Singular perturbation, zero

The original model regarding surface intermediates is a system of ordinary differential equations. It corresponds to the detailed mechanism under an assumption that the surface diffusion factor can be neglected. Physico-chemical status of the QSSA is based on the presence of the small parameter, i.e. the total amount of the surface active sites is small in comparison with the total amount of gas molecules. Mathematically, the QSSA is a zero-order approximation of the original (singularly perturbed) system of differential equations by the system of the algebraic equations (see in detail Yablonskii et al., 1991). Then, in our analysis... 

Modeling problems nearly always contain parameters, which are connected to the physicochemical dynamics of the system. These parameters may take a range of values. The solution obtained when the parameter is zero is called the base case. If one of the parameters is small, the behavior of the system either deviates slightly from the base case, or it can take a trajectory that is remote from the base case. The analysis of systems having the former behavior is called regular perturbation, whereas that of the latter is referred to as singular perturbation. 

Now, all the tools of the standard singular perturbation theory can be applied to the analysis of the system. These tools include its decomposition to slow and fast subsystems and reduction of the near steady-state dynamics to dynamics on invariant slow manifolds (slow curves in the present model). The zero approximation, e = 0 k 00), of the slow invariant curve is given... 

It is known that, in the case of singularly perturbed elliptic equations for which (as the parameter s equals zero) the equation does not contain any derivatives with respect to the space variable, the principal term in the singular part of the solution is described by an ordinary differential equation similar to Eq. (1.16a) (see, e.g., [3-6]). Thus, it can be expected that, when solving singularly perturbed elliptic and parabolic equations using classical difference schemes, one faces computational problems similar to the computational problems for the boundary value problem (1.16). 

In Section II (see Sections II.B, II.D) we considered finite difference schemes in the case when the unknown function takes given values on the boundary. The boundary value problem for the singularly perturbed parabolic equation on a rectangle, that is, a two-dimensional problem, is described by Eqs. (2.12), while the boundary value problem on a segment, that is, a one-dimensional problem, is described by equations (2.14). In Section II.B classical finite difference schemes were analyzed. It was shown that the error in the approximate solution, as a function of the perturbation parameter, is comparable to the required solution for any fine grid. For the above mentioned problems special finite difference schemes were constructed. The error in the approximate solution obtained by the new scheme does not depend on the parameter value and tends to zero as the number of grid nodes increases. 

In general, if all (n = l,. .., A7e) are distinct, then A will be full rank, and thus a = A 1 /3 as shown in (B.32). However, if any two (or more) (< />) are the same, then two (or more) columns of Ai, A2, and A3 will be linearly dependent. In this case, the rank of A and the rank of W will usually not be the same and the linear system has no consistent solutions. This case occurs most often due to initial conditions (e.g., binary mixing with initially only two non-zero probability peaks in composition space). The example given above, (B.31), illustrates what can happen for Ne = 2. When ((f)) = ()2, the right-hand sides of the ODEs in (B.33) will be singular nevertheless, the ODEs yield well defined solutions, (B.34). This example also points to a simple method to overcome the problem of the singularity of A due to repeated (< />) it suffices simply to add small perturbations to the non-distinct perturbed values need only be used in the definition of A, and that the perturbations should leave the scalar mean (4>) unchanged. 

X(K), the perturbation characteristic, is shown in Fig. 5.2d. It is negative and approaches zero for K > 2kF. At 2kF, there is a logarithmic singularity responsible for the Friedel oscillations of < eff(r)./(X), the local-field correction, takes care of the modifications due to correlation and exchange. For crystalline... 

Problem 3.3. Equation (3.45) determining r (d) has a singularity at the Brewster angle and cannot be used directly. The reflection coefficient can be evaluated as follows. Let us introduce a small perturbation in the system by setting e = 1 + a, with a eventually tending to zero. Using Eq. (3.22) for 0 = arctan(n) one finds in the lowest nonvanishing order in a... 

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