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Differential geometric approach to singular perturbatio ns

F. Jamitzky

A differential geometric approach to singular perturbation theory is presented. It is shown that singular perturbation problems such as multiple-scale and boundary layer problems can be treated more easily on a differential geometric basis. A general method is proposed based on differential forms and Lie-derivatives. Examples from multiple scale theory, boundary layer theory and WKB-theory are given and it is demonstrated that without the a priori knowledge of the scaling behaviour of the problem the correct asymptotic expansion can be derived with the aid of differential forms. The method is well suited for a mechanical implementation in computer algebra programs.
     Regular perturbation theory is often not applicable to various problems due to resonance effects or the cancellation of degrees of freedom. In order to obtain a uniformly valid asymptotic expansion of a solution for these singular problems a whole bunch of methods has been developed such as boundary layer expansions, multiple scale methods, asymptotic matching, stretched coordinates, averaging and WKB-expansions. Although these methods work well for the respective cases, a general theory that unifies all these methods is still lacking.
     A physical system often involves multiple temporal or spatial scales on which characteristics of the system change. In some cases the long time behaviour of the system can depend on slowly changing time scales which have to be identified in order to apply multiple scale theory. The choice of the slowly or fast changing scales is a nontrivial task, which presumes a good understanding of the physical behaviour of the system and can sometimes only be justified by the final result. A geometrical approach is presented which allows a mechanical algorithm for the determination of the rapid or slow scales on which the system changes. For that purpose the system of ordinary differential equations is interpreted as a system of differential forms. It is shown that a small perturbation of the system of differential forms can be viewed as a deformation of the solution manifold in configuration space. By searching for a Lie-derivative that transforms the unperturbed differential forms into the perturbed forms a mapping in configuration space is defined that can be used to map the solution of the unperturbed system into the an approximate solution of the perturbed system in order to obtain uniformly valid asymptotic expansions of the solution and thus determining the secular scales. The method is exemplified for common problems like multiple scale problems, boundary layer problems and WKB-expansions. We have given three classical examples from singular perturbation theory and have shown how easily uniformly valid approximations can be obtained with the new method. The method is based on very general geometrical considerations and thus can be easily extended to more complicated problems involving partial differential equations (work in progress).