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1. Feedback Stabilization of a 1-D Linear Reaction–Diffusion Equation With Delay Boundary Control.

Author

Prieur, Christophe and Trelat, Emmanuel

Subjects
*DIFFERENTIAL equations, *FEEDBACK control systems, *BOUNDARY value problems, *COMPUTER simulation, *NUMERICAL analysis
Abstract

The goal of this paper is to design a stabilizing feedback boundary control for a reaction–diffusion partial differential equation (PDE), where the boundary control is subject to a constant delay while the equation may be unstable without any control. For this system, which is equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed by splitting the infinite-dimensional system into two parts: a finite-dimensional unstable part and a stable infinite-dimensional part. A finite-dimensional delayed controller is computed for the unstable part, and it is shown that this controller stabilizes the whole PDE. The proof is based on an explicit expression of the classical Artstein transformation combined with an adequately designed Lyapunov function. A numerical simulation illustrates the constructive feedback design method. [ABSTRACT FROM AUTHOR]

Published
2019
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2. Existence and stability of nonconstant positive steady states of morphogenesis models.

Author

Chen, Haohao, Tong, Bo, and Wang, Qi

Subjects
*BOUNDARY value problems, *NUMERICAL analysis, *DIFFERENTIAL equations, *LINEAR differential equations, *COMPUTER simulation
Abstract

In this paper, we study a one-dimensional morphogenesis model considered by C. Stinner et al. (Math. Meth. Appl. Sci. 2012; 35: 445-465). Under homogeneous boundary conditions, we prove the existence of nonconstant positive steady states through local bifurcation theories. Then we rigorously study the stability of these nonconstant solutions when the sensitivity functions are chosen to be linear and logarithmic, respectively. Finally, we present numerical solutions to illustrate the formation of stable inhomogeneous spatial patterns. Our numerical simulations show that this model can develop very complicated and interesting structures even over one-dimensional finite domains. [ABSTRACT FROM AUTHOR]

Published
2015
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3. A hybrid model for opinion formation.

Author

Borra, Domenica and Lorenzi, Tommaso

Subjects
*FINITE difference method, *FINITE differences, *NUMERICAL analysis, *DIFFERENTIAL equations, *BOUNDARY value problems, *COMPUTER simulation
Abstract

This paper presents a hybrid model for opinion formation in a large group of agents exposed to the persuasive action of a small number of strong opinion leaders. The model is defined by coupling a finite difference equation for the dynamics of leaders opinion with a continuous integro-differential equation for the dynamics of the others. Such a definition stems from the idea that the leaders are few and tend to retain original opinions, so that their dynamics occur on a longer time scale with respect to the one of the other agents. A general well-posedness result is established for the initial value problem linked to the model. The asymptotic behavior in time of the related solution is characterized for some general parameter settings, which mimic distinct social scenarios, where different emerging behaviors can be observed. Analytical results are illustrated and extended through numerical simulations. [ABSTRACT FROM AUTHOR]

Published
2013
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4. A highly accurate alternating 6-point group method for the dispersive equation.

Author

Wang, Wen-Qia and Zhang, Qingjie

Subjects
*BOUNDARY value problems, *COMPUTER simulation, *ALGORITHMS, *EQUATIONS, *NUMERICAL analysis, *DIFFERENTIAL equations
Abstract

In this paper, we construct a group of Saul'yev type asymmetric difference formulas for the dispersive equation. Based on these formulas we derive a new alternating 6-point group algorithm to solve dispersive equations with periodic boundary conditions. The algorithm has a high-order accuracy in space and an unconditional stability. The theoretical results are conformed to the numerical simulation. A comparison of this algorithm with the previous Alternating Group Explicit method is presented. [ABSTRACT FROM AUTHOR]

Published
2010
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