Multiple Asymptotic Expansions and Singular Problems

Applications of multiple asymptotic expansions to singular differential equations have been investigated by means of four examples. The techniques of multiple asymptotic expansions are first applied to an equation with an essential singularity in the leading coefficient. The results are then compared to the results obtained by the techniques of H. Schmidt. Then two linear problems of singular perturbation are investigated. For a boundary value problem it is shown that the technique of multiple asymptotic expansions yields the same result as the two-variable expansion technique, and for an initial value problem it is shown that this technique improves upon the two-variable technique. Finally a nonlinear boundary value problem of singular perturbation is considered. It has been shown that, whenever the calculations are not too onerous, the technique of multiple asymptotic expansions yields new insights into the nature of the solution. All of the derivations are formal.