Hyperbolic type

Phenomena of propagation such as vibrations are characterized by equations of hyperbolic type which are essentially different in their properties from other classes such as those which describe equihb-rium (elhptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows ... 

Natural-draft, or hyperbolic-type, towers have been in use since about 1916 in Europe and have become standard equipment for the watercooling requirements of British power stations. They are primarily... 

In this chapter we study the stability with respect to the initial data and the right-hand side of two-layer and three-layer difference schemes that are treated as operator-difference schemes with operators in Hilbert space. Necessary and sufficient stability conditions are discovered and then the corresponding a priori estimates are obtained through such an analysis by means of the energy inequality method. A regularization method for the further development of various difference schemes of a desired quality (in accuracy and economy) in the class of stability schemes is well-established. Numerous concrete schemes for equations of parabolic and hyperbolic types are available as possible applications, bring out the indisputable merit of these methods and unveil their potential. 

In the study of nonstationary processes described by partial differential equations of parabolic and hyperbolic types... 

Example 6 To avoid generality for which we have no real need, the object of investigation is the equation of hyperbolic type in the rectangle Go = 0 < < 1, a = 1,2. The boundary conditions of the first kind... 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

Phenomena of propagation such as vibrations are characterized by equations of hyperbolic type which are essentially different in their... 

Tower design relying on natural air movement, esp. hyperbolic type. Nozzles ... 

It will be shown in another chapter that a possible type of distribution generated by certain processes is of the hyperbolic type (see Chapter 22). The number of particles increases with decreasing size until we reach the dimensions of molecules, atoms, and electrons. There appears to be no definite limit to which material may be subdivided. As the particles increase in size they become relatively fewer in number. The hyperbolic curve is a close approximation of the subdivision of matter in the universe. 

The viscoelastic constitutive equations are of hyperbolic type and it is well-known that numerical solutions require special care. The classical Galerkin method was very shown to be inadequate for such problems. Special techniques were developed based on the very general... 

Bradshaw et al. (B3) use Eqs. (40) to derive a differential equation for the turbulent shear stress t. The transport velocity Qa is taken as (Tmei/p), where Tm x is the maximum value of riy) in the boundary layer. G and I are prescribed as functions of the position across the boundary layer, and o is essentially taken as constant. Together with Eqs. (10a,b), Eq. (36) gives a closed set of equations for U, V, and t this system is of hyperbolic type, with three real characteristic lines. Bradshaw et al. construct a numerical solution using the method of characteristics it can also be done using small streamwise steps with an explicit difference scheme (Nl A. J. Wheeler and J. P. Johnston, private communications). There is a great physical appeal to the characteristics, especially since it is found that the solutions along the outward-going characteristic dominates the total solution. This... 

Therefore, the state of two points a and b in any time step, considering the corrected leakage effect and hence by notice to the equations that govern the problem, being hyperbolic types of equations, during the time of the leakage, the effect is transferred permanently as a third boundary addition to the upstream and downstream boundaries to the solution field. 

In Figure 4 the overall polymerization time is plotted as a function of one or the other active species concentration. A hyperbolic type dependence of tp on [active species] is evident, with a very sharp decrease of tp in the concentration range between 0.3 and 0.7 mole %, and a much slower decrease at higher concentrations. At the highest levels of active species concentrations,tp is very low (ca. 3 min) and this value compares rather well with the usual reaction times for the RIM technology. Non-equimolar concentration conditions roughly follow the same pattern, as evidenced from the data quoted in Table V, and allow to underline the prominent role of [I] on tp, whereas [A] has a much lower relevance on it. 

The determination of more comprehensive coking mechanisms and rate equations requires simultaneous treatment of all experimental data to enable all the relevant parameters related to coking to be considered. After analysing the experimental data, numerical values of the rate and adsorption equilibrium constants were determined by statistical tests, and models were rejected if a negative constant was estimated at more than one temperature. It was found that the hyperbolic type of decay, as described in Equation (1), gives the best fit from the 9 models tested because it gives the least error from the sum of squares analysis [8],... 

The proposed methodology for computer-aided optimal design in the development of YSZ-based gas sensors comprises three phases. Firstly, the complete mathematical model with distributed temporal and spatial parameters for electrochemical gas sensors is presented as a system of the differential equations in private derivatives of parabolic and hyperbolic types. The complexity of physical and chemical interactions, represented in this model, allows performing a mathematical description of the electrochemical gas sensors toward standardization of the calculating procedures. The complete mathematical model and the algorithm of transfer from the complete... 

The problem is reduced to solving a linear partial differential equation of hyperbolic type (the telegraph equation [482]) in the region t > 0, 0 < Y < oo. Specifically, we have the equation... 

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