The Asymptotic Expansion Method via Symbolic Computation

The origin of symbolic manipulation derives from the sheer magnitude of the work involved in the building of perturbation theories, which made inevitable that scientific community became interested in the possibility of constructing those theories with the help of computers. Perturbation theories for differential equations containing a small parameter are quite old. The small perturbation theory originated by Sir Isaac Newton has been highly developed by many others, and an extension of this theory to the asymptotic expansion, consisting of a power series expansion in the small parameter, was devised by Poincaré 1892 1 . The main point is that for the most of the differential equations, it is not possible to obtain an exact solution. In cases where equations contain a small parameter, we can consider it as a perturbation parameter to obtain an asymptotic expansion of the solution. In practice, the work involved in the application of this approach to compute the solution to a differential equation cannot be performed by hand, and algebraic processors seem to be a very useful tool. As explained in 2 , the first symbolic processors were developed toworkwith Poisson series, that is, multivariate Fourier series whose coefficients are multivariate Laurent series, ∑