Matching conditions, asymptotic solutions

The solution of Equation 4.28 should be matched with the outer solution (4.23). Matching of asymptotic solutions gives the following condition ... 

Completion of a solution by matched asymptotic expansions entails employing matching conditions. Although there are many ways to effect matching, the most infallible approach currently available is to investigate a parametric limit in an intermediate variable [35]. Thus we consider oo with f]t held fixed, where t] = s(p)t]/t(p) = ( — o)/KPX with t(p) 0 and KP)/s(P) 00 in the limit. The general matching condition is then written as... 

The condition (3 235) implies that aPe for Pe 1. We have seen that the coefficient m must be positive for the Taylor analysis to apply, but it is otherwise undetermined. In a full matched asymptotic analysis, it would be obtained by matching with the solutions for smaller 7, but these solutions are not available here. Fortunately, the final result for the Taylor dispersion coefficient and the governing equation for (0) are independent of m provided only that it is positive so that the analysis is valid. We can now determine 9 by solving either (3-231) or (3-236). We choose to solve Eq. (3-236) for 9. Because 3 (9)/dz is independent of r, we can simply integrate twice with respect to r. The general solution is ... 

Now, we may note, by analogy to the analysis of Section C for the specific case of a solid sphere, that the asymptotic form (9-154) of the first term in the inner expansion for r 1 is all that we need to obtain a complete solution for the first term, 0, in the expansion for the outer region. Indeed, if we examine the governing equation and boundary condition (9-135) and the matching condition derived from (9 154),... 

It is only the boundary conditions that are significantly different. The problem is considered now on the unbounded interval — < p < < , which corresponds to a neighborhood of the time in which explosive growth occurs l -H 1. For the functions x, y, z (p e), the asymptotics at minus infinity (as p— -< ) are determined. This determination is done by matching them with the slow asymptotic solution (5.8). 

In a similar way an asymptotic solution can be constructed for the case of system (7.12) containing both fast variables (function u) and slow variables (function u). However, the addition of the slow variables leads to a situation where the asymptotics can be constructed only to the zeroth- and first-order approximations. This is related to the fact that the initial and boundary conditions for the function V2(x, t) [the coefficient of in the regular series for v(x, t, e)] are not matched at the corner points (0,0) and (1,0). As a result, the function V2(x, t) is not smooth in Cl. The derivatives and d V2/dx are unbounded in the vicinities of the... 

Equations for each of the perturbation functions xu yh Xu Yl are derived by substituting the asymptotic expansions into the initial differential system, by matching terms with the same power in e, and finally by writing the proper initial and boundary layer conditions. The zeroth-order outer approximation is the solution to the system... 

Notice that when r - oo,v - S0/(Km + S0). This is exactly the value of v that we arrived at for r = 0. Thus as r -> oc (on the fast timescale), v approaches the derived initial condition for the slow timescale (r = 0). Hence, the entire transient for Michaelis-Menten kinetics can be represented by combining the short timescale result, Equation (4.29), with the long timescale result, the solution to Equation (4.25). The two results match seamlessly at r = oo and x = 0. This is known as asymptotic matching in singular perturbation analysis [110]. 

The constant A cannot be determined from the boundary condition at the wall but must be obtained from the matching requirement that (4-27) reduce to the form of the core solution (4-17) in the region of overlap between the boundary layer and the interior region. Now, any arbitrarily large, but finite, value of Y will fall within the boundary-layer domain on the other hand, the corresponding value of y can be made arbitrarily small in the asymptotic limit R0J - oo. Thus the condition of matching is often expressed in the form... 

This solution obviously does not satisfy no-sfip conditions on the sidewalls,

oo by use of the method of matched asymptotic expansions, which does satisfy boundary conditions on the walls. Be as detailed and explicit as possible, including actually setting up the equations and boundary conditions for the solution in the regions near the walls. 

The approximate solution of the thermal and diffusion problem can be found by the method of matched asymptotic expansions (see Section 4.4) with the stream functions (5.10.6) one must retain only the zero and the first terms of the expansions with respect to low Peclet numbers and use the boundary conditions (5.10.5) and (5.11.2) to obtain the following values of the constant B and the force acting on the drop ... 

The solution holds under conditions Sf/R l and Ca 1. A more detailed asymptotic analysis of the solution for the above-considered problem can be found in [17]. The formal procedure of matching asymptotic expansions demonstrates that the solution in the form (17.30) is applicable for Ca 0. In the case of finite values of Ca < 1, the expression (17.30) is modified [17] ... 

This statement can be proved by applying the method of successive approximations to the equation for the remainder w-u-Ufy and using the maximum principle to estimate these successive approximations. A detailed proof is presented in [47]. Under more severe constraints of matching of the initial and boundary conditions (10.6) in the corner points (0, y, 0) and (l,y, 0), it is possible to construct asymptotic approximations for the solution with greater accuracy. 

Now we make use of the finite range of the potential U(R). At internuclear distances i > a we will consider the potential to be zero so that the asymptotic form Eqn. (10.40) is the relevant solution to the Schrodinger equation. At shorter distances we assume a well-defined potential form with known solutions TZi for Eqn. (10.30). At the boundary R = a we apply joining conditions to the interior and exterior solutions. Smooth joining requires that the solutions and their derivatives match at the boundary R = a. A convenient way to express this matching is the logarithmic derivative... 

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