Singular perturbation

Lam S H and Goussis D A 1988 Understanding complex chemical kinetics with computational singular perturbation 22nd Int. Symp. on Combustion ed M C Salamony (Pittsburgh, PA The Combustion Institute) pp 931-41... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

By assumption, the mass ratio = m/M is a small parameter. Thus, rescaling the Schrbdinger equation properly in time and potential transforms it into the singularly perturbed equation... 

Bornemann, F. A. Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems. Manuscript (1997) 146pp... 

Moet H.J.K. (1982) Asymptotic analysis of the boundary in singularly perturbed elliptic variational inequalities. Lect. Notes Math. 942, 1-17. 

Schuss Z. (1976) Singular perturbations and the transition from thin plate to membrane. Proc. Am. Math. Soc. 58, 139-147. 

Some modification of the describing monotone difference scheme for divergent second-order equations was made by Golant (1978) and Ka-retkina (1980). In Andreev and Savin (1995) this scheme applies equally well to some singular-perturbed problems. Various classes of monotone difference schemes for elliptic equations of second order were composed by Samarskii and Vabishchevich (1995), Vabishchevich (1994) by means of the regularization principle with concern of difference schemes. 

Alekseevskii, M. (1984) Difference schemes of higher-order accuracy for some singular-perturbed boundary-value problems. Differential Equations, 17, 1177-1183 (in Russian). 

Barcilon, V, Singular Perturbation Analysis of the Fokker-Planck Equation Kramer s Underdamped Problem, SIAM Journal of Applied Mathematics 56, 446, 1996. 

For a large the singular perturbation method should be used[5]. For this, we transform the independent variable x as x=l-t. Then Eq. (1) becomes... 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

Fox, R. O. (1989). Steady-state IEM model Singular perturbation analysis near perfectmicromixing limit. Chemical Engineering Science 44, 2831-2842. 

Using a somewhat more sophisticated singular perturbation method, Acrivos and Taylor (1962) obtained... 

Our goal is the study of reactive flows through slit channels in the regime of Taylor dispersion-mediated mixing and in this chapter we will develop new effective models using the technique of anisotropic singular perturbations. 

Recent approach using the anisotropic singular perturbation is the article by Mikelic et al. (2006). This approach gives the error estimate for the approximation and, consequently, the rigorous justification of the proposed effective models. It uses the strategy introduced by Rubinstein and Mauri (1986) for obtaining the effective models. 

In the article Balakotaiah and Chang (1995) the surface reactions are much faster and do not correspond to our problem. In order to compare two approaches we will present in the paragraph from Section 3.4 computations with our technique for the timescale chosen in Balakotaiah and Chang (1995) and we will see that one gets identical results. This shows that our approach through the anisotropic singular perturbation reproduces exactly the results obtained using the center manifold technique. 

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... 

The mathematical theory of dissipative structures is mainly based on approximate methods such as bifurcation theory of singular perturbation theory. Situations like those described in Section VI and that permit an exact solution are rather exceptional. 

The LEN approximation will be employed in various electro-diffusional contexts in Chapters 3, 4, and 6. In particular, in Chapter 4 we shall elaborate upon the limits of applicability of the LEN approximation. It will be further treated as a leading approximation for the singularly perturbed system (1.9), (l.lld) in Chapter 5. 

As pointed out in the Introduction, it is customary in the treatment of such systems to assume local electro-neutrality (LEN), that is, to omit the singularly perturbed higher-order term in the Poisson equation (1.9c). Such an omission is not always admissible. We shall address the appropriate situations at length in Chapter 5 and partly in Chapter 4. We defer therefore a detailed discussion of the contents of the local electro-neutrality assumption to these chapters and content ourselves here with stating only that this assumption is well suited for a treatment of the phenomena to be considered in this chapter. 

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

Determining o from (3.3.35) concludes the construction of the leading term in the direct procedure of singular perturbation. Unfortunately, an essential difficulty already arises at the construction of the first correction. 

A natural candidate for Uo(x,e) is the composite leading term of the above singular perturbation procedure, with the outer and inner parts defined by (3.3.27), (3.3.32). This composite term is of the form... 

We point out that the results of locally electro-neutral studies should be extrapolated upon the nonreduced systems with a certain caution even for e very small. This is so because, to the best of the author s knowledge, no asymptotic procedure for the singularly perturbed one-dimensional system (4.1.1), (4.1.2) has been developed so far that would be uniformly valid for the entire range of the operational parameters (e.g., for arbitrary voltages and fixed charge densities). 

F. Brezzi, A. Capelo, and L. Gastaldi, A Singular Perturbation for Semiconductor Device Equations, to appear. 

The proper singular perturbation treatment has thus to take care of this initial stage. Probably the simplest way to do this is via a matched asymptotic expansion procedure, with the outer solution of the type (5.2.13), (5.2.14), valid for t = 0(1), matched with an initial layer solution that has an internal layer at x = 0. 

Because of the complex nature of the Painleve transcendents and of the resulting difficulties in satisfying the boundary conditions we shall not proceed with the exact analytical solution of b.v.p. (5.3.6) (5.3.8) any further, but rather we turn to an asymptotic and numerical study of this singular perturbation problem. 

P. A. Markowich and C. A. Ringhofer, A singularly perturbed boundary value problem modelling a semiconductor device, SIAM J. Appl. Math., 44 (1984), pp. 231-256. 

Tyson, J. J. and Keener, J. P. (1988). Singular perturbation theory of traveling waves in excitable media (a review). Physica, D 32, 327-61. (December)... 

We shall now solve the Kramers equation (7.4) approximately for large y by means of a systematic expansion in powers of y-1. Straightforward perturbation theory is not possible because the time derivative occurs among the small terms. This makes it a problem of singular perturbation theory, but the way to handle it can be learned from the solution method invented by Hilbert and by Chapman and Enskog for the Boltzmann equation.To simplify the writing I eliminate the coefficient kT/M by rescaling the variables,... 

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