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

We show the main features of the MATLAB code HOFiD_UP for solving second order singular perturbation problems. The code is based on high order finite differences, in particular on the generalized upwind method. Within its simplicity, it uses order variation and continuation for solving any difficult nonlinear scalar problem. Several numerical tests on linear and nonlinear problems are considered. The best performances are reported on problems with perturbation parameters near the machine precision, where most of the codes for two-point BVPs fail

We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order Finite Difference Schemes. We describe a code to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are proposed to emphasize the behaviour of the proposed algorithm.

Magnetic particle imaging (MPI) is a new medical imaging technique capable of recovering the distribution of superparamagnetic particles from their measured induced signals. In literature there are two main MPI reconstruction techniques: measurement-based (MB) and x-space (XS). The MB method is expensive because it requires a long calibration procedure as well as a reconstruction phase that can be numerically costly. On the other side, the XS method is simpler than MB but the exact knowledge of the field free point (FFP) motion is essential for its implementation. Our simulation work focuses on the implementation of a new approach for MPI reconstruction: it is called hybrid x-space (HXS), representing a combination of the previous methods. Specifically, our approach is based on XS reconstruction because it requires the knowledge of the FFP position and velocity at each time instant. The difference with respect to the original XS formulation is how the FFP velocity is computed: we estimate it from the experimental measurements of the calibration scans, typical of the MB approach. Moreover, a compressive sensing technique is applied in order to reduce the calibration time, setting a fewer number of sampling positions. Simulations highlight that HXS and XS methods give similar results. Furthermore, an appropriate use of compressive sensing is crucial for obtaining a good balance between time reduction and reconstructed image quality. Our proposal is suitable for open geometry configurations of human size devices, where incidental factors could make the currents, the fields and the FFP trajectory irregular.