A generalized multiple scales approach to a class of linear differential equations

Abstract A technique for approximating uniformly the solutions for a class of ordinary linear differential equations with variable coefficients is developed. The coefficients are taken to contain a small or large parameter in a simple way. In particular, the coefficients vary on a single scale and are small and rapidly varying or large and slowly varying. The method employed is the following (“extension”). The ordinary differential equation is replaced by a set of partial differential equations that determine the unknown function in terms of a set of independent “scales.” The partial differential equations, in conjunction with the requirement of uniformity of the approximation in an interval, help us establish the functional dependence of the scales in terms of the original independent variable (“scale functions”). With the use of two scales, we obtain an approximation to the amplitude and phase of each of the independent solutions of nth-order equations that improves perturbative and frozen approximations. In particular, “whipping tail” effects are eliminated. Under appropriate conditions, for second-order equations, the Liouville—Green (or WKBJ) approximation is readily recovered as a special case of our method. Several examples are given. It is essential, for the success of the approximation, that the scale functions be nonlinear as well as, in general, complex. Thus, the present approach generalizes earlier “time scale” analyses in several respects.