The method of nonlocal transformations: Applications to singularly perturbed boundary-value problems

We present a method of numerical integration of singularly perturbed boundary-value problems with a small parameter, based on introducing a new nonlocal independent variable. As a result, we obtain more suitable problems that allow the application of standard fixed-step numerical methods. Two test boundary-value problems for second-order ODEs that have monotonic and non-monotonic exact or asymptotic solutions, expressed in elementary functions, are considered. Comparison of numerical, exact, and asymptotic solutions showed the high efficiency of the method based on generalized nonlocal transformations for solving singular boundary-value problems with a small parameter.