System Of Hyperbolic Differential Equations Research Articles

From Individual to Collective Behavior in Bacterial Chemotaxis

Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these very different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time- and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time, we obtain a single second-order hyperbolic equation for the population density, and using a parabolic scaling, we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement.

Forced oscillation for certain systems of hyperbolic differential equations

In this paper, sufficient conditions are established for the forced oscillation of certain systems of hyperbolic differential equations with deviating arguments. The main result is illustrated by an example.

Oscillation Properties for Systems of Hyperbolic Differential Equations of Neutral Type

Sufficient conditions are established for the oscillations of systems of hyperbolic differential equations of the form∂2∂t2ptuix,t+∑r=1dλrtuix,t−τr =aitΔuix,t+∑j=1m∑k=1saijktΔujx,ρkt −qix,tuix,t−∑j=1m∑h=1lqijhx,tujx,σht, x,t∈Ω×0,∞≡G,i=1,2,…,m,where Ω is a bounded domain in Rn with a piecewise smooth boundary ∂Ω, and Δ is the Laplacian in Euclidean n-space Rn.

Fixing Einstein's Equations

Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations. The physical and geometrical content of the original theory remain unchanged, but dynamical evolution is made sound. Here we present a new hyperbolic formulation in terms of $g_{ij}$, $K_{ij}$, and $\bGam_{kij}$ that is strikingly close to the space-plus-time (``3+1'') form of Einstein's original equations. Indeed, the familiarity of its constituents make the existence of this formulation all the more unexpected. This is the most economical first-order symmetrizable hyperbolic formulation presently known to us that has only physical characteristic speeds, either zero or the speed of light, for all (non-matter) variables. This system clarifies the relationships between Einstein's original equations and the Einstein-Ricci and Frittelli-Reula hyperbolic formulations of general relativity and establishes links to other hyperbolic formulations.

Construction of the solutions of linear hyperbolic systems using laguerre geometric-optic expansions

We discuss a technique for expanding the solution of a linear hyperbolic system of differential equations in a series of Laguerre functions behind the wave front. Solving the original linear system is reduced to solving the eikonal equation that describes the behavior of the wave front and a sequence of transfer systems that describe the Laguerre spectrum. The application of the technique is illustrated using the examples of one-dimensional acoustic and thermoelastic waves.

On an inverse initial-boundary value problem for a one-dimensional hyperbolic system of differential equations

In this paper, the inverse boundary value problem of the hyperbolic system of first-order differential equations is discussed. The estimate of the solution and the quantitative analysis about its stability are obtained, and some stability criteria are established.

A dynamic model for a Timoshenko beam in an elastic-plastic state

In this paper, equations which describe the propagation of elastic-plastic waves of combined stress resultants in a Timoshenko beam under symmetrical bending and tension or compression are derived. It is assumed that the beam material exhibits a weak work-hardening and strain-rate independent behaviour. The normality rule is postulate for the plastic strain rates of an infinitesimal volume element. The wave propagation is described by a hyperbolic system of differential equations, the state variables of which are the stress resultants and the strain rates of an infinitesimal beam element. It is shown that the actual distribution of shear stress in the plastic range need not be taken into account by means of a plastic shear correction factor.

Application of two-point implicit central-difference methods to hyperbolic systems

This paper presents a general solution algorithm for the set of difference equations that arise when two-point central differences are used to approximate the flux difference terms in systems of hyperbolic differential equations. The general algorithm eliminates the weak points associated with the non-standard algorithm reported by Wornom and Hafez [ AIAA Jl 24(2), 215 (1986)]. The disadvantages of their algorithm relate to its implementation. It consists of separate algorithms for subsonic, supersonic, sonic and shock cells, applied individually, which presents a major bookkeeping problem when multiple sonic and shock cells are present. The general algorithm eliminates this problem and introduces an improved shock treatment which produces shocks with at most one interior shock point.

A boundary value problem for a system of hyperbolic differential equations in ℝm

A boundary value problem for a system of hyperbolic differential equations in ℝm

Dynamical analysis of a compressible elastic rod in the current configuration

The dynamical behavior of a compressible elastic rod, left in a nonequilibrium state, is described in the current configuration in a context of large strains and velocities. The phenomenon is governed by a nonlinear hyperbolic system of differential equations in which the unknown are the mass density, the velocity and the stress fields. Some particular cases are simulated from the solution of the associated Riemann problems.

Modeling and numerical simulation of some dynamical phenomena involving elastic-nonlinear bars

A mathematical model is proposed to analyze the dynamical behavior of elastic-nonlinear bars in which the stress-strain curve has no inflection points. This phenomenon is governed by nonlinear hyperbolic systems of differential equations that may present shock waves in their solution. We present the complete solution of the general associated Riemann problem and some typical results obtained numerically by Glimm's method for a finite bar left initially in a nonequilibrium state.

Systems of three differential equations of hydrodynamic type with hexagonal 3-web of characteristics on the solutions

In this article we obtain complete description of the class of hyperbolic systems of three differential equations of hydrodynamic type, for which the characteristics form hexa- gonal 3-web on each solution. For systems in Riemann invariants the hexagonality condition turns out to be equivalent to the weak nonlinearity and semi-Hamiltonianness, which enables us to integrate these systems by the generalized method of hodograph (i). However, in the nondiagonalizable case the hexagonality conditions are apparently in no way connected with semi-Hamilto~iaru~ess and describe the class of the systems that are reduced by the transfor- mations with respect to the solution to constant eigenvalues. The geometry of the charac- teristics on the solutions of quasilinear hyperbolic systems of differential equations has been studied earlier in (2) by the method of Cartan exterior forms. In (3) a classification of the particular solutions of equations for an incompressible ideal liquid and also for simultaneous nonstationa~y flow of a polytropic gas, on which the characteristics form hexa- gonal 3-web, has been carried out. It4is convenient to reduce the system, under consideration, of hydrodynamic type u i -

A class of initially isotropic elastic materials

Basic results of Biot and Hill concerning (i) the elastic moduli of initially isotropic materials and (ii) Cauchy symmetry for large deformation are combined. The result is a system of hyperbolic differential equations to be satisfied by the elastic potential as a function of the principal stretches. Imposing these restrictions, the elastic potential for any strain is shown to be a functional of the elastic potential for pure dilatation. The theory is to some extent corroborated by experimental results on foam rubber by Blatz & Ko ( Trans. Soc. Rheol . 6, 223 (1962)).

On the stability of one-dimensional stationary solutions of hyperbolic systems of differential equations containing points at which one characteristic velocity becomes zero

The stability of the stationary solutions of hyperbolic systems of partial differential equations containing a point at which one of the characteristic velocities becomes zero, is investigated. The functions sought are assumed to be time and coordinate dependent, and their number is arbitrary. The study of stability carried out below is based on the results obtained in /1, 2/, according to which the behaviour of the unsteady perturbations near the critical point is described by a single non-linear partial differential equation irrespective of the number of equations in the initial system. The equation is written in terms of a function analogous to the Riemann invariant connected with the vanishing characteristic velocity. The equation is used below to examine all possible cases of continuous solutions of an arbitrary hyperbolic system of equations with continuous and discontinuous right-hand sides, and conditions are formulated under which the growth of perturbations near the critical point at which one of the characteristic velocities becomes zero, leads to the instability of the whole solution in toto. The investigation is carried out taking into account the onset and development of the perturbations connected with other characteristic velocities which have a constant sign within the region considered.

Solvability of a class of boundary-value problems for systems of hyperbolic differential equations

Solvability of a class of boundary-value problems for systems of hyperbolic differential equations

Construction of a Curvilinear Grid

The construction of overlapping grids is explained and applied to a system of hyperbolic differential equations.

The asymptotic characteristic initial value problem for Einstein’s vacuum field equations as an initial value problem for a first-order quasilinear symmetric hyperbolic system

The asymptotic characteristic initial value problem for Einstein’s vacuum field equations where data are given on an incoming null hypersurface and on part of past null infinity is reduced to a characteristic initial value problem for a first-order quasilinear symmetric hyperbolic system of differential equations for which existence and uniqueness of solutions can be shown. It is delineated how the same method can be applied to the standard Cauchy problems for Einstein’s vacuum and conformal vacuum equations.

On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations

The regular characteristic initial value problem for Einstein’s vacuum field equations where data are given on two intersecting null hypersurfaces is reduced to a characteristic initial value problem for a symmetric hyperbolic system of differential equations. This is achieved by making use of the spin-frame formalism instead of the harmonic gauge condition. The method is applied to the asymptotic characteristic initial value problem for Einstein’s vacuum field equations, where data are given on part of past null infinity and on an incoming null-hypersurface. A uniqueness theorem for this problem is proved by showing that a solution of the problem must satisfy a regular symmetric hyperbolic system of differential equations in a neighbourhood of past null infinity.

Dynamics of stress redistribution in fiber fracture in composites

1. On the basis of a unidimensional model, an analysis was presented of dynamic effects in stress redistribution both within a fractured fiber and in fibers adjacent to this fiber. 2. The process of movement of the fractured fiber and the first three belts of adjacent fibers is described for different stages of matrix shear strain and the case of fiber separation from the matrix (delamination) by systems of hyperbolic differential equations, the solution of which was obtained by the method of characteristics with numerical realization on a computer. 3. Analysis of stress redistribution in the fibers adjacent to the fractured fiber revealed overloading waves which are formed in the fracture plane and, travelling along the fibers, may result in their fracture a substantial distance from the fracture plane. 4. Development of the delamination process leads to a reduction in the overloads on the adjacent fibers.

Generalized Runge-Kutta methods for coupled systems of hyperbolic differential equations

Runge-Kutta formulas are discussed for the integration of systems of differential equations. The parameters of these formulas are square matrices with component-dependent values. The systems considered are supposed to originate from hyperbolic partial differential equations, which are coupled in a special way. In this paper the discussion is concentrated on methods for a class of two coupled systems. For these systems first and second order formulas are presented, whose parameters are diagonal matrices. These formulas are further characterized by their low storage requirements, by a reduction of the computational effort per timestep, and by their relatively large stability interval along the imaginary axis. The new methods are compared with stabilized Runge-Kutta methods having scalar-valued parameters. It turns out that a gain factor of 2 can be obtained.