scholarly journals A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling

2011 ◽  
Vol 22 (4) ◽  
pp. 291-316 ◽  
Author(s):  
K. ANGUIGE
Keyword(s):  
Cell Adhesion ◽  
Discrete Model ◽  
Continuum Limit ◽  
Numerical Solutions ◽  
Problem Formulation ◽  
Dimensional Model ◽  
One Dimensional ◽  
Spatial Oscillations ◽  
Cell To Cell Adhesion ◽  
The Continuum

We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion

2010 ◽  
Vol 21 (2) ◽  
pp. 109-136 ◽  
Author(s):  
K. ANGUIGE
Keyword(s):  
Cell Adhesion ◽  
Existence Theory ◽  
Dimensional Model ◽  
One Dimensional ◽  
Stefan Problems ◽  
Non Linear ◽  
Long Time ◽  
Multi Phase ◽  
Cell To Cell Adhesion ◽  
And Diffusion

We consider a family of multi-phase Stefan problems for a certain one-dimensional model of cell-to-cell adhesion and diffusion, which takes the form of a non-linear forward–backward parabolic equation. In each material phase the cell density stays either high or low, and phases are connected by jumps across an ‘unstable’ interval. We develop an existence theory for such problems which allows for the annihilation of phases and the subsequent continuation of solutions. Stability results for the long-time behaviour of solutions are also obtained, and, where necessary, the analysis is complemented by numerical simulations.

Instability and Collapse in Discrete Wave Equations

2005 ◽  
Vol 5 (3) ◽  
pp. 223-241
Author(s):  
A. Carpio ◽  
G. Duro
Keyword(s):  
Continuum Limit ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Blow Up ◽  
Theoretical Predictions ◽  
Initial States ◽  
The Impact ◽  
The Continuum

AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

scholarly journals Dissociation and recombination in the electrolyte flow model

2021 ◽  
Vol 2090 (1) ◽  
pp. 012076
Author(s):  
A Shobukhov ◽  
H Koibuchi
Keyword(s):  
Flow Model ◽  
Numerical Solutions ◽  
Stable Equilibrium ◽  
Steady State Solution ◽  
System Of Equations ◽  
Constant Flow ◽  
Dimensional Model ◽  
Electrolyte Flow ◽  
One Dimensional ◽  
Dilute Aqueous Solution

Abstract We propose a one-dimensional model for the dilute aqueous solution of NaCl which is treated as an incompressible fluid placed in the external electric field. This model is based on the Poisson-Nernst-Planck system of equations, which also contains the constant flow velocity as a parameter and considers the dissociation and the recombination of ions. We study the steady-state solution analytically and prove that it is a stable equilibrium. Analyzing the numerical solutions, we demonstrate the importance of dissociation and recombination for the physical meaningfulness of the model.

scholarly journals Quantitative Spectroscopy of Wolf-Rayet Stars

1988 ◽  
Vol 108 ◽  
pp. 133-140
Author(s):  
W. Schmutz
Keyword(s):  
Representative Point ◽  
Dimensional Model ◽  
Model Calculations ◽  
One Dimensional ◽  
The Past ◽  
First Results ◽  
The One ◽  
Expanding Atmospheres ◽  
New Generation ◽  
The Continuum

Advances in theoretical modeling of rapidly expanding atmospheres in the past few years made it possible to determine the stellar parameters of the Wolf-Rayet stars. This progress is mainly due to the improvement of the models with respect to their spatial extension: The new generation of models treat spherically-symmetric expanding atmospheres, i.e. the models are one-dimensional. Older models describe the wind by only one representative point. The older models are in fact ‘core-halo’ approximations. They have been introduced by Castor and van Blerkom (1970), and were extensively employed in the past (cf. e.g. Willis and Wilson, 1978; Smith and Willis, 1982). First results from new one-dimensional model calculations are published by Hillier (1984), Schmutz (1984), Hamann (1985), Hillier (1986), and Schmutz et al. (1987a); more detailed results are presented by Schmutz and Hamann (1986), Hamann and Schmutz (1987), Hillier (1987a,b), Wessolowski et al. (1987), Hillier (1987c) and Hamann et al. (1987). These results demonstrate that the step from zero- to one-dimensional calculations is essential. The important point is that the complicated interrelation between NLTE-level populations and radiation field is treated adequately (Schmutz and Hamann, 1986; Hillier, 1987). For this interrelation it is crucial to model consistently not only the line-formation region, but also the layers where the continuum is emitted. In fact, it is the core-halo approximation that causes the one-point models to fail in certain aspects.

scholarly journals SUPERLOOP EQUATIONS AND TWO-DIMENSIONAL SUPERGRAVITY

1992 ◽  
Vol 07 (21) ◽  
pp. 5337-5367 ◽  
Author(s):  
L. ALVAREZ-GAUMÉ ◽  
H. ITOYAMA ◽  
J.L. MAÑES ◽  
A. ZADRA
Keyword(s):  
Partition Function ◽  
Discrete Model ◽  
Continuum Limit ◽  
Basic Assumption ◽  
Scaling Limit ◽  
Integrable Hierarchy ◽  
Two Dimensional ◽  
Virasoro Constraints ◽  
Double Scaling Limit ◽  
The Continuum

We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m) minimal superconformal models coupled to 2D supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of nonlinear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin.

Exchange renormalization in one dimensional classical antiferromagnets in the continuum limit

1993 ◽  
Vol 87 (12) ◽  
pp. 1141-1143
Author(s):  
A.S.T. Pires ◽  
M.E. Gouvêa ◽  
S.L. Menezes
Keyword(s):  
Continuum Limit ◽  
One Dimensional ◽  
The Continuum

scholarly journals One-dimensional model for a laser-ablated slab under acceleration

2004 ◽  
Vol 22 (2) ◽  
pp. 183-188 ◽  
Author(s):  
J. RAMÍREZ ◽  
R. RAMIS ◽  
J. SANZ
Keyword(s):  
Subsonic Flow ◽  
Numerical Solutions ◽  
Critical Surface ◽  
Slab Surface ◽  
Dimensional Model ◽  
Initial Value ◽  
Sonic Point ◽  
One Dimensional ◽  
Ablation Pressure ◽  
Simple Dependence

A one-dimensional model for a laser-ablated slab under acceleration g is developed. A characteristic value gc is found to separate two solutions: Lower g values allow sonic or subsonic flow at the critical surface; for higher g the sonic point approaches closer and closer to the slab surface. A significant reduction in the ablation pressure is found in comparison to the g = 0 case. A simple dependence law between the ablation pressure and the slab acceleration, from the initial value g0 to infinity, is identified. Results compared well with fully hydrodynamic computer simulations with Multi2D code. The model has also been found a key step to produce indefinitely steady numerical solutions to study Rayleigh–Taylor instabilities in heat ablation fronts, and validate other theoretical analysis of the problem.

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