In the recent years considerable attention has been focused on the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. However, due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we discuss several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.

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We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in saturated porous media such as concrete.\ud Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy