest pseudo-differential operators (Calderon–Zygmund operators [3, 4]) and corresponding equations. Also certain results are related to approximate solutions.
There are few variants of the theory of discrete boundary value problems (see, for example [5, 6]), but these theories are related especially to partial differential operators and do not use the harmonic analysis technique. Since the classical theory of pseudo-differential operators is based on the Fourier transform we will use the discrete Fourier transform and discrete analogue of pseudo-differential operators which will include discrete analogues of partial differential and some integral convolution operators.
Given function u d of a discrete variable , h > 0, we define its discrete Fourier transform by the series:
We will remind here some definitions of functional spaces  and will consider discrete analogue of the Schwartz space . Let us denote and introduce the following.
One can define some discrete operators for such functions u d .
If is a periodic function in with the basic cube of periods then we consider it as a symbol. We will introduce a digital pseudo-differential operator in the following way.
This case is very different from , and an ellipticity condition is not sufficient for a solvability. A principal role for the solvability takes a concept of the periodic factorization which is defined for an elliptic symbol.
To describe a solvability picture for equation (3) we introduce the following notations. Let us denote .
We will use a special periodic factorization of an elliptic symbol :
The special index æ of periodic factorization determines the solvability for equation (3), and for special cases we will describe obtained results [7, 8]. These cases are distinct. So, if |æ − s| < 1/2 then we have the unique solution:
for equation (3). But if then there are a lot of solutions,
where is an arbitrary polynomial of order n of variables , satisfying the condition (2), are arbitrary functions from .
We will consider the pseudo-differential equation:
We assume that the symbol of the operator A satisfies the condition:
with respect to the last variable with the index æ .
Since we know solvability conditions for pseudo-differential equations in and  we will select such discrete pseudo-differential operators which reserve all needed properties of their continuous analogues.
Let P h be a restriction operator on , i.e. for
We tried this projector for simplest pseudo-differential operators, namely Calderon–Zygmund operators, these operators can be treated as pseudo-differential operators of order 0, and we obtained very acceptable results [3, 10–12]. But now we will use another restriction operator.
A construction for the restriction operator Q h for functions is the following. We take the Fourier transform , then its restriction on and periodically continue it onto a whole . Further we apply the inverse discrete Fourier transform and obtain a discrete function which is denoted by . In our opinion the projector Q h is more convenient than P h although the projectors P h and Q h are almost the same according to the following result.
If we put strong enough restrictions on a right-hand side and factorization elements then one can give a comparison between discrete and continuous solutions.
We have non-uniqueness of a solution for equation (3) for the case . We consider here the case n = 1.
To obtain the unique solution one needs some additional conditions. Discrete analogues of Dirichlet or Neumann conditions give a very simple case. We will consider here the discrete Dirichlet condition:
The condition (8) in Fourier images takes the form:
Let us denote,
Thus, we obtain the following result.
To obtain some comparison between discrete and continuous solutions we will remind how the continuous solution looks. The continuous analogue of the discrete boundary value problem is the following:
assuming that . Let us note that this is simplest variant of Shapiro–Lopatinskii condition .
We have the following discrete solution:
This paper is one of first steps for studying discrete boundary value problems and their connections with classical theory of boundary value problems for elliptic pseudo-differential equations. We intend to study more general situations in forthcoming papers and to obtain approximation estimates for comparison of discrete and continuous solutions.
Research supported by the State contract of the Russian Ministry of Education and Science (contract No 1.7311.2017/8.9).
- Eskin GI (1981), Boundary value problems for elliptic pseudodifferential equations, AMS, Providence. [Google Scholar]
- Taylor ME (1980), Pseudo-differential operators, Princeton Univ. Press, Princeton. [Google Scholar]
- Vasil’ev AV, Vasil’ev VB (2015), Periodic Riemann problem and discrete convolution equations. Diff Equ 51, 5, 652–660. [CrossRef] [Google Scholar]
- Vasilyev AV, Vasilyev VB (2015), Discrete singular integrals in a half-space. Current trends in analysis and its applications, Proc. 9th ISAAC Congress, Kraków, Poland, Birkhäuser, Basel, pp. 663–670. [Google Scholar]
- Vasilyev AV, Vasilyev VB (2017), Two-scale estimates for special finite discrete operators. Math Model Anal 22, 3, 300–310. [CrossRef] [Google Scholar]
- Samarskii AA (2001), The theory of difference schemes, CRC Press, Boca Raton. [CrossRef] [Google Scholar]
- Vasilyev AV, Vasilyev VB (2018), Pseudo-differential operators and equations in a discrete half-space. Math Model Anal 23, 3, 492–506. [CrossRef] [Google Scholar]
- Vasilyev AV, Vasilyev VB (2018), On some discrete boundary value problems in canonical domains, in: Differential and Difference Equations and Applications. Springer Proc. Math. Stat, Vol. 230, Springer, Cham, pp. 569–579. [CrossRef] [Google Scholar]
- Ryaben’kii VS (2002), Method of difference potentials and its applications, Springer-Verlag, Berlin-Heidelberg. [CrossRef] [Google Scholar]
- Vasilyev VB (2017), Discreteness, periodicity, holomorphy, and factorization, in: C Constanda, M Dalla Riva, PD Lamberti, P Musolino (Eds.), Integral Methods in Science and Engineering, Theoretical Technique, Vol. 1. Birkhauser, Cham, Switzerland, pp. 315–324. [CrossRef] [Google Scholar]
- Vasilyev AV, Vasilyev VB (2016), Difference and discrete equations on a half-axis and the Wiener-Hopf method. Azerb J Math 6, 1, 79–86. [Google Scholar]
- Vasilyev AV, Vasilyev VB (2016), On solvability of some difference-discrete equations. Opusc Math 36, 4, 525–539. [CrossRef] [Google Scholar]
- Vasilyev AV, Vasilyev VB (2016), On finite discrete operators and equations. Proc Appl Math Mech 16, 1, 771–772. [Google Scholar]
- Vasilyev VB (2017), The periodic Cauchy kernel, the periodic Bochner kernel, and discrete pseudo-differential operators. AIP Conf Proc 1863, 140014-1–140014-4. [Google Scholar]
- Vasilyev VB (2018), Discrete pseudo-differential operators and boundary value problems in a half-space and a cone. Lobachevskii J Math 39, 2, 289–296. [CrossRef] [Google Scholar]
- Vasilyev VB (2018), On discrete pseudo-differential operators and equations. Filomat 32, 3, 975–984. [CrossRef] [Google Scholar]
- Vasilyev VB (2018), On some approximate calculations for certain pseudo-differential equations. Bull Karaganda Univ Math 3, 91, 9–16. [CrossRef] [Google Scholar]
Cite this article as: Tarasova O.A & Vasilyev V.B 2019. To the theory of discrete boundary value problems. 4open, 2, 17.
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