System Of First-order Hyperbolic Equations Research Articles

Inverse scattering problem for nonstrict hyperbolic system on the half-axis with a nonzero boundary condition

Abstract The paper considers the scattering problem for the first-order system of hyperbolic equations on the half-axis with a nonhomogeneous boundary condition. This problem models the phnomennon of wave propagation in a nonstationary medium where an incoming wave unaffected by a potential field. The scattering operator on the half-axis with a nonzero boundary condition is defined and the uniqueness of the inverse scattering problem (the problem of finding the potential with respect to scattering operator) is studied.

An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

On the problem of modeling the acoustic radiation pattern of source for the 2D first-order system of hyperbolic equations

In this paper the problem of modeling the acoustic radiation pattern of source is considered. The problem is formulated in the form of control problem for the 2D first-order system of hyperbolic equations. This problem is related to ultrasound tomography for early breast cancer detection. The modelling of the acoustic radiation patterns of sources allows us to improve the resolution of acoustic tomography.

A new continuum model for general relativistic viscous heat-conducting media.

The lack of formulation of macroscopic equations for irreversible dynamics of viscous heat-conducting media compatible with the causality principle of Einstein's special relativity and the Euler-Lagrange structure of general relativity is a long-lasting problem. In this paper, we propose a possible solution to this problem in the framework of SHTC equations. The approach does not rely on postulates of equilibrium irreversible thermodynamics but treats irreversible processes from the non-equilibrium point of view. Thus, each transfer process is characterized by a characteristic velocity of perturbation propagation in the non-equilibrium state, as well as by an intrinsic time/length scale of the dissipative dynamics. The resulting system of governing equations is formulated as a first-order system of hyperbolic equations with relaxation-type irreversible terms. Via a formal asymptotic analysis, we demonstrate that classical transport coefficients such as viscosity, heat conductivity, etc., are recovered in leading terms of our theory as effective transport coefficients. Some numerical examples are presented in order to demonstrate the viability of the approach. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

R2* relaxometry in sclerotic hippocampi of patients with temporal lobe epilepsy

Chapter 7 focuses on the computation of one-dimensional channel flow by the numerical method of characteristics. Techniques for construction of the characteristic network are presented. The methods of specified space and time intervals are developed, and are used to solve the dam-break problem. The chapter also addresses the computation of one-dimensional channel flow by finite-difference methods. Conservative and non-conservative formulations are explored. The two-step Lax-Wendroff and Preissmann methods are analyzed. Shock capturing and non-reflecting boundary conditions are described in detail. Fourier analysis of systems of first-order hyperbolic equations is carried out, and the conditions for stability of explicit schemes are examined. Finally, this chapter extends the finite element and finite-volume methods to open-channel flow. The emphasis is on monotonicity preserving methods and shock capturing. The Riemann problem is introduced, and flux limiting is used to construct TVD schemes for open-channel flow.

Global existence of smooth solutions to relativistic string equations in Schwarzschild space-time for small initial data

This paper is concerned with the motion of relativistic strings in Schwarzschild space-time. In a general framework, we first analyze the basic equations for the motion of a [Formula: see text]-dimensional extended object in a general enveloping space-time [Formula: see text], which is a given Lorentzian manifold, and then we investigate some important properties enjoyed by the equations for the motion of relativistic strings in Schwarzschild space-time. In particular, the equations are shown to form a totally linearly degenerate system of first-order hyperbolic equations in [Formula: see text] dimensions. Based on this observation and under suitable assumptions, we are able to prove the global existence of smooth solutions to the Cauchy problem for the equations of motion of relativistic strings (in Schwarzschild space-time) with sufficiently small arc length.

Optimization of one class of hyperbolic systems with smooth controls

In this paper we consider the optimal control problem for a system of first-order hyperbolic equations. The right-hand side of this system is determined by a control system of ordinary differential equations. Admissible controls belong to the class of smooth functions. We obtain an optimality condition and propose a general scheme of improvement methods for the mentioned problem.

COLLATION AND UPWINDING FOR THERMAL FLOW IN PIPELINES: THE LINEARIZED CASE

Simulating thermal effects in pipeline flow involves solving a coupled non-linear system of first-order hyperbolic equations. The advection term has two large eigenvalues of opposite signs, corresponding to the propagation of high-speed sound waves, and one eigenvalue close to or even equal to zero, representing the much slower fluid flow velocity, which transports temperature. Standard collocation methods work well for isothermal flow in pipelines, but the stagnating eigenvalue causes difficulties when thermal effects are included. In a companion paper we formulate and analyse a new numerical method for the non-linear system which arises in thermal modelling. The new method applies to general coupled systems of non-linear first-order hyperbolic partial differential equations with one degenerate eigenvalue. In the present paper we focus on a linearized constant coefficient form of the thermal flow equations. This substantially simplifies presentation of the error analysis for the numerical scheme. We also include numerical results for the method applied to the fully non-linear system. Both the error analysis and the numerical experiments show that the difficulties that come from the application of standard collocation can be overcome by using upwinded piecewise constant functions for the degenerate component of the solution.

The relationship between immiscible and miscible displacement in porous media

Displacement processes in porous media share several common characteristics. Fundamental among them is the representation of the momentum balance via Darcy’s law and its multiphase extension (Bear, 1972). Such description leads to systems of equations the solution of which parallels the behavior of the solution of processes in multicomponent chromatography. For instance, it has been recognized that noncapillary, multiphase displacement in one-dimensional (rectilinear or radial) porous media can be solved by techniques identical to those used in chromatographic transport (Helfferich, I98 I; Rhee et al., 1986, and references therein). This similarity arises naturally for the case of one-dimensional flow geometries and in the absence of dissipative terms (diffusion, dispersion, or capillarity), where both multiphase and multicomponent chromatography processes are formulated by systems of first-order hyperbolic equations. In this note we pursue further this relationship in mathematical representation between multiphase flow and chromatographic processes in porous media. Specifically, we consider two-phase, immiscible displacement and single-phase, miscible displacement in the presence of equilibrium adsorption. It is demonstrated that the two problems are mathematically equivalent, regardless of the dimensionality of the flow system and the presence of dissipative effects. Such an analogy is of practical importance in the analysis of various process characteristics, for instance in describing the evolution of unstable two-dimensional disturbances during viscous fingering. The latter is a topic of active current investigations (Homsy, 1987, and references therein), and of fundamental importance in process performance. Any similarities between seemingly different processes would be of considerable help in an effort to reduce complexity and to uncover common mechanisms. An extension to multiphase immiscible displacement and multicomponent miscible displacement is also briefly discussed. Mathematical Description

Pseudo-characteristic method of lines solution of first-order hyperbolic equation systems*1

Pseudo-characteristic method of lines solution of first-order hyperbolic equation systems*1

Difference schemes with a splitting operator for systems of first-order hyperbolic equations

WE construct difference schemes with a splitting operator which approximate with order O(τ + ¦h¦ 2) a first-order system of hyperbolic equations and a system of equations of the mixed type for which a Cauchy problem periodic in the space variables is established. Stability in L 2 on a layer is proved for the solution of these schemes. In this note we consider a difference scheme with a splitting operator, approximating with order O(τ + ¦h¦ 2) a system of first-order hyperbolic equations for which we establish a Cauchy problem, periodic in the space variables, and for the solution of which stability is proved. The proposed scheme is similar to the difference scheme for parabolic equations studied in [1,2], and can easily be transferred to the case of a system of equations of the mixed type, considered in [3, 4]. Difference schemes for first-order systems and equations of hyperbolic type were considered in [5–7] and in a number of other papers.