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1
Content available Critical cases in neutral functional differential equations, arising from hydraulic engineering
Răsvan Vladimir
Opuscula Mathematica
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2022
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Vol. 42, no. 4
605--633
EN
This paper starts from several applications described by initial/boundary value problems for 1D (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties.
2
Content available remote Acoustic Reverse-time Migration using Optimal Staggered-grid Finite-difference Operator Based on Least Squares
Yan H., Yang L., Liu H.
Acta Geophysica
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2015
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Vol. 63, no. 3
715--734
EN
Reverse-time migration (RTM) directly solves the two-way wave equation for wavefield propagation; therefore, how to solve the wave equation accurately and quickly is very important for RTM. The conventional staggered-grid finite-difference (SFD) operators are usually based on the Taylor-series expansion theory. If they are used to solve wave equation on a larger frequency content, a strong dispersion will occur, which directly affects the seismic image quality. In this paper, we propose an optimal SFD operator based on least squares to solve acoustic wave equation for prestack RTM, and obtain a new antidispersion RTM algorithm that can use short spatial difference operators. The synthetic and real data tests demonstrate that the least squares SFD (LSSFD) operator can mitigate the numerical dispersion, and the acoustic RTM using the LSSFD operator can effectively improve image quality comparing with that using the Taylor-series expansion SFD (TESFD) operator. Moreover, the LSSFD method can adopt a shorter spatial difference operator to reduce the computing cost.
3
Content available remote On divided difference operators in function algebras
Multarzyński P.
Demonstratio Mathematica
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2008
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Vol. 41, nr 2
273-289
EN
In this paper we study divided difference operators of any order acting in function algebras. In the definition of difference quotient operators we use a tension structure defined on the set of points on which depend the functions of the algebras considered. In the paper we mention the oportunity for partial difference quotient operators as well as for some purely algebraic definition of divided difference operators in terms of the suitable Leibniz product rules.
4
Content available remote On the finite spectra of the difference operator nabla over the sequence space lp, (1
Akhmedov A., Basar F.
Demonstratio Mathematica
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2006
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Vol. 39, nr 3
585-595
EN
In the present paper, the norm of the bounded linear operator nabla acting on the sequence space Lp has been found and the fine spectrum with respect to Goldberg's classification of the difference operator nabla over the sequence space Lp has been determined, where 1 < p < oo.
5
Content available remote A note on Ostrowski type inequalities
Pachpatte B.
Demonstratio Mathematica
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2002
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Vol. 35, nr 1
27-30
EN
In the present note we establish two new integral inequalities of the Os-trowski type involving a function of one independent variable. The discrete analogues of the main results are also given.
6
Content available An algebraic analysis approach to 2-D discrete problems
Przeworska-Rolewicz D., Wysocki H.
Control and Cybernetics
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2001
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Vol. 30, no 2
149-158
EN
A model of algebraic analysis for the 2-index sequences (of the type 2-D) is considered. For difference operators of the form D{x_m,n}:= {x_m+1,n+1 - A_m,nX_m,n} the right inverses and the corresponding initial operators are constructed. Having already known the initial operators, one can determine solutions of the corresponding initial value problems.
7
Content available remote Universa funkcyjne i różnicowe
Pawłuszewicz E.
Zeszyty Naukowe Politechniki Białostockiej. Matematyka, Fizyka, Chemia
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1999
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Z. 19
37-48
PL
W pracy przedstawione są podstawowe idee i fakty dotyczące uniwersów funkcyjnych, czyli zbiorów funkcji częściowo określonych, zawierających zero i domkniętych ze względu na operację podstawiania i amalgamacji. Są rozważone pewne klasy uniwersów funkcyjnych istotnych z punktu widzenia zastosowań, mianowicie uniwersa generowane globalnie i uniwersa różnicowe związane z nieliniowym dyskretnym układem sterowania.
EN
In the paper there are presented the basic ideas and facts about function universes, i. e. sets of partially defined functions that contain zero and that are close with respect to substitutions and amalgamations. There are also considered some special classes of function universes: globaly generated universes and differential universes associated with given nonlinear discrete-time control system.
8
Content available remote A nonstandard difference Sturm-Liouville operator
Javorskij J., Ljance V.
Bulletin of the Polish Academy of Sciences. Mathematics
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1998
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Vol. 46, no 3
285-290
EN
After Nelson's Radically Elementary Probability Theory [1] a natural question arises: whether a hyperfinite-dimensional space is sufficiently rich to be used for the same goal as an infinite-dimensional one. Here a hyperfinite 3-diagonal matrix is investigated, which spectral properties are simular to the Naimarks's singular nonselfadjoint Sturm-Liouville differential operator on semi-axis [2, 3].
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