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NHM offers a strong combination of three features: Interdisciplinary character, specific focus, and deep mathematical content. Also, the journal aims to create a link between the discrete and the continuous communities, which distinguishes it from other journals with strong PDE orientation.

NHM publishes original contributions of high quality in networks, heterogeneous media and related fields. NHM is thus devoted to research work on complex media arising in mathematical, physical, engineering, socio-economical and bio-medical problems.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 4 issues a year in March, June, September and December.
  • Publishes online only.
  • Indexed in Science Citation Index-Expanded, ISI Alerting Services, CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
  • Archived in Portico and CLOCKSS.
  • NHM is a publication of the American Institute of Mathematical Sciences. All rights reserved.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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A Godunov type scheme for a class of LWR traffic flow models with non-local flux
Jan Friedrich, Oliver Kolb and Simone Göttlich
2018, 13(4) : 531-547 doi: 10.3934/nhm.2018024 +[Abstract](642) +[HTML](307) +[PDF](497.72KB)

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide \begin{document}$L^∞$\end{document} and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect
Linglong Du
2018, 13(4) : 549-565 doi: 10.3934/nhm.2018025 +[Abstract](362) +[HTML](198) +[PDF](481.77KB)

In this paper, we investigate the existence and long time behavior of the solution for the nonlinear visco-elastic damped wave equation in \begin{document}$\mathbb{R}^n_+$\end{document}, provided that the initial data is sufficiently small. It is shown that for the long time, one can use the convected heat kernel to describe the hyperbolic wave transport structure and damped diffusive mechanism. The Green's function for the linear initial boundary value problem can be described in terms of the fundamental solution (for the full space problem) and reflected fundamental solution coupled with the boundary operator. Using the Duhamel's principle, we get the \begin{document}$ L^p $\end{document} decaying rate for the nonlinear solution \begin{document}$\partial_{{\bf x}}^{\alpha}u$\end{document} for \begin{document}$|\alpha|\le 1$\end{document}.

Influence prediction for continuous-time information propagation on networks
Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha and Haomin Zhou
2018, 13(4) : 567-583 doi: 10.3934/nhm.2018026 +[Abstract](332) +[HTML](219) +[PDF](672.08KB)

We consider the problem of predicting the time evolution of influence, defined by the expected number of activated (infected) nodes, given a set of initially activated nodes on a propagation network. To address the significant computational challenges of this problem on large heterogeneous networks, we establish a system of differential equations governing the dynamics of probability mass functions on the state graph where each node lumps a number of activation states of the network, which can be considered as an analogue to the Fokker-Planck equation in continuous space. We provides several methods to estimate the system parameters which depend on the identities of the initially active nodes, the network topology, and the activation rates etc. The influence is then estimated by the solution of such a system of differential equations. Dependency of the prediction error on the parameter estimation is established. This approach gives rise to a class of novel and scalable algorithms that work effectively for large-scale and dense networks. Numerical results are provided to show the very promising performance in terms of prediction accuracy and computational efficiency of this approach.

On boundary optimal control problem for an arterial system: First-order optimality conditions
Ciro D'Apice, Olha P. Kupenko and Rosanna Manzo
2018, 13(4) : 585-607 doi: 10.3934/nhm.2018027 +[Abstract](288) +[HTML](151) +[PDF](466.33KB)

We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that the control object is described by an initial-boundary value problem for \begin{document}$ 1D $\end{document} system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principle part and the Robin-type of boundary conditions. The main question we study in this part of the paper is about the existence of optimal solutions and first-order optimality conditions.

Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface
Markus Gahn, Maria Neuss-Radu and Peter Knabner
2018, 13(4) : 609-640 doi: 10.3934/nhm.2018028 +[Abstract](237) +[HTML](187) +[PDF](626.09KB)

In this paper, we consider a system of reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with thickness of order $ε$ and a periodic heterogeneous structure. The equations inside the layer depend on $ε$ and the diffusivity inside the layer on an additional parameter $γ ∈ [-1, 1]$. On the bulk-layer interface, we assume a nonlinear Neumann-transmission condition depending on the solutions on both sides of the interface. For $\epsilon \to0 $, when the thin layer reduces to an interface $Σ$ between two bulk domains, we rigorously derive macroscopic models with effective conditions across the interface $Σ$. The crucial part is to pass to the limit in the nonlinear terms, especially for the traces on the interface between the different compartments. For this purpose, we use the method of two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the thin layer.

Optimal model switching for gas flow in pipe networks
Fabian Rüffler, Volker Mehrmann and Falk M. Hante
2018, 13(4) : 641-661 doi: 10.3934/nhm.2018029 +[Abstract](249) +[HTML](168) +[PDF](589.29KB)

We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.

Fluvial to torrential phase transition in open canals
Maya Briani and Benedetto Piccoli
2018, 13(4) : 663-690 doi: 10.3934/nhm.2018030 +[Abstract](299) +[HTML](143) +[PDF](1320.7KB)

Network flows and specifically water flow in open canals can be modeled bysystems of balance laws defined ongraphs.The shallow water or Saint-Venant system of balance laws is one of the most used modeland present two phases: fluvial or sub-critical and torrential or super-critical.Phase transitions may occur within the same canal but transitions relatedto networks are less investigated.In this paper we provide a complete characterization of possible phase transitionsfor a case study of a simple scenariowith two canals and one junction.However, our analysis allows the study of more complicate networks.Moreover, we provide some numerical simulations to show the theory at work.

Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs
Joachim von Below and José A. Lubary
2018, 13(4) : 691-717 doi: 10.3934/nhm.2018031 +[Abstract](213) +[HTML](140) +[PDF](534.93KB)

We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction-diffusion-equations on the edges of a finite metric graph \begin{document}$ G$\end{document} under continuity and Kirchhoff flow transition conditions at the vertices.

Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity
M.A.J Chaplain and G. Lolas
2006, 1(3) : 399-439 doi: 10.3934/nhm.2006.1.399 +[Abstract](1522) +[PDF](993.7KB) Cited By(67)
Gas flow in pipeline networks
Mapundi K. Banda, Michael Herty and Axel Klar
2006, 1(1) : 41-56 doi: 10.3934/nhm.2006.1.41 +[Abstract](1201) +[PDF](3881.2KB) Cited By(64)
Distributed model predictive control of irrigation canals
Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky and Bart De Schutter
2009, 4(2) : 359-380 doi: 10.3934/nhm.2009.4.359 +[Abstract](1462) +[PDF](281.3KB) Cited By(63)
Spreading speed revisited: Analysis of a free boundary model
Gary Bunting, Yihong Du and Krzysztof Krakowski
2012, 7(4) : 583-603 doi: 10.3934/nhm.2012.7.583 +[Abstract](1279) +[PDF](737.9KB) Cited By(56)
On the variational theory of traffic flow: well-posedness, duality and applications
Carlos F. Daganzo
2006, 1(4) : 601-619 doi: 10.3934/nhm.2006.1.601 +[Abstract](1390) +[PDF](337.5KB) Cited By(53)
K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases
José Ignacio Alvarez-Hamelin, Luca Dall'Asta, Alain Barrat and Alessandro Vespignani
2008, 3(2) : 371-393 doi: 10.3934/nhm.2008.3.371 +[Abstract](1246) +[PDF](739.3KB) Cited By(52)
Coupling conditions for gas networks governed by the isothermal Euler equations
Mapundi K. Banda, Michael Herty and Axel Klar
2006, 1(2) : 295-314 doi: 10.3934/nhm.2006.1.295 +[Abstract](1122) +[PDF](1720.1KB) Cited By(48)
Large time behavior of nonlocal aggregation models with nonlinear diffusion
Martin Burger and Marco Di Francesco
2008, 3(4) : 749-785 doi: 10.3934/nhm.2008.3.749 +[Abstract](1138) +[PDF](412.0KB) Cited By(45)
Explicit solutions of some linear-quadratic mean field games
Martino Bardi
2012, 7(2) : 243-261 doi: 10.3934/nhm.2012.7.243 +[Abstract](1327) +[PDF](433.5KB) Cited By(38)
A Well Posed Riemann Problem for the $p$--System at a Junction
Rinaldo M. Colombo and Mauro Garavello
2006, 1(3) : 495-511 doi: 10.3934/nhm.2006.1.495 +[Abstract](991) +[PDF](212.0KB) Cited By(30)
On a vorticity-based formulation for reaction-diffusion-Brinkman systems
Verónica Anaya, Mostafa Bendahmane, David Mora and Ricardo Ruiz Baier
2018, 13(1) : 69-94 doi: 10.3934/nhm.2018004 +[Abstract](993) +[HTML](637) +[PDF](6157.76KB) PDF Downloads(97)
Functional model for extensions of symmetric operators and applications to scattering theory
Kirill D. Cherednichenko, Alexander V. Kiselev and Luis O. Silva
2018, 13(2) : 191-215 doi: 10.3934/nhm.2018009 +[Abstract](869) +[HTML](190) +[PDF](571.43KB) PDF Downloads(86)
Uniform stability and mean-field limit for the augmented Kuramoto model
Seung-Yeal Ha, Jeongho Kim, Jinyeong Park and Xiongtao Zhang
2018, 13(2) : 297-322 doi: 10.3934/nhm.2018013 +[Abstract](844) +[HTML](155) +[PDF](494.85KB) PDF Downloads(78)
A Godunov type scheme for a class of LWR traffic flow models with non-local flux
Jan Friedrich, Oliver Kolb and Simone Göttlich
2018, 13(4) : 531-547 doi: 10.3934/nhm.2018024 +[Abstract](642) +[HTML](307) +[PDF](497.72KB) PDF Downloads(75)
A two-dimensional data-driven model for traffic flow on highways
Michael Herty, Adrian Fazekas and Giuseppe Visconti
2018, 13(2) : 217-240 doi: 10.3934/nhm.2018010 +[Abstract](911) +[HTML](243) +[PDF](7915.97KB) PDF Downloads(74)
Sparse stabilization of dynamical systems driven by attraction and avoidance forces
Mattia Bongini and Massimo Fornasier
2014, 9(1) : 1-31 doi: 10.3934/nhm.2014.9.1 +[Abstract](1418) +[PDF](2266.2KB) PDF Downloads(69)
Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication
Young-Pil Choi, Seung-Yeal Ha and Jeongho Kim
2018, 13(3) : 379-407 doi: 10.3934/nhm.2018017 +[Abstract](722) +[HTML](216) +[PDF](525.83KB) PDF Downloads(65)
Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours
Roberto Alicandro, Giuliano Lazzaroni and Mariapia Palombaro
2018, 13(1) : 1-26 doi: 10.3934/nhm.2018001 +[Abstract](972) +[HTML](200) +[PDF](360.33KB) PDF Downloads(62)
Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow
Helge Holden and Nils Henrik Risebro
2018, 13(3) : 409-421 doi: 10.3934/nhm.2018018 +[Abstract](637) +[HTML](254) +[PDF](3315.05KB) PDF Downloads(51)
Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network
Francesca R. Guarguaglini
2018, 13(1) : 47-67 doi: 10.3934/nhm.2018003 +[Abstract](857) +[HTML](207) +[PDF](524.21KB) PDF Downloads(49)

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