Problems with different time scales

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 101-139 ◽  
Author(s):  
Heinz-Otto Kreiss
Keyword(s):  
Time Scales ◽  
Complex Number ◽  
Initial Value Problem ◽  
Time Scale ◽  
Homogeneous Equation ◽  
Simple Problem ◽  
Initial Value ◽  
Small Constant ◽  
Image Position ◽  
Different Time Scales

In this section we discuss a very simple problem. Consider the scalar initial value problemHere ε > 0 is a small constant and a = a1 + ia2, a1, a2 real, is a complex number with |a| = 1. We can write down the solution of (1.1) explicity. It iswhereis the forced solution andis a solution of the homogeneous equationyS varies on the time scale ‘1’ while yF varies on the much faster scale 1/ε. We say that yS, yF vary on the slow and fast scale, respectively. We use also the phrase: yS and yF are the slow and the fast part of the solution, respectively.

scholarly journals The Convolution on Time Scales

2007 ◽  
Vol 2007 ◽  
pp. 1-24 ◽  
Author(s):  
Martin Bohner ◽  
Gusein Sh. Guseinov
Keyword(s):  
Power Series ◽  
Transport Equation ◽  
Time Scales ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Time Scale ◽  
Convolution Theorem ◽  
Main Theme ◽  
Initial Value ◽  
Dynamic Version

The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. Via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. As an extensive example, we consider theq-difference equations case.

scholarly journals New properties of the time-scale fractional operators with application to dynamic equations

2021 ◽  
Vol 25 (1) ◽  
pp. 123-136
Author(s):  
Cherif Benaissa ◽  
Ladrani Zohra
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Initial Value Problem ◽  
Time Scale ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Fractional Operators ◽  
Initial Value

We introduce new properties of Riemann-Liouville fractional integral and derivative on time scales. As well as sufficient conditions for existence and uniqueness of solution to an initial value problem for a class differential equations on time scales.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

On Nagumo's Condition

1972 ◽  
Vol 15 (4) ◽  
pp. 609-611 ◽  
Author(s):  
Thomas Rogers
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Uniqueness Theorem ◽  
Initial Value Problem ◽  
Initial Value ◽  
Image Position

The classical uniqueness theorem of Nagumo [1] for ordinary differential equations is as follows.Theorem. If f(t, y) is continuous on 0≤t≤1, -∞<y<∞ and ifthen there is at most one solution to the initial value problem y'=f(t, y), y(0)=0.

scholarly journals Stationarity Testing of Accumulated Ethernet Traffic

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zhiping Lu ◽  
Ming Li ◽  
Wei Zhao
Keyword(s):  
Time Scales ◽  
Unit Root ◽  
Time Scale ◽  
Unit Root Tests ◽  
Quantitative Results ◽  
Different Time Scales

We investigate the stationarity property of the accumulated Ethernet traffic series. We applied several widely used stationarity and unit root tests, such as Dickey-Fuller test and its augmented version, Phillips-Perron test, as well as the Kwiatkowski-Phillips-Schmidt-Shin test and some of its generalizations, to the assessment of the stationarity of the traffic traces at the different time scales. The quantitative results in this research provide evidence that when the time scale increases, the accumulated traffic series are more stationary.

L∞C(ℝn)-decay of classical solutions for nonlinear Schrödinger equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 309-327 ◽  
Author(s):  
Nakao Hayashi ◽  
Masayoshi Tsutsumi
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Classical Solutions ◽  
Initial Value ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Sign Conditions ◽  
Image Position

SynopsisWe study the initial value problem for the nonlinear Schrödinger equationUnder suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.

EMERGING PROPERTIES IN POPULATION DYNAMICS WITH DIFFERENT TIME SCALES

1995 ◽  
Vol 03 (02) ◽  
pp. 591-602 ◽  
Author(s):  
PIERRE AUGER ◽  
JEAN-CHRISTOPHE POGGIALE
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Time Scale ◽  
Population Level ◽  
Nonlinear Process ◽  
Linear Process ◽  
Fast Time ◽  
Slow Part ◽  
Spatial Patches ◽  
Different Time Scales

The aim of this work is to show that at the population level, emerging properties may occur as a result of the coupling between the fast micro-dynamics and the slow macrodynamics. We studied a prey-predator system with different time scales in a heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to the growth and the interactions between the species. Preys go on the spatial patches on which some resources are located and can be caught by the predators on them. The efficiency of the predators to catch preys is patch-dependent. Preys can be more easily caught on some spatial patches than others. Perturbation theory is used in order to aggregate the initial system of ordinary differential equations for the patch sub-populations into a macro-system of two differential equations governing the total populations. Firstly, we study the case of a linear process of migration for which the aggregated system is formally identical to the slow part of the full system. Then, we study an example of a nonlinear process of migration. We show that under these conditions emerging properties appear at the population level.

scholarly journals Multi-Time Scale Economic Optimization Dispatch of the Park Integrated Energy System

2021 ◽  
Vol 9 ◽  
Author(s):  
Peng Li ◽  
Fan Zhang ◽  
Xiyuan Ma ◽  
Senjing Yao ◽  
Zhuolin Zhong ◽  
...  
Keyword(s):  
Renewable Energy ◽  
Time Scales ◽  
Time Scale ◽  
Energy Systems ◽  
Energy System ◽  
Optimal Operation ◽  
Utilization Efficiency ◽  
Integrated Energy System ◽  
Operation Cost ◽  
Different Time Scales

The park integrated energy system (PIES) plays an important role in realizing sustainable energy development and carbon neutral. Furthermore, its optimization dispatch can improve the energy utilization efficiency and reduce energy systems operation cost. However, the randomness and volatility of renewable energy and the instability of load all bring challenges to its optimal operation. An optimal dispatch framework of PIES is proposed, which constructs the operation models under three different time scales, including day-ahead, intra-day and real-time. Demand response is also divided into three levels considering its response characteristics and cost composition under different time scales. The example analysis shows that the multi-time scale optimization dispatch model can not only meet the supply and demand balance of PIES, diminish the fluctuation of renewable energy and flatten load curves, but also reduce the operation cost and improve the reliability of energy systems.

scholarly journals Modelling of UV radiation variations at different time scales

2008 ◽  
Vol 26 (3) ◽  
pp. 441-446 ◽  
Author(s):  
J. L. Borkowski
Keyword(s):  
Uv Radiation ◽  
Time Scales ◽  
Time Scale ◽  
Global Radiation ◽  
Regression Method ◽  
Ozone Content ◽  
Explanatory Variables ◽  
Solar Uv ◽  
Solar Global Radiation ◽  
Different Time Scales

Abstract. Solar UV radiation variability in the period 1976–2006 is discussed with respect to the relative changes in the solar global radiation, ozone content, and cloudiness. All the variables were decomposed into separate components, representing variations of different time scales, using wavelet multi-resolution decomposition. The response of the UV radiation to the changes in the solar global radiation, ozone content, and cloudiness depends on the time scale, therefore, it seems reasonable to model separately the relation between UV and explanatory variables at different time scales. The wavelet components of the UV series are modelled and summed to obtain the fit of observed series. The results show that the coarser time scale components can be modelled with greater accuracy than fine scale components and the fitted values calculated by this method are in better agreement with observed values than values calculated by the regression method, in which variables were not decomposed. The residual standard error in the case of modelling with the use of wavelets is reduced by 14% in comparison to the regression method without decomposition.

Data-driven stochastic model for cross-interacting processes with different time scales

2021 ◽  
Author(s):  
Andrey Gavrilov ◽  
Aleksei Seleznev ◽  
Dmitry Mukhin ◽  
Alexander Feigin
Keyword(s):  
Time Series ◽  
Stochastic Model ◽  
Time Scales ◽  
Time Scale ◽  
Sampling Interval ◽  
Bayesian Optimization ◽  
Data Driven ◽  
Middle Pleistocene ◽  
Model Based ◽  
Different Time Scales

&lt;p&gt;The problem of modeling interaction between processes with different time scales is very important in geoscience. In this report, we propose a new form of empirical evolution operator model based on the analysis of multiple time series representing processes with different time scales. We assume that the time series are given on the same time interval.&lt;/p&gt;&lt;p&gt;To construct the model, we extend the previously developed general form of nonlinear stochastic model based on artificial neural networks and designed for the case of time series with constant sampling interval [1]. This sampling interval is related to the main time scale of the process under consideration, which is described by the deterministic component of the model, while the faster time scales are modeled by its stochastic component, possibly depending on the system&amp;#8217;s state. This model also includes slower processes in the form of weak time-dependence, as well as external forcing. The structure of the model is optimized using Bayesian approach [1]. The model has proven its efficiency in a number of applications [2-4].&lt;/p&gt;&lt;p&gt;The idea of modeling time series with different time scales is to formulate the above-described model individually for each time scale, and then to include the parameterized influence of the other time scales in it. Particularly, the influence of &amp;#8220;slower&amp;#8221; time series is included in the form of parameter trends, and the influence of &amp;#8220;faster&amp;#8221; time series is included by time-averaging their statistics. The algorithm and first results of comparison between the new model and the model without cross-interactions will be discussed.&lt;/p&gt;&lt;p&gt;The work was supported by the Russian Science Foundation (Grant No. 20-62-46056).&lt;/p&gt;&lt;p&gt;1. Gavrilov, A., Loskutov, E., &amp; Mukhin, D. (2017). Bayesian optimization of empirical model with state-dependent stochastic forcing. Chaos, Solitons &amp; Fractals, 104, 327&amp;#8211;337. http://doi.org/10.1016/j.chaos.2017.08.032&lt;/p&gt;&lt;p&gt;2. Mukhin, D., Kondrashov, D., Loskutov, E., Gavrilov, A., Feigin, A., &amp; Ghil, M. (2015). Predicting Critical Transitions in ENSO models. Part II: Spatially Dependent Models. Journal of Climate, 28(5), 1962&amp;#8211;1976. http://doi.org/10.1175/JCLI-D-14-00240.1&lt;/p&gt;&lt;p&gt;3. Gavrilov, A., Seleznev, A., Mukhin, D., Loskutov, E., Feigin, A., &amp; Kurths, J. (2019). Linear dynamical modes as new variables for data-driven ENSO forecast. Climate Dynamics, 52(3&amp;#8211;4), 2199&amp;#8211;2216. http://doi.org/10.1007/s00382-018-4255-7&lt;/p&gt;&lt;p&gt;4. Mukhin, D., Gavrilov, A., Loskutov, E., Kurths, J., &amp; Feigin, A. (2019). Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition. Scientific Reports, 9(1), 7328. http://doi.org/10.1038/s41598-019-43867-3&lt;/p&gt;

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