S. K. Lucas

Evaluating infinite integrals involving products of Bessel functions of arbitrary order

The difficulties involved with evaluating infinite integrals involving products of Bessel functions are considered, and a method for evaluating these integrals is outlined. The method makes use of extrapolation on a sequence of partial sums, and requires rewriting the product of Bessel functions as the sum of two more well-behaved functions. Numerical results are presented to demonstrate the efficiency of this method, where it is shown to be significantly superior to standard infinite integration routines.

A mathematical study of peristaltic transport of a casson fluid

In this paper, the peristaltic flow of rheologically complex physiological fluids when modelled by a non-Newtonian Casson fluid in a two-dimensional channel is considered. A perturbation series method of solution of the stream function for zeroth and first order in amplitude ratio is sought. Of interest is the difference between peristaltic transport of Newtonian and non-Newtonian fluids. It is found that Newtonian fluid is an important subclass of non-Newtonian fluids that may adequately represent some physiological phenomena. Analytical and numerical solutions are found for the zeroth and first order in stream function and compared to well-documented research in the literature. It is shown that for a Casson fluid, when certain approximations are made in the most generalized form of constitutive equation, the fluid may be adequately represented as an improvement of a Newtonian fluid.

Evaluating infinite integrals involving Bessel functions of arbitrary order

The evaluation of integrals of the formIn = R 1 0 f(x)Jn(x)dx is considered. In the past, the method of dividing an oscillatory integral at its zeros, forming a sequence of partial sums, and using extrapolation to accelerate convergence has been found to be the most ecien t technique available where the oscillation is due to a trigonometric function or a Bessel function of order n = 0; 1. Here, we compare various extrapolation techniques as well as choices of endpoints in dividing the integral, and establish the most ecien t method for evaluating innite integrals involving Bessel functions of any order n, not just zero or one. We also outline a simple but very eectiv e technique for calculating Bessel function zeros.

Switching-time computation for bang-bang control laws

We obtain improvements and give extensions to the switching time computation (STC) method, which is used to compute the switching times for bang-bang control laws. We first report a considerable computational improvement on the STC method as applied to a nonlinear system with one control input. The second contribution of this paper is the extension and implementation of the STC method for two control inputs. In bang-bang control calculations it is a usual practice to consider all possible combinations of the switchings and then carry out the computations with a large set of switching parameters. We introduce a novel scheme for the switching times for two inputs which results in far fewer switching parameters to calculate.

An integral equation solution for the steady-state current at a periodic array of surface microelectrodes

An integral equation approach is developed for the problem of determining the steady-state current to a periodic array of microelectrodes imbedded in an otherwise insulating plane. The formulation accounts for both surface electrode and bulk fluid reactions and evaluates Greens functions for periodic systems using convergence acceleration techniques. Numerical results are presented for disc-shaped microelectrodes at the center of rectangular periodic cells for a large range of dimensionless surface and bulk reaction rates and periodic cell sizes.

Grobner Basis Representations of Sudoku.

Summary In this paper we use GrÖbner bases to explore the inherent structure of Sudoku puzzles and boards. In particular, we develop three different ways of representing the constraints of Shidoku with a system of polynomial equations. In one case, we will explicitly show how a GrÖbner basis can be used to obtain a more meaningful representation of the constraints. The GrÖbner basis representation can be used to find puzzle solutions or count numbers of boards.

A BOUNDARY INTEGRAL METHOD APPLIED TO THE 3D WATER CONING PROBLEM

Often in oil reservoirs a layer of water lies under the layer of oil. The suction pressure due to a distribution of oil wells will cause the oil‐water interface to rise up towards the wells. A three‐dimensional boundary integral formulation is presented for calculating the steady interface shape when the oil wells are represented by point sinks. Sophisticated integration techniques are implemented in an effort to obtain accurate results. In particular, the efficiency of various integration methods are compared for this problem, including QUADPACK routines, adaptive methods based on the IMT rule, the Kronrod rule, the method of degenerate quadrilaterals, and the Gauss‐Rational rule for infinite integrals. Numerical results for various general multi‐sink distributions are discussed, as are some further results for the axisymmetric single well problem.

Capillary wave scattering from a surfactant domain

The study of capillary wave scattering by a circular region with different interfacial properties from the rest of an otherwise homogeneous interface is motivated by experiments on wave attenuation at a monolayer‐covered air–water interface where domains of one surface phase are dispersed in a second surface phase. Here the scattering function is calculated for an incident wave of frequency ω (wavevector k0) scattering from an isolated circular domain of radius a with surface tension σ1 which is imbedded in an otherwise infinite interface of surface tension σ0. The underlying fluid is treated as irrotational and the three‐dimensional flow problem coupling the heterogeneous surface to the underlying liquid is reduced to a set of dual integral equations, which are solved numerically. With this solution the scattering amplitudes and the total scattering cross sections are calculated as a function of the surface tension ratio σ0/σ1 and incident wavenumber k0a. The analogous problem of a discontinuous change i...

COMPUTATIONS FOR TIME-OPTIMAL BANG–BANG CONTROL USING A LAGRANGIAN FORMULATION

Abstract In this paper an algorithm is proposed to solve the problem of time-optimal bang–bang control of nonlinear systems from a given initial state to a given terminal state. The problem is reduced to the problem of minimising a Lagrangian subject to an equality constraint defined by the terminal state. Then a solution is obtained by solving a system of nonlinear equations. Examples are given so as to illustrate the algorithm presented.

Algorithm 935: IIPBF , a MATLAB toolbox for infinite integral of products of two Bessel functions

A <tt>MATLAB</tt> toolbox, <tt>IIPBF</tt>, for calculating infinite integrals involving a product of two Bessel functions <i>J</i>_<i>a</i>(ρ<i>x</i>)<i>J</i>_<i>b</i>(τ <i>x</i>), <i>J</i>_<i>a</i>(ρ <i>x</i>)<i>Y</i>_<i>b</i>(τ <i>x</i>) and <i>Y</i>_<i>a</i>(ρ<i>x</i>)<i>Y</i>_<i>b</i>(τ <i>x</i>), for non-negative integers <i>a</i>,<i>b</i>, and a well-behaved function <i>f</i>(<i>x</i>), is described. Based on the Lucas algorithm previously developed for <i>J</i>_<i>a</i>(ρ <i>x</i>)<i>J</i>_<i>b</i>(τ <i>x</i>) only, <tt>IIPBF</tt> recasts each product as the sum of two functions whose oscillatory behavior is exploited in the three-step procedure of adaptive integration, summation, and extrapolation. The toolbox uses customised <tt>QUADPACK</tt> and <tt>IMSL</tt> functions from a <tt>MATLAB</tt> conversion of the <tt>SLATEC</tt> library. In addition, <tt>MATLAB</tt>s own <tt>quadgk</tt> function for adaptive Gauss-Kronrod quadrature results in a significant speed up compared with the original algorithm. Usage of <tt>IIPBF</tt> is described and eighteen test cases illustrate the robustness of the toolbox; five additional ones are used to compare <tt>IIPBF</tt> with the <tt>BESSELINT</tt> code for rational and exponential forms of <i>f</i>(<i>x</i>) with <i>J</i>_<i>a</i>(ρ<i>x</i>)<i>J</i>_<i>b</i>(τ <i>x</i>). Reliability for a broad range of values of ρ and τ for the three different product types as well as different orders in one case is demonstrated. An electronic appendix provides a novel derivation of formulae for five cases.