Recent Advances in Computational and Applied Mathematics

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.

We present a survey on collocation based methods for the numerical integration of Ordinary Differential Equations (ODEs) and Volterra Integral Equations (VIEs), starting from the classical collocation methods, to arrive to the most important modifications appeared in the literature, also considering the multistep case and the usage of basis of functions other than polynomials.

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.

A mathematical model describing the melt spinning process of polymer fibers is considered. Newtonian and non-Newtonian models are used to describe the rheology of the polymeric material. Two key questions related to the industrial application of melt spinning are considered: the optimization and the stability of the process. Concerning the optimization question, the extrusion velocity of the polymer at the spinneret as well as the velocity and temperature of the quench air serve as control variables. A constrained optimization problem is derived and the first-order optimality system is set up to obtain the adjoint equations. Numerical solutions are carried out using a steepest descent algorithm. Concerning the stability with respect to variations of the velocity and temperature of the quench air, a linear stability analysis is carried out. The critical draw ratio, indicating the onset of instabilities, is computed numerically solving the eigenvalue problem for the linearized equations.

The main goal of the paper is to deal with a special orthogonal system of polynomial solutions of the Riesz system in ℝ³. The restriction to the sphere of this system is analogous to the complex case of the Fourier exponential functions {e ^{in θ} } _n≥0 on the unit circle and has the additional property that also the scalar parts of the polynomials form an orthogonal system. The properties of the system will be applied to the explicit calculation of conjugate harmonic functions with a certain regularity.

The CP methods have some salient advantages over other methods, viz.: (i) the accuracy is uniform with respect to the energy E; (ii) there is an easy control of the error; (iii) the step widths are unusually big and the computation is fast; (iv) the form of the algorithm allows a direct evaluation of collateral quantities such as normalisation constant, Prüfer phase, or the derivative of the solution with respect to E; (v) the algorithm is of a form which allows using parallel computation.

In this work specially tuned Symplectic Partitioned Runge-Kutta (SPRK) methods have been considered for the numerical integration of problems with periodic or oscillatory solutions. The general framework for constructing exponentially/trigonometrically fitted SPRK methods is given and methods with corresponding order up to fifth have been constructed. The trigonometrically-fitted methods constructed are of two different types, fitting at each stage and Simos’s approach. Also, SPRK methods with minimal phase-lag are derived as well as phase-fitted SPRK methods. The methods are tested on the numerical integration of Kepler’s problem, Stiefel-Bettis problem and the computation of the eigenvalues of the Schrödinger equation.

In this paper we present an overview about our recent results on the analytic treatment of the Klein-Gordon equation on some conformally flat 3-tori and on 3-spheres.

In the first part of this paper we consider the time independent Klein-Gordon equation (Δ−α ²)u=0 (α∈ℝ) on some conformally flat 3-tori associated with a representative system of conformally inequivalent spinor bundles. We set up an explicit formula for the fundamental solution associated to each spinor bundle. We show that we can represent any solution to the homogeneous Klein-Gordon equation on such a torus as finite sum over generalized 3-fold periodic or resp. antiperiodic elliptic functions that are in the kernel of the Klein-Gordon operator. Furthermore, we prove Cauchy and Green type integral formulas and set up an appropriate Teodorescu and Cauchy transform for the toroidal Klein-Gordon operator on this spin tori. These in turn are used to set up explicit formulas for the solution to the inhomogeneous Klein-Gordon equation (Δ−α ²)u=f on the 3-torus attached to the different choices of different spinor bundles. In the second part of the paper we present a unified approach to describe the solutions to the Klein-Gordon equation on 3-spheres. We give an explicit representation formula for the solutions in terms of hypergeometric functions and monogenic homogeneous polynomials.

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential in the number of degrees of freedom. hp-FEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate can not simultaneously determine whether it is better to do the refinement by h or by p. Several strategies for making this determination have been proposed over the years. In this paper we summarize these strategies and demonstrate the exponential convergence rates with two classic test problems.

Vectorization is very important to the efficiency of computation in the popular problem-solving environment Matlab. Here we develop an explicit Runge–Kutta (7,8) pair of formulas that exploits vectorization. Conventional Runge–Kutta pairs control local error at the end of a step. The new method controls the extended local error at 8 points equally spaced in the span of a step. This is a byproduct of a control of the residual at these points. A new solver based on this pair, odevr7, not only has a very much stronger control of error than the recommended Matlab solver ode45, but on standard sets of test problems, it competes well at modest tolerances and is notably more efficient at stringent tolerances.

We consider classes of fluid flow problems under given initial value and boundary value conditions on the sphere and on ball shells in ℝ³. Our attention is focused to the forecasting equations and the deduction of a suitable quaternionic operator calculus.

The construction of symmetric and symplectic exponentially-fitted Runge-Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions is reconsidered. In previous papers fourth-order and sixth-order symplectic exponentially-fitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new such integrators are constructed by making use of the six-step procedure of Ixaru and Vanden Berghe (Exponential Fitting, Kluwer Academic, Dordrecht, 2004). Numerical experiments for some oscillatory problems are presented and compared to the results obtained by previous methods.