scholarly journals A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices

2020 ◽  
Vol 81 (4-5) ◽  
pp. 981-1028
Author(s):  
Shin-Ichiro Ei ◽  
Hiroshi Ishii ◽  
Makoto Sato ◽  
Yoshitaro Tanaka ◽  
Miaoxing Wang ◽  
...  
Keyword(s):  
Planar Cell Polarity ◽  
Evolution Equations ◽  
Shift Operator ◽  
Continuation Method ◽  
Continuous Model ◽  
Discrete Models ◽  
Nonlocal Operators ◽  
Size And Shape ◽  
Differentiation Wave ◽  
Singular Limit Analysis

Abstract In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta–Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model.

scholarly journals Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1172 ◽  
Author(s):  
Joel Alba-Pérez ◽  
Jorge E. Macías-Díaz
Keyword(s):  
Anomalous Diffusion ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Continuous Model ◽  
Discrete Models ◽  
Extended Form ◽  
Original Population ◽  
Predator Prey ◽  
Structure Preserving ◽  
Spatial Dimensions

In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator–prey systems with anomalous diffusion.

scholarly journals From the discrete to the continuous coagulation–fragmentation equations

2002 ◽  
Vol 132 (5) ◽  
pp. 1219-1248 ◽  
Author(s):  
Philippe Laurençot ◽  
Stéphane Mischler
Keyword(s):  
A Priori ◽  
Continuous Model ◽  
Weak Compactness ◽  
Discrete Models ◽  
Fragmentation Model ◽  
Weak Stability ◽  
Stability Principle

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

Accuracy of Discrete Models for the Solution of the Inverse Dynamics Problem for Flexible Arms, Feasible Trajectories

1997 ◽  
Vol 119 (3) ◽  
pp. 396-404 ◽  
Author(s):  
H. C. Moulin ◽  
E. Bayo
Keyword(s):  
High Frequency ◽  
Inverse Dynamics ◽  
Numerical Procedure ◽  
Continuous System ◽  
Continuous Model ◽  
Discrete Models ◽  
Frequency Range ◽  
Model Order ◽  
Inverse Dynamics Problem ◽  
Tip Mass

The inverse dynamics problem for a single link flexible arm is considered. The tracking order of consistent and lumped finite element models is derived and compared with the tracking order of the continuous model when there is no tip-mass. These comparisons show that discrete models fail to identify the tracking order of a modelled continuous system. A frequency domain analysis shows that an increase in the model order extends the well-modelled low-frequency range and, at the same time, increases the inadequacy in the high-frequency range. As a result, inverse dynamics solutions computed with discrete models do not converge to the continuous solution as the model order increases. The use of high-frequency filters allows us to construct a convergent numerical procedure. A conjecture about the tracking order is presented when there is a tip mass. It is shown that the same results are obtained if superposition of modes rather than finite elements is used.

scholarly journals Evidence for the continuous latent structure of mania in the Epidemiologic Catchment Area from multiple latent structure and construct validation methodologies

2010 ◽  
Vol 41 (3) ◽  
pp. 575-588 ◽  
Author(s):  
J. J. Prisciandaro ◽  
J. E. Roberts
Keyword(s):  
Predictive Validity ◽  
Catchment Area ◽  
Latent Class ◽  
Continuous Model ◽  
Externalizing Disorders ◽  
Discrete Models ◽  
Latent Structure ◽  
Factor Analyses ◽  
Diagnostic Systems ◽  
Epidemiologic Catchment Area

BackgroundAlthough psychiatric diagnostic systems have conceptualized mania as a discrete phenomenon, appropriate latent structure investigations testing this conceptualization are lacking. In contrast to these diagnostic systems, several influential theories of mania have suggested a continuous conceptualization. The present study examined whether mania has a continuous or discrete latent structure using a comprehensive approach including taxometric, information-theoretic latent distribution modeling (ITLDM) and predictive validity methodologies in the Epidemiologic Catchment Area (ECA) study.MethodEight dichotomous manic symptom items were submitted to a variety of latent structural analyses, including factor analyses, taxometric procedures and ITLDM, in 10105 ECA community participants. In addition, a variety of continuous and discrete models of mania were compared in terms of their relative abilities to predict outcomes (i.e. health service utilization, internalizing and externalizing disorders, and suicidal behavior).ResultsTaxometric and ITLDM analyses consistently supported a continuous conceptualization of mania. In ITLDM analyses, a continuous model of mania demonstrated 6.52:1 odds over the best-fitting latent class model (LCM) of mania. Factor analyses suggested that the continuous structure of mania was best represented by a single latent factor. Predictive validity analyses demonstrated a consistent superior ability of continuous models of mania relative to discrete models.ConclusionsThe present study provided three independent lines of support for a continuous conceptualization of mania. The implications of a continuous model of mania are discussed.

scholarly journals Integrated Heat and Electricity Dispatch for District Heating Networks with Constant Mass Flow: A Generalized Phasor Method

2020 ◽  
Author(s):  
Yuwei Chen ◽  
Qinglai Guo ◽  
Hongbin Sun ◽  
Zhaoguang Pan
Keyword(s):  
Mass Flow ◽  
Electrical Power ◽  
Thermal Inertia ◽  
District Heating ◽  
Continuous Model ◽  
Discrete Models ◽  
Convex Programming Problem ◽  
Algebraic Equations ◽  
Constant Mass ◽  
Electrical Power System

Using thermal inertia in district heating systems (DHSs) to improve the dispatch flexibility and economy of integrated heat and electricity systems (IHESs) is a research hotspot and difficulty. In most existing studies, the partial differential equations (PDEs) of thermal inertia are approximated by discrete-time models, making it difficult to accurately describe the continuous dynamic processes. In this paper, we propose a novel generalized phasor method (GPM) for thermal inertia in DHSs with constant mass flow. Based on the analytical solution of the PDEs and the Fourier transform, the intractable PDEs are transformed into a series of complex algebraic equations represented by phasors. The GPM has higher accuracy compared to traditional discrete models because it is essentially a continuous model in the time domain. Then, we present a different representation of an integrated heat and electricity dispatch (IHED) model combining a DHS model in phasor form and a traditional electrical power system model. The IHED model is a convex programming problem and can be easily solved. The effectiveness of the proposed GPM and dispatch model is verified in three test systems. Compared with traditional methods for modeling the thermal inertia, the proposed GPM is more accurate.

scholarly journals From the discrete to the continuous coagulation–fragmentation equations

2002 ◽  
Vol 132 (5) ◽  
pp. 1219-1248 ◽  
Author(s):  
Philippe Laurençot ◽  
Stéphane Mischler
Keyword(s):  
A Priori ◽  
Continuous Model ◽  
Weak Compactness ◽  
Discrete Models ◽  
Fragmentation Model ◽  
Weak Stability ◽  
Stability Principle

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

scholarly journals Integrated Heat and Electricity Dispatch for District Heating Networks with Constant Mass Flow: A Generalized Phasor Method

2020 ◽  
Author(s):  
Yuwei Chen ◽  
Qinglai Guo ◽  
Hongbin Sun ◽  
Zhaoguang Pan
Keyword(s):  
Mass Flow ◽  
Electrical Power ◽  
Thermal Inertia ◽  
District Heating ◽  
Continuous Model ◽  
Discrete Models ◽  
Convex Programming Problem ◽  
Algebraic Equations ◽  
Constant Mass ◽  
Electrical Power System

Using thermal inertia in district heating systems (DHSs) to improve the dispatch flexibility and economy of integrated heat and electricity systems (IHESs) is a research hotspot and difficulty. In most existing studies, the partial differential equations (PDEs) of thermal inertia are approximated by discrete-time models, making it difficult to accurately describe the continuous dynamic processes. In this paper, we propose a novel generalized phasor method (GPM) for thermal inertia in DHSs with constant mass flow. Based on the analytical solution of the PDEs and the Fourier transform, the intractable PDEs are transformed into a series of complex algebraic equations represented by phasors. The GPM has higher accuracy compared to traditional discrete models because it is essentially a continuous model in the time domain. Then, we present a different representation of an integrated heat and electricity dispatch (IHED) model combining a DHS model in phasor form and a traditional electrical power system model. The IHED model is a convex programming problem and can be easily solved. The effectiveness of the proposed GPM and dispatch model is verified in three test systems. Compared with traditional methods for modeling the thermal inertia, the proposed GPM is more accurate.

scholarly journals Integrated Heat and Electricity Dispatch for District Heating Networks with Constant Mass Flow: A Generalized Phasor Method

2020 ◽  
Author(s):  
Yuwei Chen ◽  
Qinglai Guo ◽  
Hongbin Sun ◽  
Zhaoguang Pan
Keyword(s):  
Mass Flow ◽  
Electrical Power ◽  
Thermal Inertia ◽  
District Heating ◽  
Continuous Model ◽  
Discrete Models ◽  
Convex Programming Problem ◽  
Algebraic Equations ◽  
Constant Mass ◽  
Electrical Power System

Using thermal inertia in district heating systems (DHSs) to improve the dispatch flexibility and economy of integrated heat and electricity systems (IHESs) is a research hotspot and difficulty. In most existing studies, the partial differential equations (PDEs) of thermal inertia are approximated by discrete-time models, making it difficult to accurately describe the continuous dynamic processes. In this paper, we propose a novel generalized phasor method (GPM) for thermal inertia in DHSs with constant mass flow. Based on the analytical solution of the PDEs and the Fourier transform, the intractable PDEs are transformed into a series of complex algebraic equations represented by phasors. The GPM has higher accuracy compared to traditional discrete models because it is essentially a continuous model in the time domain. Then, we present a different representation of an integrated heat and electricity dispatch (IHED) model combining a DHS model in phasor form and a traditional electrical power system model. The IHED model is a convex programming problem and can be easily solved. The effectiveness of the proposed GPM and dispatch model is verified in three test systems. Compared with traditional methods for modeling the thermal inertia, the proposed GPM is more accurate.

Modal interactions and energy transfers in large-amplitude vibrations of functionally graded microcantilevers

2017 ◽  
Vol 24 (17) ◽  
pp. 3882-3893 ◽  
Author(s):  
Hamed Farokhi ◽  
Mergen H Ghayesh ◽  
Alireza Gholipour ◽  
Shahid Hussain
Keyword(s):  
Large Amplitude ◽  
Nonlinear Oscillations ◽  
Couple Stress Theory ◽  
Continuation Method ◽  
Continuous Model ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Modal Interactions ◽  
Energy Transfers ◽  
Highly Nonlinear

Modal interactions and internal energy transfers are investigated in the large-amplitude oscillations of a functionally graded microcantilever with an intermediate spring-support. Based on the Mori–Tanaka homogenization technique and the modified couple stress theory, the energy terms of the functionally graded microsystem (kinetic and size-dependent potential energies) are developed and dynamically balanced. Large-amplitude deformations, due to having one end free, are modeled taking into account curvature-related nonlinearities and assuming an inextensibility condition. The continuous model of the functionally graded microsystem is reduced, by means of the Galerkin method, yielding an inertial- and stiffness-wise nonlinear model. Numerical simulations on this highly nonlinear reduced-order model of the functionally graded microcantilever are performed using a continuation method; a possible case of modal interactions is determined by obtaining the natural frequencies of the microsystem. The nonlinear oscillations of the microcantilever are examined, and it is shown how the energy fed to the functionally graded microsystem (from the base excitation) is transferred between different modes of oscillation.

scholarly journals A Consistent Discrete Version of a Nonautonomous SIRVS Model

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Joaquim Mateus ◽  
César M. Silva ◽  
S. Vaz
Keyword(s):  
Discrete Model ◽  
Real Data ◽  
Continuous Model ◽  
Discrete Models ◽  
Discrete Version ◽  
Continuous Family ◽  
Discretization Method ◽  
Time Step ◽  
The Stability ◽  
Disease Free

A family of discrete nonautonomous SIRVS models with general incidence is obtained from a continuous family of models by applying Mickens nonstandard discretization method. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. Concerning extinction and persistence, the consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (permanence) for the continuous model, we also have extinction (permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data.

Sign in / Sign up

Export Citation Format

Share Document