Application of non-local transformations for numerical integration of singularly perturbed boundary-value problems with a small parameter

摘要

Singularly perturbed boundary-value problems for second-order ODEs of the form epsilon y(xx)" = F(x, y, y(x)') with epsilon - 0 are considered. We present a new method of numerical integration of such problems, based on introducing a new non-local independent variable xi which is related to the original variables x and), by the equation xi(x)' = g(x, y, y(x)',xi). With a suitable choice of the regularizing function g, this method leads to more appropriate problems that allow the application of standard numerical methods with fixed stepsize of xi (in the whole range of variation of the independent variable x, including both the boundary-layer region and the outer region). It is shown that methods based on piecewise-uniform grids are a particular (degenerate) case of the method of non-local transformations with a piecewise-smooth regularizing function of special form. A number of linear and non-linear test problems with a small parameter (including convective heat and mass transfer type problems) that have exact or asymptotic solutions (both monotonic and non-monotonic), expressed in elementary functions, are presented. Comparison of numerical, exact, and asymptotic solutions showed the high efficiency of the method of non-local transformations for solving singularly perturbed problems with boundary layers. In addition to non local transformations, examples of the use of point (local) transformations for numerical integration of singularly perturbed boundary-value problems are also given.