Asymptotic approximation initial value problem

In [157] the authors present an initial-value methodology for the numerical approximation of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations. These problems have a boundary layer at one end (left or right) point. The techniaque which used by the authors is to reduce the original problem to an asymptotically equivalent first order initial-value problem. This is done with the... 

The next problem is the determination of the initial value of the three-particle density operator. Using the generalized Bogolyubov asymptotic condition in the approximation (3.39), we obtain finally... 

Now, the dynamics of changes in bubble radius with time, starting from some initial radius that differs slightly from an equilibrium value, is a problem that is ideally suited to solution by means of a regular asymptotic approximation. Of course, the governing equation is still the Rayleigh Plesset equation. Before beginning our analysis, we follow... 

Of great importance is the fact that the quasi-steady-state approximation is the solution asymptote of the initial system at e -> 0, but it is applied at finite e. To establish a starting value from which this approximation can be used with the prescribed accuracy is a rather difficult problem in each particular case. 

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