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1
Content available The heat equation on time scales
Cuchta Tom, Ferreira Rui A. C.
Opuscula Mathematica
|
2023
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Vol. 43, no. 4
475--491
EN
We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study.
2
Content available On oscillatory behaviour of third-order half-linear dynamic equations on time scales
Grace Said R., Chhatria Gokula Nanda
Opuscula Mathematica
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2022
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Vol. 42, no. 6
849--865
EN
In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case T = R or T = Z.
3
Content available Analogy of classical and dynamic inequalities merging on time scales
Sahir Muhammad Jibril Shahab
Journal of Mathematics and Applications
|
2020
|
Vol. 43
139--152
EN
In this paper, we present analogues of Radon’s inequality and Nesbitt’s inequality on time scales. Furthermore, we find refinements of some classical inequalities such as Bergström’s inequality, the weighted power mean inequality, Cauchy–Schwarz’s inequality and Hölder’s inequality. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues.
4
Generalized multivariate Fink-type identity and some related results on time scales with applications
Hussain Sabir, Latif Muhammad
Journal of Applied Analysis
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2019
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Vol. 25, nr 2
189--203
EN
Generalized Fink-type identity for multi-variables to an arbitrary time scales is obtained, giving some multi-variate Ostrowski, Iyengar and Grüss-type inequalities unifying the corresponding continuous and discrete version. Some new applications to generalized polynomials are also obtained.
5
Content available remote A nonstandard Volterra integral equation on time scales
Reinfelds Andrejs, Christian Shraddha
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
503--510
EN
This paper introduces the more general result on existence, uniqueness and boundedness for solutions of nonstandard Volterra type integral equation on an arbitrary time scales. We use Lipschitz type function and the Banach’s fixed point theorem at functional space endowed with a suitable Bielecki type norm. Furthermore, it allows to get new sufficient conditions for boundedness and continuous dependence of solutions.
6
Oscillatory behavior of higher order nonlinear homogeneous neutral delay dynamic equations with positive and negative coefficients
Panigrahi S., Reddy P. R.
Journal of Applied Analysis
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2018
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Vol. 24, nr 2
139--154
EN
In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solutions of the higher order nonlinear neutral delay dynamic equation with positive and negative coefficients. The results of this paper extend and generalize the results of [S. Panigrahi and P. Rami Reddy, Oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. AMath. Anal. 20 (2013), 143-163] and [S. Panigrahi, J. R. Graef and P. Rami Reddy, Oscillation results for fourth order nonlinear neutral dynamic equations, Commun. Math. Anal. 15 (2013), 11-28]. Examples are included to illustrate the validation of the results.
7
Content available remote Opial and Pólya type inequalities via convexity
Saker S. H., Abdou D. M., Kubiaczyk I.
Fasciculi Mathematici
|
2018
|
Nr 60
145--159
EN
In this paper, we prove some new dynamic inequalities related to Opial and Pólya type inequalities on a time scale T. We will derive the integral and discrete inequalities of Polya’s type as special cases and also derive several classical integral inequalities of Opial’s type that has been obtained in the literature as special cases. The main results will be proved by using the chain rule, Hölder’s inequality and Jensen’s inequality, Taylor formula on time scales.
8
Content available remote Generalized weighted Ostrowski, Trapezoid and Grüss type inequalities on time scales
El-Deeb A. A., Elsennary H. A., Nwaeze E. R.
Fasciculi Mathematici
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2018
|
Nr 60
123--144
EN
In this article, using two parameters, we obtain generalizations of a weighted Ostrowski type inequality and its companion inequalities on an arbitrary time scale for functions whose first delta derivatives are bounded. Our work unifies the continuous and discrete versions and can also be applied to the quantum calculus case.
9
Content available remote Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales
Anderson D. R., Onitsuka M.
Demonstratio Mathematica
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2018
|
Vol. 51, nr 1
198--210
EN
We establish the Hyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, which include the continuous (step size zero) and the discrete (step size constant and nonzero) dynamic equations as important special cases. In particular, for certain parameter values in relation to the graininess of the time scale, we find the minimum HUS constants. A few nontrivial examples are provided. Moreover, an application to a perturbed linear dynamic equation is also included.
10
Content available On nonoscillatory solutions of two dimensional nonlinear delay dynamical systems
Ozturk O., Akin E.
Opuscula Mathematica
|
2016
|
Vol. 36, no. 5
651--669
EN
We study the classification schemes for nonoscillatory solutions of a class of nonlinear two dimensional systems of first order delay dynamic equations on time scales. Necessary and sufficient conditions are also given in order to show the existence and nonexis-tence of such solutions and some of our results are new for the discrete case. Examples will be given to illustrate some of our results.
11
Content available Asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations
Bohner M., Grace S., Sultana N.
Opuscula Mathematica
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2014
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Vol. 34, no. 1
5--14
EN
In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales.
12
Content available remote Some new generalizations of Maroni inequality on time scales
Yin L., Zhao C.
Demonstratio Mathematica
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2013
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Vol. 46, nr 4
645--654
EN
The aim of present paper is to establish some new integral inequalities on time scales involving several functions and their derivatives which in the special cases yield the well known Maroni inequality and some of its generalizations.
13
Content available remote Existence of solutions of the dynamic Cauchy problem in Banach spaces
Cichoń M., Kubiaczyk I., Sikorska-Nowak A., Yantir A.
Demonstratio Mathematica
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2012
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Vol. 45, nr 3
561-573
EN
In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales (…), where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.
14
Content available remote Dynamic equations (…) on time scales
Sikorska-Nowak A.
Demonstratio Mathematica
|
2011
|
Vol. 44, nr 2
317-333
EN
In this paper we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem (…), where (…) denotes a mth order (…) - derivative,T denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence (…) E -a Banach space and f is a continuous function or satis .es Carathéodory's conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.
15
Content available Time, its scales and part in satellite navigation systems
Januszewski J.
Zeszyty Naukowe / Akademia Morska w Szczecinie
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2010
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nr 20 (92)
52-59
EN
There are different scales related to time; actually two the most important timescales are the TAI (Time Atomic International) and the UTC (Universal Time Coordinated). In addition to these times Satellite Navigation Systems (SNS) have developed their own system time: GPS Time (GPST), the GLONASS System Time (GLONASSST) and the Galileo System Time (GST). The sources, the generation and the relation between all these times and timescales are described in this paper. Additionally the time dilution of precision (TDOP) coefficient and the data concerning the time transmitted in navigation messages by satellites of different SNS will be presented.
PL
Z czasem związane są różne skale, ale dwie najważniejsze z nich to TAI (Międzynarodowa Skala Atomowa) i UTC (Czas Uniwersalny Skoordynowany). Nawigacyjne Systemy Satelitarne (NSS) stworzyły jednak własne skale czasu: GPST - czas systemu GPS, GLONASSST - czas systemu GLONASS i GST - czas systemu Galileo. W artykule omówiono źródła, pochodzenie i relacje zachodzące między wszystkimi ww. czasami i skalami. Dodatkowo opisano współczynnik dokładności pozycji użytkownika TDOP oraz te parametry dotyczące czasu, które są przekazywane w depeszach nawigacyjnych satelitów poszczególnych systemów.
16
On a nonlinear dynamic integrodifferential equation on time scales
Pachpatte D. B.
Journal of Applied Analysis
|
2010
|
Vol. 16, nr 2
279-294
EN
The main objective of the present paper is to study some basic qualitative properties of solutions of a certain nonlinear integrodifferential equation on time scales. The tools employed in the analysis are based on the applications of the Banach fixed point theorem and a certain inequality with explicit estimate on time scales.
17
Content available remote The delta-nabla calculus of variations
Malinowska A. B., Torres D. F. M.
Fasciculi Mathematici
|
2010
|
Nr 44
75-83
EN
The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we propose a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.
18
Content available remote Short reversals of the earth’s magnetic field and its relative paleointensity variations as chronostratigraphic tools - perspectives and limitations
Nowaczyk Norbert R.
Geologos
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2000
|
Vol. 5
5--12
EN
Geomagnetic variations in direction as well as in intensity throughout the geologic history as documented in the remanent magnetization of sediment successions generally offer a potential (the investigations are performed in correlation) for dating of these sediments. Therefore, magnetostratigraphic investigations are performed in order to monitor geomagnetic fluctuations and to correlate them to established reference datasets such as the geomagnetic polarity time scale. The last major reversal of the earth’s magnetic field occured at about 780 ka. Since then, a quite large number of so-called reversal excursions with a duration of a few thousand years interrupted the normal polarity phase of the geomagnetic field, known as the ’Brunhes Chron’, about each 50 to 100 kyr. According to absolute as well as relative paleointensity determinations the field intensity throughout the Brunhes Chron was not constant but variable at least within the range of one order of magnitude. Especially reversal excursions, as expressed by magnetization directions differing largely from the expected dipole direction for a certain site, are associated with pronounced lows in the field intensity. So, in principle, determination of the sediment’s magnetization in terms of direction and intensity of time-calibrated curves can provide a detailed age model of (late) Quaternary sediments.
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