scholarly journals On the existence of solutions for the Frenkel-Kontorova models on quasi-crystals

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jianxing Du ◽  
Xifeng Su
Keyword(s):  
Rotation Number ◽  
Numerical Experiments ◽  
Multiple Equilibria ◽  
Existence Of Solutions ◽  
Equilibrium Configuration ◽  
Dimensional Model ◽  
One Dimensional ◽  
Minimal Configuration ◽  
Quasi Crystals

<p style='text-indent:20px;'>This article focuses on recent investigations on equilibria of the Frenkel-Kontorova models subjected to potentials generated by quasi-crystals.</p><p style='text-indent:20px;'>We present a specific one-dimensional model with an explicit potential driven by the Fibonacci quasi-crystal. For a given positive number <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>, we show that there are multiple equilibria with rotation number <inline-formula><tex-math id="M2">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>, e.g., a minimal configuration and a non-minimal equilibrium configuration. Some numerical experiments verifying the existence of such equilibria are provided.</p>

scholarly journals Kompiuterinis biojutiklių su perforuota ir selektyvia membrana modeliavimas vienmačiu keturių sluoksnių modeliu

2009 ◽  
Vol 50 ◽  
pp. 328-333
Author(s):  
Karolis Petrauskas
Keyword(s):  
Numerical Experiments ◽  
Enzymatic Reaction ◽  
Diffusion Reaction ◽  
Homogeneous Layer ◽  
Dimensional Model ◽  
One Dimensional ◽  
Proposed Model ◽  
The One ◽  
Menten Kinetic ◽  
Reaction Equations

Straipsnyje pateikiamas vienmatis biojutiklio su perforuota ir selektyvia membrana modelis. Šis modelis sudarytas pakeičiant perforuotą membraną dviem homogeniškais sluoksniais atitinkamai membranos dalims, kur skylutės yra užpildytos fermento ir kur fermento nėra. Pasiūlytas modelis buvo ištirtas vykdant skaitinius eksperimentus, kad būtų nustatytos sąlygos, kuriomis jis gali būti taikomas tiksliam biojutiklio veiksmo modeliavimui. Šio modelio tikslumas buvo vertinamas lyginant juo gaunamus rezultatus su dvimačio modelio rezultatais. Pasiūlyto modelio rezultatai taip pat buvo palyginti su vienmačio modelio, kuriame perforuota membrana pakeičiama vienu homogenišku sluoksniu, rezultatais. Biojutiklis buvo modeliuojamas reakcijos-difuzijos lygtimis su netiesiniu nariu, aprašančiu fermentinės reakcijos Michaelio–Menteno kinetiką. Modelio lygčių sistema buvo sprendžiama skaitiškai, naudojant baigtinių skirtumų metodą.Computer-Aided Modeling of a Biosensor with Selective and Perforated Membranes Using a Four-Layered One-Dimensional ModelKarolis Petrauskas SummaryThis article presents a one-dimensional model for a biosensor with perforated and selective membranes. This model is constructed by replacing the perforated membrane with two homogeneous layers. These layers are used to model parts of the perforated membrane, where holes are fi lled with an enzyme and where is no enzyme in the holes, separately. The proposed model was investigated by performing numerical experiments in order to determine conditions, under which the proposed model can be used to simulate an operation of a biosensor with an outer perforated membrane precisely. A preciseness of the model was measured by comparing its results with results of the corresponding two-dimensional model. Beside the measurement of the preciseness, results of the proposed model were compared to the results of the one-dimensional model, constructed by replacing the perforated membrane with one homogeneous layer. A biosensor was modeled using diffusion-reaction equations with a nonlinear member representing the Michaelis-Menten kinetic of an enzymatic reaction. These equations were solved numerically, using the method of fi nite differences.: 18px;"> 

Analysis of Stick-Slip Motion by the Rate-Dependent Friction Model

2008 ◽  
Vol 33-37 ◽  
pp. 867-874 ◽  
Author(s):  
S. Ozaki ◽  
Koichi Hashiguchi ◽  
D.H. Chen
Keyword(s):  
Numerical Experiments ◽  
Friction Model ◽  
Dimensional Model ◽  
Mass System ◽  
Stick Slip ◽  
One Dimensional ◽  
Rate Dependent ◽  
The One ◽  
Stick Slip Motion ◽  
Slip Motion

In this study, the rate-dependent subloading-friction model, which can rationally describe the reciprocal transition of static-kinetic frictions by the unified formulation, is proposed. Then, the one-dimensional model of spring-mass system is implemented by incorporating the present friction model, and is applied to simulations of stick-slip motion. Further, we verified the validity of the present approach for the stick-slip motion by numerical experiments under various dynamic conditions.

Modelling Techniques in Applied Glaciology: Numerical Modelling of Avalanches

1983 ◽  
Vol 4 ◽  
pp. 297-297
Author(s):  
G. Brugnot
Keyword(s):  
Finite Difference Method ◽  
Transverse Velocity ◽  
Longitudinal Velocity ◽  
Momentum Conservation ◽  
Dimensional Model ◽  
One Dimensional ◽  
Modelling Techniques ◽  
Reference Grid ◽  
The One ◽  
Near Future

We consider the paper by Brugnot and Pochat (1981), which describes a one-dimensional model applied to a snow avalanche. The main advance made here is the introduction of the second dimension in the runout zone. Indeed, in the channelled course, we still use the one-dimensional model, but, when the avalanche spreads before stopping, we apply a (x, y) grid on the ground and six equations have to be solved: (1) for the avalanche body, one equation for continuity and two equations for momentum conservation, and (2) at the front, one equation for continuity and two equations for momentum conservation. We suppose the front to be a mobile jump, with longitudinal velocity varying more rapidly than transverse velocity.We solve these equations by a finite difference method. This involves many topological problems, due to the actual position of the front, which is defined by its intersection with the reference grid (SI, YJ). In the near future our two directions of research will be testing the code on actual avalanches and improving it by trying to make it cheaper without impairing its accuracy.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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