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Networks & Heterogeneous Media

June 2021 , Volume 16 , Issue 2

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Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph
Vaibhav Mehandiratta, Mani Mehra and Günter Leugering
2021, 16(2): 155-185 doi: 10.3934/nhm.2021003 +[Abstract](300) +[HTML](183) +[PDF](481.01KB)

In this paper, we study a nonlinear fractional boundary value problem on a particular metric graph, namely, a circular ring with an attached edge. First, we prove existence and uniqueness of solutions using the Banach contraction principle and Krasnoselskii's fixed point theorem. Further, we investigate different kinds of Ulam-type stability for the proposed problem. Finally, an example is given in order to demonstrate the application of the obtained theoretical results.

A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function
Raimund Bürger, Christophe Chalons, Rafael Ordoñez and Luis Miguel Villada
2021, 16(2): 187-219 doi: 10.3934/nhm.2021004 +[Abstract](378) +[HTML](160) +[PDF](2723.65KB)

The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of \begin{document}$ N $\end{document} conservation laws for \begin{document}$ N $\end{document} driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model
Abraham Sylla
2021, 16(2): 221-256 doi: 10.3934/nhm.2021005 +[Abstract](231) +[HTML](105) +[PDF](545.59KB)

In this paper, we propose a macroscopic model that describes the influence of a slow moving large vehicle on road traffic. The model consists of a scalar conservation law with a nonlocal constraint on the flux. The constraint level depends on the trajectory of the slower vehicle which is given by an ODE depending on the downstream traffic density. After proving well-posedness, we first build a finite volume scheme and prove its convergence, and then investigate numerically this model by performing a series of tests. In particular, the link with the limit local problem of [M. L. Delle Monache and P. Goatin, J. Differ. Equ. 257 (2014), 4015–4029] is explored numerically.

Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations
Mayte Pérez-Llanos, Juan Pablo Pinasco and Nicolas Saintier
2021, 16(2): 257-281 doi: 10.3934/nhm.2021006 +[Abstract](236) +[HTML](115) +[PDF](906.58KB)

In this work we study the formation of consensus in homogeneous and heterogeneous populations, and the effect of attractiveness or fitness of the opinions. We derive the corresponding kinetic equations, analyze the long time behavior of their solutions, and characterize the consensus opinion.

Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure
Benjamin Boutin, Frédéric Coquel and Philippe G. LeFloch
2021, 16(2): 283-315 doi: 10.3934/nhm.2021007 +[Abstract](223) +[HTML](109) +[PDF](1187.95KB)

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the restricted case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions.

An inverse problem for quantum trees with observations at interior vertices
Sergei Avdonin and Julian Edward
2021, 16(2): 317-339 doi: 10.3934/nhm.2021008 +[Abstract](234) +[HTML](84) +[PDF](495.25KB)

In this paper we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that positive masses may be attached to the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Dirichlet-to-Neumann map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph, the coefficients of the equations and the masses at the vertices.

2019  Impact Factor: 1.053




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