Solutions of the System

After the insertion of the boundaiw conditions the solution of the system of algebraic equations in this case gives the required nodal values of 7 (i.e. T2 to 7io) as... 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

The solution of the system may then be found by elimination or matrix methods if a solution exists (see Matrix Algebra and Matrix Computations ). 

The fact that = 1 indicates that our solution is not very good. Indeed the exact solution of the system is Xi = 1.00010 and I2 = 0.99990, so the result computed by Gauss elimination is pretty bad. 

Using linear programming, determine the MOC solution of the system. 

A numerical algorithm for the solution of the system of Eqs. (15), (19) and (51) consists of the expansion of the two-particle functions into a Fourier-Bessel series. We omit all the details of the numerical method they can be found in Refs. 55-58, 85, 86. In Fig. 3 we show a comparison of the total... 

The typical strategy employed in studying the behavior of nonlinear dissipative dynamical systems consists of first identifying all of the periodic solutions of the system, followed by a detailed characterization of the chaotic motion on the attractors. 

If the roots of the characteristic equation for the abridged system (Xt = 0) have moduli less than one, the zero solution of the system,... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

The problem here consists of finding a continuous in Qt solution of the system of parabolic equations... 

The analytic solution of the system of differential equations in eq. (39.14) can be written as follows ... 

An alternative approach to the solution of the system dynamic equations, is by the natural cause and effect mass transfer process as formulated, within the individual phase balance equations. This follows the general approach, favoured by Franks (1967), since the extractor is now no longer constrained to operate at equilibrium conditions, but achieves this eventual state as a natural consequence of the relative effects of solute accumulation, solute flow in, solute flow out and mass transfer dynamics. 

To determine the characteristics of the 2x1 phase in the system CO/NaCl(100) from general formulae (4.3.47), we equate expressions (4.3.47) and (4.3.48) thus deriving four equations in four unknown parameters, y, ij and A ty with j = S, and A. It is noteworthy that for the spectral lines associated with local vibrations S and A, the vector k assumes two values k = 0 and k = kA (kA is a symmetric point at the boundary of the first Brillouin zone). The exact solution of the system of equations provides parameter values listed in Table 4.3.187 The same parameters were previously evaluated by formulae (4.3.49) without regard for lateral interactions of low-frequency molecular modes." As a consequence, the result was physically meaningless the quantities y and t] proved to be different for vibrations S and A (also see Table 4.3). 

Solution of the system of equations in (3) and (4), subject to ary constraints imposed ty (2). The solution in same cases may be possible analytically, but, mare generally numerical methods are required. The solution may give, for example, Ft as a function of axial position or volume, or C as a function of time. 

A is an m X n matrix whose (/, j) element is the constraint coefficient aij9 and c, b, 1, u are vectors whose components are cjy bjt ljy ujy respectively. If any of the Equations (7.7) were redundant, that is, linear combinations of the others, they could be deleted without changing any solutions of the system. If there is no solution, or if there is only one solution for Equation (7.7), there can be no optimization. Thus the... 

In the above relationship, the coefficients Aj to An depend on the initial conditions of the problem and the exponential values, are determined by the parameters of the system and in fact represent the eigenvalues or roots of the characteristic solution of the system. 

It is possible to follow different paths of rearrangements and substitutions to arrive at a solution of the system of equations, but all must result in the same correct concentrations. 

The exact cross-linkages number distribution of the chains at any moment in time is determined by the solution of the system QJ. 

The solution of the system of the kinetic eauations with coefficients z gives the dependence of the average number of cross-linkages on the time. 

We now consider how Steward s algorithm can help to ascertain whether or not the system of equations describing the process is determinate. It should be noted that if a system of equations having the same number of variables as equations incorporates a subset of equations that contains fewer variables than equations, a unique solution of the system equations is unlikely to exist. We have used the words is unlikely to rather than does not because there are some special classes of equations that specify more than one variable, and if such an equation is included in the system, the system may have a subset of equations with fewer variables than equations and still be determinate. For example, consider the system of Eq. (5) ... 

It should order the equations within the irreducible subsystems so that the minimum number of variables need be iterated or specified as system inputs to obtain a complete solution of the system. 

Figure 6.1 shows the solution of the system for = 0.06 note that the distribution of the three forms of oxygen is asymmetric over the compositional space. Moreover, bridging oxygen is the only form present in the Si02 monomer. 

Because t/,, f), and Z, contain b, the solution of the system A2.65 A2.71 is obtained by an iterative procedure, inserting an approximate value of b into these terms and calculating a new b from equation A2.65. The best intercept is found by use of the equation... 

In Table 2 we present the shells which correspond to our best choice as well as the values of the scaling factors which result from the solution of the system of equations (1). The Madelung potential values on each cluster site calculated with the finite adjusted array are given in Table 3. When these values are compared to that of the Table... 

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