This paper gives an approximate solution to the Wiener-Hopf integral equation for filtering fractional Riesz-Bessel motion. This is obtained by showing that the corresponding covariance operator of the integral equation is a continuous isomorphism between appropriate fractional Sobolev spaces. The proof relies on properties of the Riesz and Bessel potentials and the theory of fractional Sobolev spaces.

We apply superadditivity and monotonicity properties associated with the Jensen discrete inequality to derive relationships between the entropy function of a probability vector and a renormalized arbitrary sub-vector. The results are extended to cover other entropy measures such as joint entropy, conditional entropy and mutual information.

Recently Dragomir and Goh have produced some interesting new bounds relating to entropy measures in information theory. We establish several refinements of their results.

Several generalizations are given of the Gauss-Winckler inequality for the moments of a probability distribution.

Closed-form analytical expressions are derived for the reflection and transmission coefficients for the problem of scattering of surface water waves by a sharp discontinuity in the surface-boundary-conditions, for the case of deep water. The method involves the use of the Havelock-type expansion of the velocity potential along with an analysis to solve a Carleman-type singular integral equation over a semi-infinite range. This method of solution is an alternative to the Wiener-Hopf technique used previously.

Lotka-Volterra systems have been used extensively in modelling population dynamics. In this paper, it is shown that chaotic behaviour in the sense of Smale can exist in timeperiodically perturbed systems of Lotka-Volterra equations. First, a slowly varying threedimensional perturbed Lotka-Volterra system is considered and the corresponding unperturbed system is shown to possess a heteroclinic cycle. By using Melnikov's method, sufficient conditions are obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra system is examined, leading finally to an application with a model for the self-organisation of macromolecules.

The first part of this paper starts with a brief discussion of some methods for solution of nonlinear equations which have interested the first author over the last twenty years or so. In the second part we discuss a recent research involvement, the success of which relies heavily on the numerical solution of nonlinear equation systems. We briefly describe path-following methods and then present an application to a simple steady-state reaction-diffusion equation arising in combustion theory. Results for some regular geometric shapes are shown and compared with those from an approximate method.

We study an optimal control problem for a quasilinear parabolic equation which has delays in the highest order spatial derivative terms. The cost functional is Lagrange type and some terminal state constraints are presented. A Pontryagin-type maximum principle is derived.

Connections between a linear partial difference equation with constant coefficients and a nonlinear partial difference equation are established by means of a comparison theorem and a continuous dependence of parameters theorem. A linearized oscillation theorem is also established as an application.

In this survey we consider a regularized Newton method for the approximate solution of the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of time-harmonic acoustic or electromagnetic plane waves. Our analysis is in two dimensions and the numerical scheme is based on the solution of boundary integral equations by a Nyström method. We include an example of the reconstruction of a planar domain with a corner both to illustrate the feasibility of the use of radial basis functions for the reconstruction of boundary curves with local features and to connect the presentation to some of the research work of Professor David Elliott.

We consider a mesh grading quadrature method for real constant-coefficient Cauchy singular integral equations of index 0. The quadrature method is based on the trapezoidal rule. A complete stability and convergence analysis is given by the use of the noncompact perturbation analysis as in Jeon [10] and Elschner and Stephan [7]. The order of convergence can be arbitrarily high if the order of mesh grading is high enough. We also provide an efficient way of evaluating asymptotics of the solution at the end points. Experimentally, we observe that the method also works well for Cauchy singular integral equations with variable coefficients.

In this paper we prove the existence of asymptotic expansions of the error of the spline collocation method applied to Fredholm integral equations of the first kind with logarithmic kernels. These expansions justify the use of Richardson extrapolation for the acceleration of convergence of the method. The results are stated and proven for a single equation, corresponding to the parameterization of a boundary integral equation on a smooth closed curve. As a byproduct we obtain the nodal superconvergence of the scheme. These results are then extended to smooth open arcs and to systems of integral equations. Finally we prove that such expansions also exist for the Sloan iteration of the numerical solution.

We consider an ordinary differential equation arising in the study of the Ricci flow on R2. The existence and uniqueness of solutions of this equation are derived. We then study the asymptotic behaviour of these solutions at ±∞.

This paper is concerned with a generalization of the Bernstein polynomials in which the approximated function is evaluated at points spaced in geometric progression instead of the equal spacing of the original polynomials.

This paper considers a nonautonomous cooperative system, in which all the parameters are time-dependent and asymptotically approach periodic functions. We prove that under some appropriate conditions any positive solutions of the system asymptotically approach the unique positive periodic solution of the corresponding periodic system.

Sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the mutualism model

where ri, Ki, αi ∈ C(R, R+) and αi > Ki, i = 1, 2, τi, σi ∈ C(R, R+), i = 1, 2, and ri, Ki, αi, τi, σi (i = 1, 2) and functions of period ω > 0.

The paper presents some additional solutions of the diffusion equation

for the case s = 2, m = −1, a case that was left open in a previous discussion. These solutions behave in a manner that is physically acceptable as the time, t, increases and as the radial coordinate, r, tends to infinity.

In 1984, Elliott and Stenger wrote a joint paper on the approximation of Hilbert transforms over analytic arcs. In the present paper we sharpen the previously obtained results of Elliott and Stenger, and we also obtain formulas for approximating Cauchy integrals over analytic arcs.