Perturbation methods singular

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... 

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

Use limiting cases, which may be much more easily soluble, to box the problem. Perturbation methods, distinguishing between regular and singular, and asymptotics are often useful (see Perturbations and Asymptotics in Chapter 3). 

From the above one can see that a main difference between the effective Hamiltonian method and the above results is T (d). It should be noted that the results given by Eqs. (3.45)—(3.53) can be obtained by using the singular perturbation method [18,20] and other methods [19,21]. 

Kevorkian, J. and Cole, J.D. (1996). Multiple Scale and Singular Perturbation Methods. New York Springer. 

Then a singular perturbation method can be used to obtain the electrochemical potential and the fluid velocity by matching the inner and the outer solutions. 

T. J. Kaper, in Analyzing Multiscale Phenomena Using Singular Perturbation Methods, J. Cronin and R. E. O Malley, Jr., eds.. Proceedings of Symposia in Applied Mathematics, Vol. 56, American Mathematical Society, Rhode Island, 1999, p. 85. 

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

G. Li and H. Rabitz, A Special Singular Perturbation Method for Kinetic Model Reduction With Application to an H2/O2 Oxidation Model, J. Chem. Phys. 105 (1996) 4065-4075. 

The influence of activity changes on the dynamic behavior of nonisothermal pseudohomogeneoiis CSTR and axial dispersion tubular reactor (ADTR) with first order catalytic reaction and reversible deactivation due to adsorption and desorption of a poison or inert compound is considered. The mathematical models of these systems are described by systems of differential equations with a small time parameter. Thereforej the singular perturbation methods is used to study several features of their behavior. Its limitations are discussed and other, more general methods are developed. 

If the spectrum of linear differential operator contains a pair of pure imaginary eigenvalues then there Is a possibility of interference of two kinds of oscillations - the oscillations of catalytic reaction and the oscillations of catalyst activity- It is not possible to use the singular perturbation method for dynamic behavior investigation- This is done by the following procedure ... 

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

Static bifurcation can be studied by means of singular perturbation method, i.e. taking e - 0. The problem of finding static bifurcation points turns to the steady state problem similar to the CSTRf where is expressed by (16). From this and from the monotonicity of the function G follows that for a given value of bifurcation parameter Da the maximal number of static bifurcation points corresponds to that of the systems with = 1 and Da <0, Da > ... 

R. E. O Malley Jr. Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York, 1991. 

See, for example, the discussions (12, K2) of unsteady low Reynolds number flows based on singular perturbation methods. 

Employing singular perturbation methods to solve the Navier-Stokes equations, Saffman obtains, for the drag, torque about the sphere center, and lift force,... 

Using singular perturbation methods a solution is obtained in the form of a double expansion in the two parameters Re, and No restriction is placed on the parameter except that the-particle not be too near the wall i.e., al(R — Z>) 1, or, what is equivalent. 

This asymptotic expansion works for all eigenvalues (mr), except the first one, that is, = 0- For this eigenvalue, the second and subsequent terms are more singular than the first one. This problem of growing in singularity is a common problem in perturbation methods and will be dealt with in the next homework problem. 

In this chapter, we will present several alternatives, including polynomial approximations, singular perturbation methods, finite difference solutions and orthogonal collocation techniques. To successfully apply the polynomial approximation, it is useful to know something about the behavior of the exact solution. Next, we illustrate how perturbation methods, similar in scope to Chapter 6, can be applied to partial differential equations. Finally, finite difference and orthogonal collocation techniques are discussed since these are becoming standardized for many classic chemical engineering problems. 

Numerical procedures, as we show presently, pose little difficulty when nondimensional parameters are of order of unity. However, some difficulty in convergence arises when parameters become small or large. In this case, the singular perturbation method can be tried to isolate the sharpness of the solution behavior. It is difficult to imagine, however, that numerical methods could ever lead to the simple result Xir) = 1 - v. ... 

We end this chapter by noting that the application of the singular perturbation method to partial differential equations requires more ingenuity than its application to ODEs. However, as in the case of ODEs, the singular perturbation solutions can be used as a tool to explore parametric dependencies and, as well, as a valuable check on numerical solutions. The book by Cole (1968) provides a complete and formal treatment of partial differential equations by the singular perturbation method. 

---