scholarly journals Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions

2012 ◽  
Vol 44 (3) ◽  
pp. 749-773 ◽  
Author(s):  
Alexandre Genadot ◽  
Michèle Thieullen
Keyword(s):  
Ion Channels ◽  
Markov Process ◽  
Time Scale ◽  
Evolution Equations ◽  
Scale Model ◽  
Infinite Dimensions ◽  
Fast Analysis ◽  
Piecewise Deterministic Markov Process ◽  
Averaged Model ◽  
Fully Coupled

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

scholarly journals Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions

2012 ◽  
Vol 44 (03) ◽  
pp. 749-773 ◽  
Author(s):  
Alexandre Genadot ◽  
Michèle Thieullen
Keyword(s):  
Ion Channels ◽  
Markov Process ◽  
Time Scale ◽  
Evolution Equations ◽  
Scale Model ◽  
Infinite Dimensions ◽  
Fast Analysis ◽  
Piecewise Deterministic Markov Process ◽  
Averaged Model ◽  
Fully Coupled

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

MMC-LES of a syngas mixing layer using an anisotropic mixing time scale model

2018 ◽  
Vol 189 ◽  
pp. 311-314 ◽  
Author(s):  
Son Vo ◽  
Andreas Kronenburg ◽  
Oliver T. Stein ◽  
Matthew J. Cleary
Keyword(s):  
Time Scale ◽  
Mixing Time ◽  
Mixing Layer ◽  
Scale Model ◽  
Mixing Time Scale

scholarly journals Fully Coupled Multi-Scale Model for Gas Extraction from Coal Seam Stimulated by Directional Hydraulic Fracturing

2019 ◽  
Vol 9 (21) ◽  
pp. 4720 ◽  
Author(s):  
Ge ◽  
Zhang ◽  
Sun ◽  
Hu
Keyword(s):  
Coal Seam ◽  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Scale Model ◽  
Gas Extraction ◽  
Coal Seams ◽  
Directional Hydraulic Fracturing ◽  
Coal Mine Methane ◽  
Multi Scale ◽  
Fully Coupled

Although numerous studies have tried to explain the mechanism of directional hydraulic fracturing in a coal seam, few of them have been conducted on gas migration stimulated by directional hydraulic fracturing during coal mine methane extraction. In this study, a fully coupled multi-scale model to stimulate gas extraction from a coal seam stimulated by directional hydraulic fracturing was developed and calculated by a finite element approach. The model considers gas flow and heat transfer within the hydraulic fractures, the coal matrix, and cleat system, and it accounts for coal deformation. The model was verified using gas amount data from the NO.8 coal seam at Fengchun mine, Chongqing, Southwest China. Model simulation results show that slots and hydraulic fracture can expand the area of gas pressure drop and decrease the time needed to complete the extraction. The evolution of hydraulic fracture apertures and permeability in coal seams is greatly influenced by the effective stress and coal matrix deformation. A series of sensitivity analyses were performed to investigate the impacts of key factors on gas extraction time of completion. The study shows that hydraulic fracture aperture and the cleat permeability of coal seams play crucial roles in gas extraction from a coal seam stimulated by directional hydraulic fracturing. In addition, the reasonable arrangement of directional boreholes could improve the gas extraction efficiency. A large coal seam dip angle and high temperature help to enhance coal mine methane extraction from the coal seam.

Adaptive Decentralized Tracking Control for Nonlinear Large-Scale Systems

2020 ◽  
Vol 28 (Supp02) ◽  
pp. 93-110
Author(s):  
Karthik Chandran ◽  
Weidong Zhang ◽  
Rajalakshmi Murugesan ◽  
S. Prasanna ◽  
A. Baseera ◽  
...  
Keyword(s):  
Time Scale ◽  
Large Scale ◽  
State Feedback ◽  
Reference Model ◽  
Linear Time ◽  
State Feedback Control ◽  
Scale Model ◽  
Model Reference ◽  
Model Reference Adaptive ◽  
Decentralized Model

Decentralized Model Reference Adaptive Control problems are investigated for a class of linear time-invariant two time-scale model, having fast and slow dynamics and unmatched interconnections. Design of full state feedback controller is a critical task to the system having interrelated dynamics and nonlinear interconnections of time varying lags, however, can be addressed by singular perturbation procedures and time scale modeling. In this research, full order observer-based state feedback control is designed to two-time scale system to ensure the stability of the closed loop system. Then, decentralized model reference adaptive controller with the novel reference model is designed to individual subsystems. It is found that the proposed design enforces the system states to track the reference state asymptotically. To investigate the proposed design, an example of drying process is considered. Simulation results are analyzed to confirm the efficacy of the proposed control scheme.

scholarly journals Simplified stochastic soil-moisture models: a look at infiltration

2006 ◽  
Vol 10 (6) ◽  
pp. 861-871 ◽  
Author(s):  
J. R. Rigby ◽  
A. Porporato
Keyword(s):  
Soil Moisture ◽  
Time Scale ◽  
Rectangular Pulse ◽  
Short Time Scale ◽  
Soil Moisture Dynamics ◽  
Physically Based ◽  
Probabilistic Structure ◽  
Averaged Model ◽  
Moisture Dynamics ◽  
Daily Time

Abstract. A simplified, vertically-averaged model of soil moisture interpreted at the daily time scale and forced by a stochastic process of instantaneous rainfall events is compared with a vertically-averaged model which uses a non-overlapping rectangular pulse rainfall model and a more physically based description of infiltration. The models are compared with respect to the importance of short time-scale (intra-storm) variable infiltration in determining the probabilistic structure of soil-moisture dynamics at the daily time-scale. Differences in approach to infiltration modelling show only minor effects on the probabilistic structure of soil-moisture dynamics as simulated in the two models. The partitioning of losses during a single rainfall event are examined closely and the conditions under which surface-controlled runoff is significant, as a proportion of total losses, are delineated.

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