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 investigate the numerical solution of regular and singular Sturm-Liouville problems by means of finite difference schemes of high order. In particular, a set of difference schemes is used to approximate each derivative independently so to obtain an algebraic problem corresponding to the original continuous differential equation. The endpoints are treated depending on their classification and in case of limit points, no boundary condition is required. Several numerical tests are finally reported on equispaced grids show the convergence properties of the proposed approach.

In this note we show how a simple stepsize variation strategy improves the solution algorithm of regular Sturm-Liouville problems. We suppose the eigenvalue problem is approximated by variable stepsize finite difference schemes and the obtained algebraic eigenvalue problem is solved by a matrix method estimating the first eigenvalues and eigenvectors of sparse matrices. The variable stepsize strategy is based on an equidistribution of the error (approximated by two methods with different orders). The results show a marked reduction of the number of points and, consequently, a much lower computational cost, with respect to the algorithm obtained using constant stepsize.

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. 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.

The morphology dependent resonances (MDR) are of growing interest due to their extremely high quality factor. The quality factor, or Q factor, is a dimensionless parameter that indicates the energy loss relative to the stored energy within a resonant element. The higher the Q, the lower the rate of energy loss and as a result the slower the oscillations will die out. An overview about the current state of research on the MDR can be found in [1]. The numerical simulation of these phenomena is not straightforward, even in the case of symmetry allowing the separation of variables in the modeling equations. Below, we report on a progress made in the numerical simulation of the so called ‘whispering gallery’ modes occurring inside a prolate spheroid. The approach presented here is also applicable for any other separable geometry.

In this work, partition equilibriums and extraction rates of different polycyclic aromatic hydrocarbons (PAHs) have been calculated by multivariate nonlinear regression from data obtained after microextraction by packed sorbent (MEPS) of 16 PAHs from water samples. The MEPS gas chromatography-mass spectrometry (MEPS-GC-MS) method has been optimized investigating the partitioning parameters for a priori prediction of solute sorption equilibrium, recoveries, pre-concentration effects in aqueous and solvent systems. Finally, real samples from sea, agricultural irrigation wells, streams and tap water were analyzed. Detection (S/N ≥ 3) and quantification (S/N ≥ 10) limits were strictly dependent on the volume of water and methanol used during the extraction process. Under the experimental conditions used, these values range from 0.5 to 2 ng L^(-1) and from 1.6 to 6.2 ng L^(-1), respectively. The reasonably good correlation between the logarithm of the partition MEPS-water constants (log K_(meps/water) ) and the logarithm of the octanol-water partition coefficients (log K_(ow) ) (R^2 = 0.807) allows a rough estimation of K ow from the measure of K_(meps/water).

We consider the issue of energy conservation in the numerical solution of Hamiltonian systems coupled with boundary conditions and discuss a few examples arising from astrodynamics.

We consider the issue of energy conservation in the numerical solution of Hamiltonian systems coupled with boundary conditions and discuss a few examples arising from astrodynamics.

We introduce new methods for the numerical solution of general Hamiltonian boundary value problems. The main feature of the new formulae is to produce numerical solutions along which the energy is precisely conserved, as is the case with the analytical solution. We apply the methods to locate periodic orbits in the circular restricted three body problem by using their energy value rather than their period as input data. We also use the methods for solving optimal transfer problems in astrodynamics.

The numerical solution of second order ordinary differential equations with initial conditions is here approached by approximating each derivative by means of a set of finite difference schemes of high order. The stability properties of the obtained methods are discussed. Some numerical tests, reported to emphasize pros and cons of the approach, motivate possible choices on the use of these formulae.

In this work, two different extraction procedures for the analysis of different polycyclic aromatic hydrocarbons (PAHs) in water by microextraction by packed sorbent (MEPS) have been compared in terms of sensitivity, reliability and time of analysis. The first method, called “draw-eject”, consists of a sequence of cycles of aspirations and injections in the same vial; the second one, called “extract-discard”, consists of a similar cycle sequence, but the aspired sample in this case is discarded into waste. The relevant partition equilibriums and extraction rates have been calculated by multivariate regression from the data obtained after MEPS gas chromatography–mass spectrometry (MEPS-GC–MS) analysis of 16 PAHs from water samples. Partitioning parameters for a priori prediction of solute sorption equilibrium, recoveries and preconcentration effects in aqueous and solvent systems have been calculated and compared for the two extraction procedures. Finally, real samples from sea, agricultural irrigation wells, streams and tap water were analyzed.

In this paper, we discuss the progress in the numerical simulation of the so-called `whispering gallery' modes (WGMs) occurring inside a prolate sphero\-idal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version) is separable in spheroidal coordinates. However, the numerical simulation of `whispering gallery' phenomena is not straightforward. The separation of variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm--Liouville problem. We discuss an efficient and reliable technique for the numerical solution of this problem which enables calculation of highly localized WGMs inside a spheroid. The presented approach is also applicable to other separable geometries. We illustrate the performance of the method by means of numerical experiments.

In the given paper, we confront three finite difference approximations to the Navier-Stokes equations for the two-dimensional viscous incomressible fluid flows. Two of these approximations were generated by the computer algebra assisted method proposed based on the finite volume method, numerical integration, and difference elimination. The third approximation was derived by the standard replacement of the temporal derivatives with the forward differences and the spatial derivatives with the central differences. We prove that only one of these approximations is strongly consistent with the Navier-Stokes equations and present our numerical tests which show that this approximation has a better behavior than the other two.

In the recent years considerable attention has been focused on the numerical computation of the eigenvalues and eigenfunctions of the nite (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, e cient 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.

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we present 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.

Bibliometric indexes are customary used in evaluating the impact of scientific research, even though it is very well known that in different research areas they may range in very different intervals. Sometimes, this is evident even within a single given field of investigation making very difficult (and inaccurate) the assessment of scientific papers. On the other hand, the problem can be recast in the same framework which has allowed to efficiently cope with the ordering of web-pages, i.e., to formulate the PageRank of Google. For this reason, we call such problem the PaperRank problem, here solved by using a similar approach to that employed by PageRank. The obtained solution, which is mathematically grounded, will be used to compare the usual heuristics of the number of citations with a new one here proposed. Some numerical tests show that the new heuristics is much more reliable than the currently used ones, based on the bare number of citations. Moreover, we show that our model improves on recently proposed ones.

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

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.

We propose an efficient and reliable technique to calculate highly localized Whispering Gallery Modes (WGMs) inside an oblate spheroidal cavity. The idea is to first separate variables in spheroidal coordinates and then to deal with two ODEs, related to the angular and radial coordinates solved using high order finite difference schemes. It turns out that, due to solution structure, the efficiency of the calculation is greatly enhanced by using variable stepsizes to better reflect the behaviour of the evaluated functions. We illustrate the approach by numerical experiments.

We propose an efficient and reliable technique to calculate highly localized Whispering Gallery Modes (WGMs) inside an oblate spheroidal cavity. The idea is to first separate variables in spheroidal coordinates and then to deal with two ODEs, related to the angular and radial coordinates solved using high order finite difference schemes. It turns out that, due to solution structure, the efficiency of the calculation is greatly enhanced by using variable stepsizes to better reflect the behaviour of the evaluated functions. We illustrate the approach by numerical experiments.