Basic system of equations

Let us transform these equations into an equivalent more practical set  

Taking into account that electroneutrality (equation (1.8)) implies that Zz c = 0 and introducing the electric conductivity o, 

This equation shows that the current density is due to the gradient of the potential U and to the gradients of the concentrations c.. This means that Ohm s law does not hold. There is also a diffusion current. 

For homogeneous reactions in the bulk solution, the term FZz A vanishes. By virtue of the assumption of 

Finally we can substitute equation (1.4-) in (1.6) and make use of equation (1,10) with the result 

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One can consider equations (3.37), (3.39) and (3.41) to be a basic system of equations for description of dynamics of entangled systems. The system can be investigated analytically in linear approximation as will be demonstrated in the ensuing chapters. However, to study these non-linear equations in complete form, one has to use numerical methods of simulation of the stochastic processes for the particle coordinates. 

It is natural to equate the dynamical variations of the basic operators 6Xk and 5Yk, obtained with the scaling (7) and Thouless (25) perturbed wave functions. This provides the additional relation between the amplitudes and deformations Qk and pk and finally result in the system of equations for the unknowns pk and p. ... 

Within the quantum formulation of OCT, the basic variational procedure leads to a set of equations for the optimal laser held, which include two Schrodinger equations to describe the dynamics starting from the initial and the target state wavepackets. The optimal laser held is given by the imaginary part of the correlation function for these two wavepackets. This system of equations of optimal control must generally be solved iteratively, making it an extremely computationally expensive approach for multidimensional systems. 

Typke has introduced the rs-fit method [7] where Kraitchman s basic principles are retained. A system of equations is set up for all available isotopomers of a parent (not necessarily singly substituted) and is solved by least-squares methods for the Cartesian coordinates (referred to the PAS of the parent) of all atomic positions that have been substituted on at least one of the isotopomers The positions of unsubstituted atoms need not be known and cannot be determined. The method is presented here with two recent improvements true derivatives are used for the Jacobian matrix X, and the problem of the observations and theircovariances, which is rather elaborate, is fully worked out. The equations are always given for the general asymmetric rotor, noting that simplifications occur in more symmetric situations, e.g. for linear molecules, which could nonetheless be treated within the framework presented. 

We will first study a solution of the system of equations using an integral method, which leads to a simple and closed approximate solution. Integral methods are applied to many other boundary layer problems, in particular those involving compressible flows. However, with the introduction of electronic computers they have lost their importance. The basic idea behind the integral methods is that we do not need a complete solution of the boundary layer equation and are content instead with a solution that satisfies the equations in an integral (average) fashion over the entire boundary. 

Returning to Eq.(2-141), the following basic system of differential equations are obtained ... 

The solution is represented by a column vector that is equal to the matrix product A C. In order for a matrix to possess an inverse, it must be nonsingular, which means that its determinant does not vanish. If the matrix is singular, the system of equations cannot be solved because it is either linearly independent or inconsistent. We have already discussed the inversion of a matrix in Chapter 9. The difficulty with carrying out this procedure by hand is that it is probably more work to invert an n by n matrix than to solve the set of equations by other means. However, with access to Mathematica, BASIC, or another computer language that automatically inverts matrices, you can solve such a set of equations very quickly. 

In order to present here some basic results of the weakly stability analysis, we consider below a general reaction-diffusion system of equations. We assume that the problem has a one-dimensional traveling wave solution that loses stability in the same way as the gasless combustion wave as discussed in greater detail below. A study of a general reaction-diffusion system rather than a specific model is useful, because it allows us to focus on general properties of the solution, independent of a particular model. 

Scheme 3 summarizes the way in which the stochastic probabihties are generated from the rates of the different reactions. The basic assumption here is that the relative probabihties of elementary reactions at the microscopic level, Tii/Tij, are equal to their relative reaction rates (macroscopic), rj/rj. Thus, the relative reaction rates (Eq. 1 of Scheme 3) for all pairs of the considered reactive events together with the probability normalization condition (Eq. 2 of Scheme 3) constitute the system of equations that can be solved for the absolute probabihties of aU the events at a given stage. With this assumption, one can use the experimentally determined reaction rates or the theoreticaUy calculated relative rate constants, obtained from the energetics of the elementary reactions with the standard Eyring exponential equation. The Eyring equation introduces as well a temperature dependence of ah the relative probabihties (as in Eq. 3 of Scheme 3). 

For the separation of the liquid feed M (composition x ) into two streams of composition Xg (product, distillate) and Xf, (bottom product), the required number of theoretical plates can be determined graphically with the aid of the McCabe-Thiele method, which was used in the past because no computers were available to solve the extensive systems of equations for mass balances and equilibrium relationships. This method is no longer of practical importance, but it is an excellent didactic aid for understanding the basic principles of rectification. 

Referring to fig. 3.13> the BEM applied on the basic cell (1,2,2 ,1 ) yields a system of equations that can be reordered such that the potential and the normal electric field along the boundary 1-1 ... 

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