zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Well-posed solvability of the boundary value problem for difference equations of elliptic type. (English) Zbl 0818.65046
The paper is devoted to the construction and investigation of difference schemes of high order accuracy for approximately solving the boundary value problem (*) $-v''(t) + A v(t) = f(t)$, $(0 \leq t \leq 1)$, $v(0) = v\sb 0$, $v(1) = v\sb 1$, in an arbitrary Banach space, where $A$ is an unbounded strongly positive operator. The author investigates the solvability of two steps of the difference schemes for approximately solving the abstract boundary value problem (*) reproduced by Taylor’s expansion in three points. The study is based upon stability and coercive stability of this difference scheme.

MSC:
 65J10 Equations with linear operators (numerical methods) 65L10 Boundary value problems for ODE (numerical methods) 65L12 Finite difference methods for ODE (numerical methods) 34G10 Linear ODE in abstract spaces
Full Text:
References:
 [1] Krein, S. G.: Linear differential equations in Banach spaces. (1967) · Zbl 0193.09302 [2] Grisvard, P.: Elliptic problems in nonsmooth domains. (1986) · Zbl 0622.34066 [3] Ashyralyev, A.: On difference schemes of a high order accuracy for elliptical equations. Application of methods of functional analysis for non-classical equations of mathematical physics. 3-14 (1989) [4] Ashyralyev, A.; Sobolevskii, P. E.: On one class of two steps difference schemes of a high order accuracy of elliptical equations in a Hilbert space. Numerical methods of solving equations of transfer, Tartu. Materials conference, 18-22 (1990) [5] Sobolevskii, P. E.: On coercive solvability difference equations. Soviet math. Dokl. 12, No. 6, 1802-1805 (1971) · Zbl 0246.39002 [6] Sobolevskii, P. E.: Operator theory in function spaces. 304-337 (1977) [7] Sobolev, S. L.: Applications of functional analysis in mathematical physics. (1950) · Zbl 0041.52307 [8] Sobolevskii, P. E.: On elliptic equations in a Banach space. Differential’nye uraneniya 4, No. 7, 1346-1348 (1968) · Zbl 0174.06701 [9] Triebel, H.: Interpolation theory, function spaces, differential operators. (1977) · Zbl 0351.46024 [10] Smirnitskii, Yu.A.; Sobolevskii, P. E.: Positivity of difference operators. Vyclisl. systems 81, 120-133 (1980) [11] Ashyralyev, A.; Sobolevskii, P. E.: Interpolation theory of linear operators and stability of difference schemes. Soviet math. Dokl. 29, No. 2, 365-367 (1984) · Zbl 0598.65038