Search results for "Boundary value problem"

Existence and uniqueness results for a nonlinear evolution equation arising in growing cell populations

2014

Abstract The present paper is concerned with a nonlinear initial–boundary value problem derived from a model introduced by Rotenberg (1983) describing the growth of a cell population. Each cell of this population is distinguished by two parameters: its degree of maturity μ and its maturation velocity v . At mitosis, the daughter cells and mother cells are related by a general reproduction rule. We prove existence and uniqueness results in the case where the total cross-section and the boundary conditions are depending on the total density of population. Local and nonlocal reproduction rules are discussed.

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.

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…

Functional a posteriori error equalities for conforming mixed approximations of elliptic problems

2014

Optimal control in models with conductive‐radiative heat transfer

2003

In this paper an optimal control problem for the elliptic boundary value problem with nonlocal boundary conditions is considered. It is shown that the weak solutions of the boundary value problem depend smoothly on the control parameter and that the cost functional of the optimal control problem is Frechet differentiable with respect to the control parameter. Optimalus valdymas modeliuose su laidžiu-radioaktyviu šilumos pernešimu Santrauka Darbe nagrinejamas nelokalaus kraštinio uždavinio optimalaus valdymo uždavinys. Parodyta, kad silpnasis kraštinio uždavinio sprendinys tolydžiai priklauso nuo valdomojo parametro, taigi, optimalaus valdymo tikslo funkcija yra diferencijuojama Freše prasme…

On FE-grid relocation in solving unilateral boundary value problems by FEM

1992

We consider FE-grid optimization in elliptic unilateral boundary value problems. The criterion used in grid optimization is the total potential energy of the system. It is shown that minimization of this cost functional means a decrease of the discretization error or a better approximation of the unilateral boundary conditions, Design sensitivity analysis is given with respect to the movement of nodal points. Numerical results for the Dirichlet-Signorini problem for the Laplace equation and the plane elasticity problem with unilateral boundary conditions are given. In plane elasticity we consider problems with and without friction. peerReviewed

A fast Fourier transform based direct solver for the Helmholtz problem

2018

This article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT‐based direct solver is O(N log N) operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed…

The FLO Diffusive 1D-2D Model for Simulation of River Flooding

2016

An integrated 1D-2D model for the solution of the diffusive approximation of the shallow water equations, named FLO, is proposed in the present paper. Governing equations are solved using the MArching in Space and Time (MAST) approach. The 2D floodplain domain is discretized using a triangular mesh, and standard river sections are used for modeling 1D flow inside the section width occurring with low or standard discharges. 1D elements, inside the 1D domain, are quadrilaterals bounded by the trace of two consecutive sections and by the sides connecting their extreme points. The water level is assumed to vary linearly inside each quadrilateral along the flow direction, but to remain constant …

A boundary condition for arbitrary shaped inlets in lattice-Boltzmann simulations

2009

We introduce a mass-flux-based inlet boundary condition for the lattice-Boltzmann method. The proposed boundary condition requires minimal amount of boundary data, it produces a steady-state velocity field which is accurate close to the inlet even for arbitrary inlet geometries, and yet it is simple to implement. We demonstrate its capability for both simple and complex inlet geometries by numerical experiments. For simple inlet geometries, we show that the boundary condition provides very accurate inlet velocities when Re less than or similar to 1. Even with moderate Reynolds number, the inlet velocities are accurate for practical purposes. Furthermore, the potential of our boundary condit…

Optimality of Increasing Stability for an Inverse Boundary Value Problem

2021

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for the Schrödinger equation. The rigorous justification of increasing stability for the IBVP for the Schrödinger equation were established by Isakov [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631--640] and by Isakov et al. [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131--141]. In [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631--640] and [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131--141], the authors showed that the stability of this IBVP increases …