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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]

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]

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]

We present Dirichlet to Dirichlet boundary conditions for the heat equation in one, two, and three dimensions. These boundary conditions contain temporal convolution integrals with nonsingular kernels, allowing for an accurate and simple numerical approximation and enabling their straightforward coupling to any numerical scheme. The stability of these boundary conditions is proven using the Kreiss theory. [ABSTRACT FROM AUTHOR]

The semidiscretise Method of Lines (MOL) is used for the numerical analysis of transient phenomena occurring during the operation of natural gas pipelines and pipe networks. The Method of Lines discretises the spatial terms of the governing equations, reducing the partial to an ordinary differential equations system which can be solved by various techniques. For comparison and validation purposes, transient phenomena are also simulated here by means of the Method of Characteristics (MOC) which is a reliable but rather old method. The governing equations system is based on the fundamental conservation laws of mass, momentum, and energy for the unsteady one dimensional compressible pipe flow. Experimental results representing realistic transient pipeline phenomena have been obtained from the laboratory test facility. The agreement between numerical results of both methods and measurements is satisfactory especially when experimental data are introduced as time dependent boundary conditions. In addition the Method of Lines proved efficient and applicable to rapid as well as to slow transients which is a very common unsteady behaviour of natural gas pipeline flows. [ABSTRACT FROM AUTHOR]