Mixed problems for hyperbolic equations with a small parameter
Model of pulverized coal combustion in a furnace
We describe behavior of the air-coal mixture using the Navier–Stokes equations for gas and particle phases, accompanied by a turbulence model. The undergoing chemical reactions are described by the Arrhenian kinetics (reaction rate proportional to where is temperature). We also consider the heat transfer via conduction and radiation. Moreover we use improved turbulence-chemistry interactions for reaction terms. The system of PDEs is discretized using the finite volume method (FVM) and an advection...
Modeling of vibration for functionally graded beams
In this study, a vibration problem of Euler-Bernoulli beam manufactured with Functionally Graded Material (FGM), which is modelled by fourth-order partial differential equations with variable coefficients, is examined by using the Adomian Decomposition Method (ADM).The method is one of the useful and powerful methods which can be easily applied to linear and nonlinear initial and boundary value problems. As to functionally graded materials, they are composites mixed by two or more materials at a...
Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory
A matrix framework is developed for single and multispan micro-cantilevers Timoshenko beam models of use in atomic force microscopy (AFM). They are considered subject to general forcing loads and boundary conditions for modeling tipsample interaction. Surface effects are considered in the frequency analysis of supported and cantilever microbeams. Extensive use is made of a distributed matrix fundamental response that allows to determine forced responses through convolution and to absorb non-homogeneous...
Modelling of singularities in elastoplastic materials with fatigue
The hypothesis that, on the macroscopic level, the accumulated fatigue of an elastoplastic material with kinematic hardening can be identified from the mathematical point of view with the dissipated energy, is used for the construction of a new constitutive elastoplastic fatigue model. Its analytical investigation characterizes conditions for the formation of singularities in a finite time. The corresponding constitutive law is then coupled with the dynamical equation of motion of a one-dimensional...
Modulation of the Camassa-Holm equation and reciprocal transformations
We derive the modulation equations (Whitham equations) for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit a bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry...
Monge-Ampère equations and surfaces with negative Gaussian curvature
In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?...
Monotone (A,B) entropy stable numerical scheme for Scalar Conservation Laws with discontinuous flux
For scalar conservation laws in one space dimension with a flux function discontinuous in space, there exist infinitely many classes of solutions which are L1 contractive. Each class is characterized by a connection (A,B) which determines the interface entropy. For solutions corresponding to a connection (A,B), there exists convergent numerical schemes based on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes, corresponding to computationally less expensive monotone...
Monotone method for nonlineer nonlocal hyperbolic problems.
Monotonicity of the number of passages in linear chains and of the basis reproduction number in epidemic models.
Monotoniesätze für hyperbolische Anfangswertaufgaben und Einschließung von Lösungen.
Motion planning for a class of boundary controlled linear hyperbolic PDE’s involving finite distributed delays
Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived employing Mikusiński’s...
Motion planning for a class of boundary controlled linear hyperbolic PDE's involving finite distributed delays
Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived employing ...
Moving Dirichlet boundary conditions
This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second...
Multidimensional decay in the van der Corput lemma
We establish a multidimensional decay of oscillatory integrals with degenerate stationary points, gaining the decay with respect to all space variables. This bridges the gap between the one-dimensional decay for degenerate stationary points given by the classical van der Corput lemma and the multidimensional decay for non-degenerate stationary points given by the stationary phase method. Complex-valued phase functions as well as phases and amplitudes of limited regularity are considered. Conditions...
Multi-dimensional Riemann problems for linear hyperbolic systems
Multi-Grid Methods for Hamiltonian-Jacobi-Bellmann Equations.
Multi-scale Modelling for Threshold Dependent Differentiation
The maintenance of a stable stem cell population in the epidermis is important for robust regeneration of the stratified epithelium. The population size is usually regulated by cell secreted extracellular signalling molecules as well as intracellular molecules. In this paper, a simple model incorporating both levels of regulation is developed to examine the balance between growth and differentiation for the stem cell population. In particular, the dynamics of a known differentiation regulator c-Myc,...
Nash equilibria for a model of traffic flow with several groups of drivers
Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i-th group consists of κi drivers, sharing the same departure and arrival costs ϕi(t),ψi(t). For any given population sizes κ1,...,κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure...
Navier–Stokes regularization of multidimensional Euler shocks