In this paper, we develop a new mesh selection strategy based on the computation of some conditioning parameters which allows to give information about the conditioning and the stiffness of the problem. The reliability of the proposed algorithm is demonstrated by some numerical experiments. We observe that "when an initial value problem is run on a computer, the results may appear plausible even if they are unreliable because of some unrecognized numerical instability" (Miller, 1967) [23]. The additional information about the behavior of the numerical solution provided by the new mesh selection algorithm are, therefore, of great interest for potential users of a numerical computer code

In this paper we present the R package deTestSet that includes challenging test problems written as ordinary differential equations (ODEs), differential algebraic equations (DAEs) of index up to 3 and implicit differential equations (IDES). In addition it includes 6 new codes to solve initial value problems (IVPs). The R package is derived from the Test Set for Initial Value Problem Solvers available at http://www.dm.uniba.it/similar to testset which includes documentation of the test problems, experimental results from a number of proven solvers, and Fortran subroutines providing a common interface to the defining problem functions. Many of these facilities are now available in the R package deTestSet, which comprises an R interface to the test problems and to most of the Fortran solvers. The package deTestSet is free software which is distributed under the GNU General Public License, as part of the R open source software project. (C) 2012 Elsevier B.V. All rights reserved.

Boundary value methods are linear multistep methods used with a fixed number of initial and final conditions that allow us to generate stable discrete boundary value schemes for the solution of initial and boundary value ordinary differential equations.

A new class of Linear Multistep Methods based on B-splines for the numerical solution of semi-linear second order Boundary Value Problems is introduced. The presented schemes are called BS2 methods, because they are connected to the BS (B-spline) methods previously introduced in the literature to deal with first order problems. We show that, when using an even number of steps, schemes with good general behavior are obtained. In particular, the absolute stability of the 2-step and 4-step BS2 methods is shown. Like BS methods, BS2 methods are of particular interest, because it is possible to associate with the discrete solution a spline extension which collocates the differential equation at the mesh points.

In this paper we will be concerned with numerical methods for the solution of nonlinear systems of two point boundary value problems in ordinary differential equations. In particular we will consider the question which codes are currently available for solving these problems and which of these codes might we consider as being state of the art. In answering these questions we impose the restrictions that the codes we consider should be widely available (preferably written in MATLAB and/or FORTRAN) they should have reached a fairly steady state in that they are seldom, if ever, updated, they try to achieve broadly the same aims and, of course, it is relatively inexpensive to purchase the site licence. In addition we will be concerned exclusively with so called boundary value (or global) methods so that, in particular, we will not include shooting codes or Shishkin mesh methods in our survey. Having identified such codes we go on to discuss the possibility of comparing the performance of these codes on a standard test set. Of course we recognise that the comparison of different codes can be a contentious and difficult task. However the aim of carrying out a comparison is to eliminate bad methods from consideration and to guide a potential user who has a boundary value problem to solve to the most effective way of achieving his aim. We feel that this is a very worthwhile objective to pursue. Finally we note that in this paper we include some new codes for BVP's which are written in MATLAB. These have not been available before and allow for the first time the possibility of comparing some powerful MATLAB codes for solving boundary value problems. The introduction of these new codes is an important feature of the present paper.

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.

Two new classes of quadrature formulas associated to the BS Boundary Value Methods are discussed. The first is of Lagrange type and is obtained by directly applying the BS methods to the integration problem formulated as a (special) Cauchy problem. The second descends from the related BS Hermite quasi-interpolation approach which produces a spline approximant from Hermite data assigned on meshes with general distributions. The second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding Lagrange type formulas with the same accuracy. (C) 2012 Elsevier B.V. All rights reserved.

The R package bvpSolve, for the numerical solution of Boundary Value Problems (BVPs) is presented. This package is free software which is distributed under the GNU General Public License, as part of the R open source software project. It includes some well known codes to solve boundary value problems of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). In addition to the packages already available for solving initial value problems, the new package now allows non expert users to efficiently solve boundary value problems in the problem solving environment R.

In this article, block BS methods are considered for the numerical solution of Volterra integro-differential equations (VIDEs). Convergence and stability properties are analyzed. A new Matlab code for the solution of VIDEs, called VIDEBS, is presented. Numerical results using a variable stepsize implementation show the effectiveness of the proposed code.

We present some numerical experiments with the Matlab code TOMG, based on symmetric block Boundary Value Methods, a class of methods for which experiments have shown that the numerical solution nearly-preserve some type of invariants over long-time integration