Hamiltonian algorithm for solving singularly perturbed equations
Perturbation theory is of central importance for approximate integration of differential equations. However, in problems with a small parameter \(\varepsilon\) multiplying the high-order derivative, arbitrarily small changes in the parameter cause finite changes in the solution. For \(\varepsilon = 0\), the order of the equation is reduced. The difference in phase trajectories of the original and the degenerate system essentially complicates the construction of approximate solutions. The difficulties encountered in solving such equations stimulated the development of a number of asymptotic methods. The known methods of inner and outer expansions, multiple-scale expansion matching, and others are quite complex in application, some methods solve only a particular boundary value problem, and no algorithms for constructing higher approximations are available. In this paper we develop an approach to solving singularly perturbed systems based on the Hamiltonian formalism. An arbitrary system of nonlinear equations is always representable in Hamiltonian form, so that all information about its analytic properties is contained in a single function -- the Hamiltonian. This substantially simplifies the computation. Moreover, insofar as it is required to remain in the group of motion of the skew-symmetric metric, we can construct a single algorithm for general solution of a system of singular equations.
scientific article; zbMATH DE number 672503