Singularly perturbed boundary-value problems: Sundman-type transformations, test problems, exact solutions, and numerical integration

Solutions of singularly perturbed boundary-value problems with a small parameter are characterized by large gradients in very narrow regions (boundary layers). This circumstance sharply limits the use of standard finite-difference methods with a fixed stepsize in such problems due to significant calculation errors or possible loss of stability. This paper presents an effective method for numerical integration of singularly perturbed boundary-value problems based on replacing the spatial variable with a new independent variable of the Sundman-type, which depends on the derivatives of the unknown function. The use of such non-local transformations, which satisfy a simple asymptotic condition, makes it possible to automatically stretch the boundary-layer region. The resulting problem turns out to be much simpler than the original one in the sense that standard (classical) numerical methods with a fixed stepsize can already be applied to solve it. Several new multiparameter nonlinear and linear singularly perturbed boundary-value problems for second-order reaction-diffusion type ODEs having monotonic and non-monotonic exact or asymptotic solutions, expressed in terms of elementary functions, are constructed. A comparison of numerical solutions with exact and asymptotic solutions is presented. The numerical results show that the method based on Sundman-type transformations for solving boundary-layer problems gives high accuracy. As a result of an extensive analysis of the obtained results, recommendations are given for the choice of regularizing functions that determine the most effective Sundman-type transformations. The difference between regularizing functions in boundary-layer problems and blow-up problems is discussed. The test problems formulated in this paper can be used to estimate the accuracy of any other numerical methods for solving two-point singularly perturbed boundary-value problems with a small parameter.