We investigate the model problem of flow of a viscous incompressible fluid past a symmetric curved surface when the flow is parallel to its axis. This problem is known to exhibit boundary layers. Also the problem does not have solutions in closed form, it is modelled by boundary-layer equations. Using a self-similar approach based on a Blasius series expansion (up to three terms), the boundary-layer equations can be reduced to a Blasius-type problem consisting of a system of eight third-order ordinary differential equations on a semi-infinite interval. Numerical methods need to be employed to attain the solutions of these equations and their derivatives, which are required for the computation of the velocity components, on a finite domain with accuracy independent of the viscosity v, which can take arbitrary values from the interval (0,1]. To construct a robust numerical method we reduce the original problem on a semi-infinite axis to a problem on the finite interval [0, K], where K = K(N) = ln N. Employing numerical experiments we justify that the constructed numerical method is parameter robust.