Small parameters asymptotic solutions

For great values of oo equation (5.1.23) is nothing but a differential equation with a small parameter multiplying the derivative, whose main term of the asymptotic expansion is the solution of the degenerate equation (o-q 1 = 0)... 

In conclusion, it must be noted that the equations to describe the transient behaviour of heterogeneous catalytic reactions, usually have a small parameter e = Altsot/Alt t. Here Atsot = bsS = the number of active sites (mole) in the system and Nfot = bg V = gas quantity (mole). Of most importance is the solution asymptotes for kinetic equations at A/,tsot/7Vtflt - 0, 6S, bg and vin/S being constant. Here we deal with the parameter SjV which is readily controlled in experiments. The case is different for the majority of the asymptotes examined. The parameters with respect to which we examine the asymptotes are difficult for control. For example, we cannot, even in principle, provide an infinite increase (or decrease) of such a parameter as the density of active sites, bs. Moreover, this parameter cannot be varied essentially without radical changes in the physico-chemical properties of the catalyst. Quasi-stationarity can be claimed when these parameters lie in a definite range which does not depend on the experimental conditions. 

A new circumstance, in comparison with Eqs. (5.22) and (5.23), is the fact that the asymptotic system (5.87), (5.88) contains an infinite number of equations (0 < K < oo, 0 < k < oo). If the solution for polarization moments is found by way of expansion into a series over a small parameter, then the argument produced in Section 5.4 can be applied here without change. However, if the system of equations (5.87) and (5.88) is solved by computer, then we are deprived, in principle, of the possibility of accounting for all emerging polarization moments. What could be done in this case ... 

In another study, the nonlinear PB equation with added salt has been approximately solved for a cylindrical polyion of radius a, by matching the near- and far-field solutions in an asymptotic expansion in terms of a small parameter, s= l/ln(l/Ka), and where k 1 is the Debye screening length [60],... 

Prom Eqn. (2.6.65), it is apparent that this is a singular perturbation problem (as the highest derivative term is multiplied by the small parameter) and then one can use matched asymptotic expansion to obtain (f> by describing the solution in terms of outer and inner solutions. 

Thus we see, as expected, that the exact solution (3-157) reduces to a self-similar form for <direct examination of the governing equation and boundary/initial conditions, (3-144) and (3-145). To determine the asymptotic form for these equations for small 7, we again introduce the small parameter e and rescale y and 7 in (3 144) and (3 145) according to (3-159) and (3-160). The result is... 

Examining these solutions, we see that the temperature becomes nonuniform and the velocity profile nonlinear. These results are to be expected from a qualitative point of view. In a sense, the most important conclusion is that the regular asymptotic expansion in terms of the small parameter Br provides a method to obtain an approximate solution of the highly nonlinear boundary-value problem to evaluate the influence of weak dissipation, which can clearly be applied to other problems. 

We could obtain a general solution of Eq. (4-94). However, there is no obvious way to apply the boundary condition at the channel wall, at least in the general form (4-95) or (4-96). The method of domain perturbations provides an approximate way to solve this problem for e << 1. The basic idea is to replace the exact boundary condition, (4-96), with an approximate boundary condition that is asymptotically equivalent for e 1 but now applied at the coordinate surface y = d/2. The method of domain perturbations leads to a regular perturbation expansion in the small parameter s. 

In the analysis of stability that follows, we seek an approximate asymptotic solution of this equation in which the small parameter is the dimensionless magnitude of the initial departure from the equilibrium radius RE. Hence we seek a solution in the form of a regular perturbation expansion ... 

The dimensionless forms of the governing equations emphasize the fact that there are really two dimensionless parameters, s and s1 Re, which must be small for the classical lubrication analysis to apply. It is convenient for present purposes, to assume that Re is fixed independent of and thus sn Re 0 as e 0. Thus in the limit Owe may seek an asymptotic solution of (5-61) and (5 65) (5 67) for the inner thin-fihn part of the flow domain in the form... 

In particular, let us start with the nondimensionalized vorticity transport equation (10-6) and attempt to obtain an approximate solution for Re 1. We expect an asymptotic expansion with Re 1 as the small parameter, but we restrict our attention here to the leading-order term in this expansion, which we can obtain by solving the limiting form of (10-6) lor Re —> oo, namely,... 

We seek the solution of the problem in the form of asymptotic expansions with respect to the small parameter e. The leading term of the expansion outside the drop is determined by the solution of the problem about the flow past a solid sphere. The leading term inside the drop corresponds to the viscous fluid... 

Here, the solution can be found in the form of a regular asymptotic expansion (4.8.12) in powers of the small parameter Pe l. The leading term of this series satisfies the equation derived in [238], The numeric solution leads to the following formula for the mean-volume dimensionless concentration inside the drop ... 

Perry, Newman and Cairns [5] obtained a numerical solution to a problem and provided asymptotic solutions for large and small proton current densities Jo - However, they did not present the expressions for the voltage current curve valid in the whole range of jo, nor the relations for the profiles of basic parameters across the CCL. Eikerling and Komyshev [7] used a similar approach and derived an analytical solution in the case of small overpotentials. In the general case they presented numerical results. 

Vasil eva, A. B., Asymptotic Behavior of Solutions to Certain Problems Involving Nonlinear Differential Equations. Containing a Small Parameter Multiplying the Highest Derivatives, Russian Mathematical Surveys, 18, 13-84 (1963). 

Compare the asymptotic solution obtained in part (a) with the exact solution for e = 0.1. Disoiss the diverging behavior of the asymptotic solution near x = 0. Does the asymptotic solution behave better near X = 0 as more terms are retained Observing the asymptotic solution, it shows that the second term is more singular (secular) than the first term, and the third term is more singular than the second term. Thus, it is seen that just like the cubic equations dealt with in Problems 6.5 and 6.6, where the cubic term is multiplied by a small parameter, this differential equation also suffers the same growth in singular behavior. 

It is possible to vary the common domain of the different asymptotic solutions, up to the order of a small parameter, as follows ... 

This situation is standard for problems with a small parameter. The interval of fitness of the trivial asymptotic solution (4.1) cannot be too long. The small terms affect the solution for long times. On the long time scale t-lie one has to construct a new asymptotic solution depending on another typical time. A suitable new independent variable is t = et as one can guess by looking at the secular terms. 

This system and other problems for singularly perturbed ordinary differential equations will be investigated in Sections II-V. Solutions with boundary and/or interior layers will be considered. Our main goal will be the construction of an approximation to the solution valid outside the boundary (interior) layer as well as within the boundary (interior) layer, that is, so-called uniform approximation in the entire t domain. This approximation will have an asymptotic character. The definition of an asymptotic approximation with respect to a small parameter will be introduced in Section LB. 

In Ref [114], an approach to the dynamics of ionic surfactant adsorption was developed, which is simpler as both concept and application, but agrees very well with the experiment. Analytical asymptotic expressions for the dynamic surface tension of ionic surfactant solutions are derived in the general case of nonstationary interfacial expansion. Because the diffusion layer is much wider than the EDL, the equations contain a small parameter. The resulting perturbation problem is singular and it is solved by means of the method of matched asymptotic expansions [115]. The derived general expression for the dynamic surface tension is simplified for two important special cases, which are considered in the following section. 

Parameters defining the small-scale cracking solution when the asymptotic solution is a quarter plane with one edge fixed and the other edge stress free (Fig. 3)... 

In the following discussion the smallness of the parameter is utilized. This small parameter is a multiplier at the highest derivative in Equation 5.57. This means that matching of asymptotic solutions can be used. Let us introduce the following local variable, z ... 

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