# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

Open Access Articles

2021, 16(4): 609-636 doi: 10.3934/nhm.2021020 +[Abstract](703) +[HTML](240) +[PDF](571.81KB)
Abstract:

This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point of view, this model consists of a 5\begin{document}$\times$\end{document}5 semi-linear hyperbolic system. In literature similar models neglect the epithelial layers. In this paper, we show rigorously that such models may be obtained by assuming that the permeabilities between lumen and epithelium are large. We show that when these permeabilities grow, solutions of the 5\begin{document}$\times$\end{document}5 system converge to those of a reduced 3\begin{document}$\times$\end{document}3 system without epithelial layers. The problem is defined on a bounded spacial domain with initial and boundary data. In order to show convergence, we use \begin{document}${{{\rm{BV}}}}$\end{document} compactness, which leads to introduce initial layers and to handle carefully the presence of lateral boundaries. We then discretize both 5\begin{document}$\times$\end{document}5 and 3\begin{document}$\times$\end{document}3 systems, and show numerically the same asymptotic result, for a fixed meshsize.

2021, 16(4): 513-533 doi: 10.3934/nhm.2021015 +[Abstract](783) +[HTML](248) +[PDF](9210.31KB)
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In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solutions are investigated, both theoretically and numerically.

2021, 16(4): 569-589 doi: 10.3934/nhm.2021018 +[Abstract](713) +[HTML](251) +[PDF](436.15KB)
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In this paper, we study the concept of approximate controllability of retarded network systems of neutral type. On one hand, we reformulate such systems as free-delay boundary control systems on product spaces. On the other hand, we use the rich theory of infinite-dimensional linear systems to derive necessary and sufficient conditions for the approximate controllability. Moreover, we propose a rank condition for which we can easily verify the conditions of controllability. Our approach is mainly based on the feedback theory of regular linear systems in the Salamon-Weiss sense.

2021, 16(4): 553-567 doi: 10.3934/nhm.2021017 +[Abstract](807) +[HTML](270) +[PDF](464.8KB)
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We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the \begin{document}${\mathrm{L}}^{\infty}$\end{document}-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.

2021, 16(4): 535-552 doi: 10.3934/nhm.2021016 +[Abstract](845) +[HTML](306) +[PDF](1039.55KB)
Abstract:

The spread of rumors is a phenomenon that has heavily impacted society for a long time. Recently, there has been a huge change in rumor dynamics, through the advent of the Internet. Today, online communication has become as common as using a phone. At present, getting information from the Internet does not require much effort or time. In this paper, the impact of the Internet on rumor spreading will be considered through a simple SIR type ordinary differential equation. Rumors spreading through the Internet are similar to the spread of infectious diseases through water and air. From these observations, we study a model with the additional principle that spreaders lose interest and stop spreading, based on the SIWR model. We derive the basic reproduction number for this model and demonstrate the existence and global stability of rumor-free and endemic equilibriums.

2021, 16(4): 591-607 doi: 10.3934/nhm.2021019 +[Abstract](840) +[HTML](245) +[PDF](351.11KB)
Abstract:

We study convex and quasiconvex functions on a metric graph. Given a set of points in the metric graph, we consider the largest convex function below the prescribed datum. We characterize this largest convex function as the unique largest viscosity subsolution to a simple differential equation, \begin{document}$u'' = 0$\end{document} on the edges, plus nonlinear transmission conditions at the vertices. We also study the analogous problem for quasiconvex functions and obtain a characterization of the largest quasiconvex function that is below a given datum.

2021, 16(1): 1-29 doi: 10.3934/nhm.2020031 +[Abstract](1149) +[HTML](343) +[PDF](681.95KB)
Abstract:

The paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of trunk to all the leaves. In a 2-dimensional setting, the solution is proved to be unique and explicitly determined.

2020, 15(3): i-i doi: 10.3934/nhm.2020020 +[Abstract](873) +[HTML](309) +[PDF](90.18KB)
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2019, 14(1): i-ii doi: 10.3934/nhm.20191i +[Abstract](5082) +[HTML](594) +[PDF](97.78KB)
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2018, 13(1): 69-94 doi: 10.3934/nhm.2018004 +[Abstract](5487) +[HTML](2123) +[PDF](6157.76KB)
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We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nédélec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

2017, 12(3): i-ii doi: 10.3934/nhm.201703i +[Abstract](2931) +[HTML](260) +[PDF](130.1KB)
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2017, 12(2): 217-243 doi: 10.3934/nhm.2017009 +[Abstract](4628) +[HTML](1723) +[PDF](819.2KB)
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We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equilibrium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are validated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates.

2017, 12(2): i-ii doi: 10.3934/nhm.201702i +[Abstract](2665) +[HTML](279) +[PDF](130.1KB)
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2016, 11(2): i-ii doi: 10.3934/nhm.2016.11.2i +[Abstract](2918) +[PDF](113.0KB)
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During last 20 years the theory of Conservation Laws underwent a dramatic development. Networks and Heterogeneous Media is dedicating two consecutive Special Issues to this topic. Researchers belonging to some of the major schools in this subject contribute to these two issues, offering a view on the current state of the art, as well pointing to new research themes within areas already exposed to more traditional methodologies.

2016, 11(1): i-ii doi: 10.3934/nhm.2016.11.1i +[Abstract](2709) +[PDF](105.3KB)
Abstract:
During last 20 years the theory of Conservation Laws underwent a dramatic developmen. Networks and Heterogeneous Media is dedicating two consecutive Special Issues to this topic. Researchers belonging to some of the major schools in this subject contribute to these two issues, offering a view on the current state of the art, as well pointing to new research themes within areas already exposed to more traditional methodologies.

2015, 10(3): i-ii doi: 10.3934/nhm.2015.10.3i +[Abstract](3700) +[PDF](105.5KB)
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This Special Issue is based on research presented at the Workshop Modeling and Control of Social Dynamics", hosted by the Center of Computational and Integrative Biology and the Department of Mathematical Sciences at Rutgers University - Camden. The Workshop is part of the activities of the NSF Research Network in Mathematical Sciences: Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # 1107444, which is also acknowledged for funding the workshop.

2015, 10(1): i-iii doi: 10.3934/nhm.2015.10.1i +[Abstract](3704) +[PDF](156.8KB)
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The real world surrounding us is full of complex systems from various types and categories. Internet, the World Wide Web, biological and biochemical networks (brain, metabolic, protein and genomic networks), transport networks (underground, train, airline networks, road networks), communication networks (computer servers, Internet, online social networks), and many others (social community networks, electric power grids and water supply networks,...) are a few examples of the many existing kinds and types of networks [1,2,3,4,6,8,9,10,11]. In the recent past years, the study of structure and dynamics of complex networks has been the subject of intense interest. Recent advances in the study of complex networked systems has put the spotlight on the existence of more than one type of links whose interplay can affect the structure and function of those systems [5,7]. In these networks, relevant information may not be captured if the single layers are analyzed separately, since these different components and units interact with others through different channels of connectivity and dependencies. The global characteristics and behavior of these systems depend on multiple dimensions of integration, relationship or cleavage of its units.

2014, 9(4): i-ii doi: 10.3934/nhm.2014.9.4i +[Abstract](2434) +[PDF](112.9KB)
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Although the concrete is a simple man-made material with initially-controlled composition (for instance, all ingredients are known beforehand, the involved chemical mechanisms are well studied, the mechanical strength of test samples is measured accurately), forecasting its behaviour for large times under variable external (boundary) conditions is not properly understood. The main reason is that the simplicity of the material is only apparent. The combination of the heterogeneity of the material together with the occurrence of a number of multiscale phase transitions either driven by aggressive chemicals (typically ions, like in corrosion situations), or by extreme heating, or by freezing/thawing of the ice lenses within the microstructure, and the inherent non-locality of the mechanical damage leads to mathematically challenging nonlinear coupled systems of partial differential equations (PDEs).

2014, 9(1): 65-95 doi: 10.3934/nhm.2014.9.65 +[Abstract](3537) +[PDF](1803.6KB)
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We numerically explore network models which are derived for the isothermal Euler equations. Previously we proved the existence and uniqueness of solutions to the generalized Riemann problem at a junction under the conditions of monotone momentum related coupling constant and equal cross-sectional areas for all connected pipe sections. In the present paper we extend this proof to the case of pipe sections of different cross-sectional areas.
We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical high-resolution Roe scheme. We propose a numerical treatment of the boundary conditions at the pipe-junction interface, consistent with the coupling conditions. In particular, mass is exactly conserved across the junction.
Numerical results are provided for two different network configurations and for three different network models. Mechanical energy considerations are applied in order to evaluate the results in terms of physical soundness. Analytical predictions for junctions connecting three pipe sections are verified for both network configurations. Long term behaviour of physical and unphysical solutions are presented and compared, and the impact of having pipes with different cross-sectional area is shown.
2013, 8(3): 745-772 doi: 10.3934/nhm.2013.8.745 +[Abstract](4356) +[PDF](1439.2KB)
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Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.

2020 Impact Factor: 1.213
5 Year Impact Factor: 1.384
2020 CiteScore: 1.9