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Second-Order Traffic Flow Models on Networks
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 258-281 |
| Abstrak | : | This paper deals with the Aw--Rascle--Zhang [A. Aw and M. Rascle, SIAM J. Appl. Math., 60 (2000), pp. 916--938; H. M. Zhang, Transp. Res. B Methodol., 36 (2002), pp. 275--290] model for traffic flow on uni-directional road networks. We construct weak solutions to Riemann problems at the junctions, which conserve the mass and the generalized momentum. In particular, we introduce a new approach to approximate the homogenized pressure through an additional equation for the propagation of a reference pressure. The resulting system of coupled conservation laws is then solved using an appropriate numerical scheme of Godunov type. Numerical simulations show that the proposed approach enables us to approximate the homogenized pressure sufficiently well. The features of the new approach are illustrated through a comparative analysis with other methods proposed in the literature for the Aw--Rascle--Zhang second-order traffic model [2, 31] and the Lighthill--Whitham--Richards model [M. J. Lighthill and G. B. Whitman, Proc. A, 229 (1955), pp. 317--345; P. I. Richards, Oper. Res., 4 (1956), pp. 42--51]. |
Spectrum Decomposition of Translation Operators in Periodic Waveguide
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 233-257 |
| Abstrak | : | Scattering problems in periodic waveguides are interesting but also challenging topics in mathematics both theoretically and numerically. As is well known, the unique solvability of these problems is not always guaranteed. To obtain a unique solution that is “physically meaningful,” the limiting absorption principle (LAP) is a commonly used method. LAP assumes that the limit of a family of solutions with absorbing media converges as the absorption parameter tends to 0 and that the limit is the “physically meaningful solution.” It is also called the LAP solution in this paper. It has been proved that the LAP holds for periodic waveguides in [V. Hoang, SIAM J. Appl. Math., 71 (2011), pp. 791--810]. In this paper, we consider the spectrum decomposition of periodic translation operators. With the curve integral formulation and a generalized residue theorem, the operator is explicitly described by its eigenvalues and generalized eigenfunctions, which are closely related to Bloch wave solutions. Then the LAP solution is decomposed into generalized eigenfunctions. This gives a better understanding of the structure of scattered fields. |
A NonLinear Transient Reaction-Diffusion Problem from Electroanalytical Chemistry
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 208-232 |
| Abstrak | : | A nonlinear reaction-diffusion partial differential equation occurring in models of transient controlled-potential experiments in electroanalytical chemistry is investigated analytically and numerically, with a view to determining a relationship between the concentration of a chemical species and its flux at a reacting electrode. It is shown that a previously known relation that holds for the steady-state case can be used as the first term in a singular perturbation expansion for the time-dependent case. However, in trying to determine the second term, so as to extend the range of validity of the solution, it is found that a phenomenon akin to switchbacking occurs, with the asymptotic details being strongly dependent on the reaction order; this appears to be a consequence of the spatial algebraic decay of the leading-order solution far from the electrode. Comparison of asymptotic results with numerical solutions obtained using finite element methods indicates a relation involving the homogeoneous reaction order for which the two-term asymptotic approximation would work best for all time. Links to problems that involve algebraically decaying boundary layers are briefly discussed. |
Controlling Propagation of Epidemics via Mean-Field Control
| Pengarang | : | Wonjun Lee |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 190-207 |
| Abstrak | : | The coronavirus disease 2019 (COVID-19) pandemic is changing and impacting lives on a global scale. In this paper, we introduce a mean-field game model in controlling the propagation of epidemics on a spatial domain. The control variable, the spatial velocity, is first introduced for the classical disease models, such as the SIR model. For this proposed model, we provide fast numerical algorithms based on proximal primal-dual methods. Numerical experiments demonstrate that the proposed model illustrates how to separate infected patients in a spatial domain effectively. |
A Macroscopic Traffic Flow Model Accounting for Bounded Acceleration
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 173-189 |
| Abstrak | : | This paper details a new macroscopic traffic flow model accounting for the boundedness of traffic acceleration, which is required for physical realism. Our approach relies on the coupling between a scalar conservation law, which refers to the seminal Lighthill--Whitham--Richards model, and a system of ordinary differential equations describing the trajectories of accelerating vehicles, which we treat as moving constraints. We propose a wave-front tracking algorithm to construct approximate solutions. We use this algorithm to prove the existence of entropy weak solutions to the associated Cauchy problem and provide some numerical simulations illustrating the solution behaviour. |
A Fragmentation Phenomenon for a Nonenergetic Optimal Control Problem: Optimization of the Total Population Size in Logistic Diffusive Models
| Pengarang | : | Idriss Mazari |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 153-172 |
| Abstrak | : | Following several recent works devoted to the analysis of spatial heterogeneity in reaction-diffusion equations, we investigate the problem of optimizing the total population size for logistic diffusive models with respect to resources distributions. Using the spatially heterogeneous Fisher-KPP equation, we obtain a surprising fragmentation phenomenon: depending on the scale of diffusivity (i.e., the dispersal rate), it is better to either concentrate or fragment resources. Our main result is that, the smaller the dispersal rate of the species in the domain, the more optimal resources distributions tend to oscillate. This is in sharp contrast with other criteria in population dynamics, such as the classical problem of optimizing the survival ability of a species, where concentrating resources is always favorable, regardless of the diffusivity. Our study is completed by numerous numerical simulations that confirm our results. |
Modeling the Effect of Surfactant on Droplet Breakup in a Turbulent Flow
| Pengarang | : | Rachel M. Philip |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 129-152 |
| Abstrak | : | In many industrial applications oil drops suspended in water are broken down by turbulent fluid motion, assisted by the addition of surfactants. In this paper, we develop a model for the interaction between droplet breakup and surfactant behavior in a homogeneous turbulent flow. We derive equations for the evolution of droplet size distribution, surfactant concentration, and surface tension. Our resulting model is analyzed using both asymptotic and numerical methods. We find that the large-time behavior is highly dependent on the history of the drop-surfactant interaction, and we identify distinct phases in the breakup dynamics. The model is then used to investigate how increasing the concentration of surfactant promotes enhanced droplet breakup and thus increases the proportion of smaller droplets. We also explore varying the method of surfactant addition and find that there may be advantages to adding the surfactant gradually rather than all at once. These results can be used to optimize surfactant application in industrial processes. |
Existence and Dynamics of Strains in a Nonlocal Reaction-Diffusion Model of Viral Evolution
| Pengarang | : | Nikolai Bessonov |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 107-128 |
| Abstrak | : | In this work, we develop a mathematical framework for predicting and quantifying virus diversity evolution during infection of a host organism. It is specified as a virus density distribution with respect to genotype and time governed by a reaction-diffusion integro-differential equation taking virus mutations, replication, and elimination by immune cells and medical treatment into account. Conditions for the existence of virus strains that correspond to localized density distributions in the space of genotypes are determined. It is shown that common viral evolutionary traits like diversification and extinction are driven by nonlocal interactions via immune responses, target-cell competition, and therapy. This provides us with a mechanistic explanation for clinically relevant properties like immune escape and drug resistance selection, and allows us to link virus genotypes to phenotypes. |
Reconstruction of the Emission Coefficient in the Nonlinear Radiative Transfer Equation
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 91-106 |
| Abstrak | : | In this paper, we investigate an inverse problem for the radiative transfer equation that is coupled with a heat equation in a nonscattering medium in ?????,???? ≥2. The two equations are coupled through a nonlinear blackbody emission term that is proportional to the fourth power of the temperature. By measuring the radiation intensity on the surface of the blackbody, we prove that the emission property of the system can be uniquely reconstructed. In particular, we design a reconstruction procedure that uses merely one set of experimental setup to fully recover the emission parameter. |
A Surprising Observation in the Quarter-Plane Diffraction Problem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 60-90 |
| Abstrak | : | In this paper, we revisit Radlow's highly original attempt at a double Wiener--Hopf solution to the canonical problem of wave diffraction by a quarter-plane. Using a constructive approach, we reduce the problem to two equations, one containing his somewhat controversial ansatz, and an additional compatibility equation. We then show that despite Radlow's ansatz being erroneous, it gives surprisingly accurate results in the far-field, particularly for the spherical diffraction coefficient. This unexpectedly good result is established by comparing it to results obtained by the recently established modified Smyshlyaev formulae. |