Hyperbolic system

Systems of Logarithms. There are two common systems of logarithms in use (1) the natural (Napierian or hyperbolic) system which uses the base e = 2.71828. . . (2) the common (Briggsian) system which uses the base 10. 

Rhee, Aris, and Amundson, First-Order Partial Differential Equations Volume 1. Theory and Application of Single Equations Volume 2. Theory and Application of Hyperbolic Systems of Quasi-Linear Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1986,1989. 

It is also shown that the. one-dimensional, unsteady flow eqs 2.2.1 and 2.2.2 form a hyperbolic system with two characteristic directions, while the steady plane flow eqs 2.2.4 2.2.5 have the toots for... 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

Fig. 4. 32. Potential singular point surfaces and stable node bifurcation behavior for an intrinsically hyperbolic system ...

Systems in the collinear eZe configuration which have tori would be the antiproton-proton-antiproton (p-p-p) system, the positronium negative ion (Pr- e-e-e)), which corresponds to the case of Z= 1, = 1, and If these systems have bound states, we can see the effect of our finding in the Fourier transform of the density of states for the spectrum. For a positronium negative ion, the EBK quantization was done [34]. Stable antisymmetric orbits were obtained and were quantized to explain some part of the energy spectrum. As hyperbolic systems, H and He have been already analyzed in Refs. 11 and 17, respectively. Thus, Li+ is the next candidate. We might see the effect of the intermittency for this system in quantum defect as shown for helium [14]. 

Typical ratios of gas velocity to solids velocity are about 400, 4200, 1200, at the top of the reactor, in the burning zone, and at the bottom of the reactor, respectively. The solids and gas velocities represent the two characteristic directions for our hyperbolic system. If we plot these velocity curves on a reactor length versus time graph, the characteristic curves for the gas will be essentially horizontal in comparison to the solids stream characteristic because of the large gas to solids velocity ratios. 

First order hyperbolic differential equations transmit discontinuities without dispersion or dissipation. Unfortunately, as Carver (10) and Carver and Hinds (11) point out, the use of spatial finite difference formulas introduces unwanted dispersion and spurious oscillation problems into the numerical solution of the differential equations. They suggest the use of upwind difference formulas as a way to diminish the oscillation problem. This follows directly from the concept of domain of influence. For hyperbolic systems, the domain of influence of a given variable is downstream from the point of reference, and therefore, a natural consequence is to use upstream difference formulas to estimate downstream conditions. When necessary, the unwanted dispersion problem can be reduced by using low order upwind difference formulas. 

Beam, R. M., and Wanning, R. R, An implicit finite difference algorithm for hyperbolic systems in conservation-law form. J. Comp. Phys. 22(1), 87 (1977). 

Such considerations where first carried out by Engquist and Majda [34] for wave equations (linear hyperbolic equations), and developed by many authors later on. In particular Halpern [35] considered the case of parabolic perturbations of hyperbolic systems. 

A good situation to first at look is the one of linear hyperbolic equations or systems. Actually the implication (S3) ==> (S2) has been proven in [59] for certain hyperbolic systems in one space variable. A more general (and simpler) proof heis been given by Renardy [60]. On the other hand, Renardy [61] has constructed a simple example (namely the wave equation Wtt = + u v + with periodic boundary conditions), showing... 

G. Schleiniger, M.C. Calderer and L.P. Cook, Embedded hyperbolic regions in a nonlinear model for viscoelastic flow, in Current Progress in Hyperbolic Systems Rie-mann Problems and Computations, W.B. Lingquist (ed.), Contemporary Mathematics 100, American Mathematical Society, Providence, 1990. 

L. Halpern, Artificial boundary conditions for incompletely parabohc perturbations of hyperbolic systems, SIAM J. Math. Anal., 87 (1985) 213-251. 

A.F. Neves, H. de Souza Ribeiro and 0. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., 67 (1986) 320-344. 

If a system is uniformly hyperbolic, every point in phase space has both stable and unstable directions, and the maximum Lyapunov exponent with respect the maximum entropy measure is positive. The system has the mixing property and is therefore ergodic. The correlation function of observables also shows exponential decay. Uniformly hyperbolicity, which is sometimes rephrased as strong chaos in physical literature, is a well-established class of systems and is controllable by means of many mathematical tools [15]. In hyperbolic systems, there are no sources to make the relaxation process slow. 

It is true that the hyperbolic system is an ideal dynamical system to understand from where randomness comes into the completely deterministic law and why the loss of memory is inevitable in the chaotic system, but generic physical and chemical systems do not belong strictly to such ideal systems. They are not uniformly hyperbolic, meaning that invariant structures are heterogeneously distributed in phase space, and there may not exist a lower bound of instability. It is believed that dynamical systems of such classes are certainly to be explored for our understanding of dynamical aspects of all relevant physical and chemical phenomena. 

However, as compared to hyperbolic systems, it is rather difficult to specify more explicitly than just to say they are nonhyperbolic systems. After developing the theory of hyperbolic systems, the studies of nonhyperbolic systems are now the next targets in mathematical studies, either of which will be hard problems however. 

In contrast to hyperbolic systems, the phase space structure in the mixed system is quite intricate and inhomogeneous, which brings about transport phenomena and relaxation processes essentially different from uniformly hyperbolic cases [3]. A remarkable fact is that qualitatively different classes of motions such as quasi-periodic motions on invariant tori and stochastic motions in chaotic seas coexist in a single phase space. The ordered motions associated with invariant tori are embedded in disordered motions in a self-similar way. The geometry of phase space then reflects the dynamics. 

The application of QBMM to Eq. (C.l) will require a closure when m(7 depends on 7 Nevertheless, the resulting moment equations (used for the QMOM or the EQMOM) and transport equations for the weights and abscissas (used for the DQMOM) will still be hyperbolic. In terms of hyperbolic conservation laws, the moments are conserved variables (which result from a linear operation on /), while the weights and abscissas are primitive variables. Because conservation of moments is important to the stability of the moment-inversion algorithms, it is imperative that the numerical algorithm guarantee conservation. For hyperbolic systems, this is most easily accomplished using finite-volume methods (FVM) (or, more specifically, realizable FVM). The other important consideration is the accuracy of the moment closure used to close the function, as will be described below. 

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