Search results for "Partial Differential Equation"
showing 10 items of 326 documents
Age-Structured Human Population Dynamics
2006
ABSTRACT A von Foerster-McKendrick model to study age-structured human population dynamics is presented in this paper. Forecasts of population density (population per age unit) depending on ages are possible using this model. The model consists of a quasi-linear first order partial differential equation for the dynamics of population density per age-unit (except for the zero-age), a boundary condition for the births flow at zero-age, and an initial condition for the population density at the initial instant. A general solution independent of the particular human-system under study is obtained based on some hypotheses about the mathematical structure of its input variables. The model has bee…
A two-point boundary value formulation of a mean-field crowd-averse game
2014
Abstract We consider a population of “crowd-averse” dynamic agents controlling their states towards regions of low density. This represents a typical dissensus behavior in opinion dynamics. Assuming a quadratic density distribution, we first introduce a mean-field game formulation of the problem, and then we turn the game into a two-point boundary value problem. Such a result has a value in that it turns a set of coupled partial differential equations into ordinary differential equations.
Opinion dynamics and stubbornness through mean-field games
2013
This paper provides a mean field game theoretic interpretation of opinion dynamics and stubbornness. The model describes a crowd-seeking homogeneous population of agents, under the influence of one stubborn agent. The game takes on the form of two partial differential equations, the Hamilton-Jacobi-Bellman equation and the Kolmogorov-Fokker-Planck equation for the individual optimal response and the population evolution, respectively. For the game of interest, we establish a mean field equilibrium where all agents reach epsilon-consensus in a neighborhood of the stubborn agent's opinion.
A Side-by-Side Single Sex Age-Structured Human Population Dynamic Model: Exact Solution and Model Validation
2008
A side-by-side single sex age-structured population dynamic model is presented in this paper. The model consists of two coupled von Foerster-McKendrick-type quasi-linear partial differential equations, two initial conditions, and two boundary conditions. The state variables of the model are male and female population densities. The solutions of these partial differential equations provide explicit time and age dependence of the variables. The initial conditions define the male and female population densities at the initial time, while the boundary conditions compute the male and female births at zero-age by using fertility rates. The assumptions of the nontime-dependence of the death and fe…
Inverse problems and invisibility cloaking for FEM models and resistor networks
2013
In this paper we consider inverse problems for resistor networks and for models obtained via the finite element method (FEM) for the conductivity equation. These correspond to discrete versions of the inverse conductivity problem of Calderón. We characterize FEM models corresponding to a given triangulation of the domain that are equivalent to certain resistor networks, and apply the results to study nonuniqueness of the discrete inverse problem. It turns out that the degree of nonuniqueness for the discrete problem is larger than the one for the partial differential equation. We also study invisibility cloaking for FEM models, and show how an arbitrary body can be surrounded with a layer …
Optimal Heating of an Indoor Swimming Pool
2020
This work presents the derivation of a model for the heating process of the air of a glass dome, where an indoor swimming pool is located in the bottom of the dome. The problem can be reduced from a three dimensional to a two dimensional one. The main goal is the formulation of a proper optimization problem for computing the optimal heating of the air after a given time. For that, the model of the heating process as a partial differential equation is formulated as well as the optimization problem subject to the time-dependent partial differential equation. This yields the optimal heating of the air under the glass dome such that the desired temperature distribution is attained after a given…
From Particle Systems to Partial Differential Equations International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019
2021
This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general…
Local regularity estimates for general discrete dynamic programming equations
2022
We obtain an analytic proof for asymptotic H\"older estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.
Analytical model for the long- distance tracer-transport in plants
2011
International audience; Recent investigations of long-distance transport in plants using non-invasive tracer techniques such as C radiolabeling monitored by positron emission tomography (PET) combined with magnetic resonance imaging (MRI) revealed the need of dedicated methods to allow a quantitative data analysis and comparison of such experiments. A mechanistic compartmental tracer transport model is presented, defined by a linear system of partial differential equations (PDEs). This model simplifies the complexity of axial transport and lateral exchanges in the transport pathways of plants (e.g. the phloem) by simulating transport and reversible exchange within three compartments using j…
Equivalence of viscosity and weak solutions for the normalized $p(x)$-Laplacian
2018
We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$ regularity for the viscosity solutions of the normalized $p(x)$-Laplace equation. As an additional application, we prove a Rad\'o-type removability theorem.