Differential equations hyperbolic form

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. 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

A simplified analysis of production decline curves was initially intended for evaluation of individual oil well production however, this type of analysis can also provide reasonable estimates when applied to multiple-well LNAPL recovery systems. This analytical method is applicable to most types of decline curves, whether they tend to follow exponential, hyperbolic, or harmonic forms. The following general differential equation is applicable to all forms of decline curves ... 

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. 

The theory for classifying linear, second-order, partial-differential equations is well established. Understanding the classification is quite important to understanding solution algorithms and where boundary conditions must be applied. Partial differential equations are generally classified as one of three forms elliptic, parabolic, or hyperbolic. Model equations for each type are usually stated as... 

By considering the combined variable z = x — xj2, we remove the mixed partial differential term from Eq. (4.293). The transformation obtained is the hyperbolic partial differential equation (4.294). This equation represents a new form of the stochastic model of the deep bed filtration and has the characteristic univocity conditions given by relations (4.295) and (4.296). The univocity conditions show that the suspension is only fed at times higher than zero. Indeed, here, we have a constant probability for the input of the microparticles ... 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

Equations (6) and (7) form a set of quasi-linear hyperbolic partial differential equations. A solution of the form ... 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

Mathematical models of catalytic systems in the general form are rather sophisticated. Often, they consist of nonlinear systems of differential equations containing both conventional equations and equations with partial derivatives of parabolic, hyperbolic, and other forms. Efficient simulation is only possible if a well developed qualitative theory of differential equations (mainly, equations with partial derivatives) and high performance programs for computational experiments exist. 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

This is a time-dependent, linear (hyperbolic) partial differential equation. The steady state solutions of this equation (defined by 9 /= 0) are of the form g(z, f) = (p z) where cp satisfies Cf(p z) = 0. These are just the first integrals of the ordinary differential equation system z = /(z). 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

Equation (9.101) can be seen to be a special differential equation with one independent variable. The number of variables in the hyperbolic PDE has thus been reduced from two to one. Comparing Equation (9.101) with the generalized form of Bessel s ... 

Linear second-order partial differential equations in two independent variables are further classified into three canonical forms elliptic, parabolic, and hyperbolic. The general form of this class of equations is... 

Bykov, V. V. [1978] On the structure of a neighborhood of a separatrix contour with a saddle-focus, in Methods of Qualitative Theory of Differential Equation (Gorky Gorky State University), 3-32 [1980] On bifurcations of dynamical systems with a separatrix contour containing a saddle-focus, ibid. 44-72 [1988] On the birth of a non-trivial hyperbolic set from a contour formed by separatrices of a saddle, ibid. 22-32. 

Such a classification can also be applied to higher order equations involving more than two independent variables. Typically elliptic equations are associated with physical systems involving equilibrium states, parabolic equations are associated with diffusion type problems and hyperbolic equations are associated with oscillating or vibrating physical systems. Analytical closed form solutions are known for some linear partial differential equations. However, numerical solutions must be obtained for most partial differential equations and for almost all nonlinear equations. 

The coefficients in the above series in if/p(r) alternate between the hyperbolic sine and cosine value of the uniform contribution, i/r0(z). In contrast to a full linear treatment, which is the usual procedure followed, the 0(if/p) term here does not vanish. As if/0(z) is large we must regard it as satisfying the nonlinear, ordinary differential form of the PB equation,... 

This Chapter is concerned with some of the mathematical tools required to describe special properties of curved surfaces. The tools are to be found in differential geometry, analytical function theory, and topology. General references can be foimd at the end of the Chapter. The reader xminterested in the mathematics can skip the equations and their development. The ideas we want to focus on will be clear enough in the text. A particular class of saddle-shaped (hyperbolic) surfaces called minimal surfaces will be treated with special attention since they are relatively straightforward to treat mathematically and do form good approximate representations of actual physical and chemical structures. 

The forces can be obtained from this analysis by simple differentiation, since the sums are absolutely convergent. Although the form in Eq. 69 has a much better convergence than the original form in Eq. 64, its main advantage is a linear computation time with respect to the number of particles N. To see this, the equation has to be rewritten using the addition theorems for the cosine and the hyperbolic cosine. First, one calculates the eight terms... 

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