FOUNDATIONS OF FRACTIONAL DYNAMICS

Fractals ◽  
1995 ◽  
Vol 03 (03) ◽  
pp. 549-556 ◽  
Author(s):  
R. HILFER
Keyword(s):  
Measure Zero ◽  
Time Derivative ◽  
Time Limit ◽  
Fractional Dynamics ◽  
Equilibrium Concept ◽  
Time Step ◽  
Time Flow ◽  
Generalized Equilibrium ◽  
Derivatives Of ◽  
Insight Into

Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated by fractional time derivatives of orders less than unity, and not by a first order time derivative. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractionally stationary states are dissipative. Fractional stationarity also provides the dynamical foundation for a previously proposed generalized equilibrium concept.

On the use of jerk and snap in condition monitoring of machinery – review and case studies

2021 ◽  
Vol 63 (8) ◽  
pp. 457-464
Author(s):  
S Lahdelma
Keyword(s):  
Condition Monitoring ◽  
Large Scale ◽  
Early Stage ◽  
Time Derivative ◽  
Additional Insight ◽  
Rolling Bearings ◽  
Monitoring Applications ◽  
Derivatives Of ◽  
Insight Into ◽  
Peak Value

The time derivatives of acceleration offer a great advantage in detecting impact-causing faults at an early stage in condition monitoring applications. Defective rolling bearings and gears are common faults that cause impacts. This article is based on extensive real-world measurements, through which large-scale machines have been studied. Numerous laboratory experiments provide additional insight into the matter. A practical solution for detecting faults with as few features as possible is to measure the root mean square (RMS) velocity according to the standards in the frequency range from 10 Hz to 1000 Hz and the peak value of the second time derivative of acceleration, ie snap. Measuring snap produces good results even when the upper cut-off frequency is as low as 2 kHz or slightly higher. This is valuable information when planning the mounting of accelerometers.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

scholarly journals Insight into the Secondary Metabolites of Geum urbanum L. and Geum rivale L. Seeds (Rosaceae)

Plants ◽  
2021 ◽  
Vol 10 (6) ◽  
pp. 1219
Author(s):  
Marek Bunse ◽  
Peter Lorenz ◽  
Florian C. Stintzing ◽  
Dietmar R. Kammerer
Keyword(s):  
Fatty Acids ◽  
Phenolic Compounds ◽  
Unsaturated Fatty Acids ◽  
Asiatic Acid ◽  
Cosmetic Products ◽  
Nutritional Properties ◽  
First Time ◽  
Derivatives Of ◽  
Geum Urbanum ◽  
Insight Into

The present study aimed at the identification and quantitation of phenolic compounds, fatty acids, and further characteristic substances in the seeds of Geum urbanum L. and Geum rivale L. For this purpose, individual components of extracts recovered with MeOH, CH2Cl2, and by cold-pressing, respectively, were characterized by HPLC-DAD/ESI-MSn and GC/MS and compared with reference compounds. For both Geum species, phenolic compounds, such as flavonoids and gallic acid derivatives, and triterpenes, such as saponins and their aglycones, were detected. Surprisingly, both Geum species revealed the presence of derivatives of the triterpenoid aglycons asiatic acid and madecassic acid, which were characterized for the first time in the genus Geum. Furthermore, the fatty acids of both species were characterized by GC–MS after derivatization. Both species showed a promising fatty-acid profile in terms of nutritional properties because of high proportions of unsaturated fatty acids. Linoleic acid and linolenic acid were most abundant, among other compounds such as palmitic acid and stearic acid. In summary, the present study demonstrates the seeds of G. urbanum and G. rivale to be a valuable source of unsaturated fatty acids and bioactive phenolics, which might be exploited for nutritional and cosmetic products and for phytotherapeutic purposes.

Fast Conjugate Heat Transfer Simulation of Long Transient Flexible Operations Using Adaptive Time Stepping

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
R. Maffulli ◽  
L. He ◽  
P. Stein ◽  
G. Marinescu
Keyword(s):  
Heat Transfer ◽  
Conjugate Heat Transfer ◽  
Time Derivative ◽  
Computational Cost ◽  
Strong Impact ◽  
Time Step ◽  
Time Stepping ◽  
Adaptive Time ◽  
Fully Coupled ◽  
Adaptive Time Stepping

The emerging renewable energy market calls for more advanced prediction tools for turbine transient operations in fast startup/shutdown cycles. Reliable numerical analysis of such transient cycles is complicated by the disparity in time scales of the thermal responses in fluid and solid domains. Obtaining fully coupled time-accurate unsteady conjugate heat transfer (CHT) results under these conditions would require to march in both domains using the time-step dictated by the fluid domain: typically, several orders of magnitude smaller than the one required by the solid. This requirement has strong impact on the computational cost of the simulation as well as being potentially detrimental to the accuracy of the solution due to accumulation of round-off errors in the solid. A novel loosely coupled CHT methodology has been recently proposed, and successfully applied to both natural and forced convection cases that remove these requirements through a source-term based modeling (STM) approach of the physical time derivative terms in the relevant equations. The method has been shown to be numerically stable for very large time steps with adequate accuracy. The present effort is aimed at further exploiting the potential of the methodology through a new adaptive time stepping approach. The proposed method allows for automatic time-step adjustment based on estimating the magnitude of the truncation error of the time discretization. The developed automatic time stepping strategy is applied to natural convection cases under long (2000 s) transients: relevant to the prediction of turbine thermal loads during fast startups/shutdowns. The results of the method are compared with fully coupled unsteady simulations showing comparable accuracy with a significant reduction of the computational costs.

scholarly journals ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xingjie Helen Li ◽  
Fei Lu ◽  
Felix X.-F. Ye
Keyword(s):  
Large Time ◽  
Gradient System ◽  
Numerical Schemes ◽  
Time Step ◽  
Inference Problem ◽  
Time Dynamics ◽  
Efficient Simulation ◽  
Time Flow ◽  
Ergodic Measures ◽  
Locally Lipschitz

<p style='text-indent:20px;'>Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.</p><p style='text-indent:20px;'>We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.</p>

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

2011 ◽  
Vol 14 (07) ◽  
pp. 979-1004
Author(s):  
CLAUDIO ALBANESE
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Fourier Transforms ◽  
Uniform Continuity ◽  
Time Step ◽  
Uniform Bounds ◽  
Kernel Convergence ◽  
Discretization Schemes ◽  
A New Technique ◽  
Derivatives Of

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

scholarly journals Improving weather and climate predictions by training of supermodels

2019 ◽  
Author(s):  
Francine Schevenhoven ◽  
Frank Selten ◽  
Alberto Carrassi ◽  
Noel Keenlyside
Keyword(s):  
Time Derivative ◽  
Learning Rule ◽  
Training Methods ◽  
Model Errors ◽  
Climate Simulations ◽  
Weather And Climate ◽  
Present Context ◽  
Training Schemes ◽  
The Individual ◽  
Derivatives Of

Abstract. Recent studies demonstrate that weather and climate predictions potentially improve by dynamically combining different models into a so called "supermodel". Here we focus on the weighted supermodel – the supermodel's time derivative is a weighted superposition of the time-derivatives of the imperfect models, referred to as weighted supermodeling. A crucial step is to train the weights of the supermodel on the basis of historical observations. Here we apply two different training methods to a supermodel of up to four different versions of the global atmosphere-ocean-land model SPEEDO. The standard version is regarded as truth. The first training method is based on an idea called Cross Pollination in Time (CPT), where models exchange states during the training. The second method is a synchronization based learning rule, originally developed for parameter estimation. We demonstrate that both training methods yield climate simulations and weather predictions of superior quality as compared to the individual model versions. Supermodel predictions also outperform predictions based on the commonly used Multi-Model Ensemble (MME) mean. Furthermore we find evidence that negative weights can improve predictions in cases where model errors do not cancel (for instance all models are warm with respect to the truth). In principle the proposed training schemes are applicable to state-of-the-art models and historical observations. A prime advantage of the proposed training schemes is that in the present context relatively short training periods suffice to find good solutions. Additional work needs to be done to assess the limitations due to incomplete and noisy data, to combine models that are structurally different (different resolution and state representation for instance) and to evaluate cases for which the truth falls outside of the model class.

scholarly journals Insight into the Presence of Stilbenes in Medicinal Plants Traditionally Used in Croatian Folk Medicine

2016 ◽  
Vol 11 (6) ◽  
pp. 1934578X1601100 ◽  
Author(s):  
Ivana Generalić Mekinić ◽  
Danijela Skroza ◽  
Ivica Ljubenkov ◽  
Višnja Katalinić
Keyword(s):  
Biological Activity ◽  
Medicinal Plants ◽  
Biological Activities ◽  
Folk Medicine ◽  
Free Form ◽  
Naturally Occurring ◽  
Derivatives Of ◽  
Insight Into ◽  
Phenolic Extracts

Over the last years, great interest has arisen concerning plant stilbenes, especially resveratrol, which has a whole spectrum of positive biological activities. In this study, we investigated the presence of resveratrol monomers (trans- and cis- form) and naturally occurring derivatives of fraas-resveratrol (piceid, astringin and isorhapontin) in phenolic extracts of twenty medicinal plants traditionally used in Croatian folk medicine. The investigated compounds were present in the samples, in free form or as glucosides, and the highest share was found in immortelle, common yarrow and Lamiaceae plants. The obtained results indicate that biological activity of selected medicinal plants can be related to the presence of this valuable group of phytochemicals.

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