Given a Fredhoim integral equation of the second kind, which is defined over a certain region ⊆ R2, we define and , two different numerical approximations to its solution, using the collocation and iterated collocation methods respectively. We describe without proof some known results concerning the general convergence properties of and when the kernel and solution of the integral equation are smooth. Then, we prove rigorously order of convergence estimates for and which are applicable in the practically siginificant case when is a rectangle, and the kernel of the integral equation is weakly singular. These estimates are illustrated by the numerical solution of a two dimensional weakly singular equation which arises in electrical engineering.

We describe a C0-collocation-like method for solving two-dimensional elliptic Dirichlet problems on rectangular regions, using tensor products of continuous piecewise polynomials. Nodes of the Lobatto quadrature formula are taken as the points of collocation. We show that the method is stable and convergent with order hr(r ≥ 1) in the H1–norm and hr+1(r ≥ 2) in the L2–norm, if the collocation solution js a piecewise polynomial of degree not greater than r with respect to each variable. The method has an advantage over the Galerkin procedure for the same space in that no integrals need be evaluated or approximated.

Computing the generalised gradient directly using its standard definition can involve forming the convex hull of a very large number of vectors. Here an alternative concise parametrization is developed for the generalised gradient of the signed rank regression family of objective functions, a class of piecewise linear functions which includes both convex and nonconvex members. The approach uses the geometry of the epigraph explicitly and this suggests extensions to more general functions. A nondegeneracy condition is assumed which is natural in optimization problems.

Random transition probability matrices with stationary independent factors define “white noise” environment processes for Markov chains. Two examples are considered in detail. Such environment processes can be used to construct several Markov chains which are dependent, have the same transition probabilities and are jointly a Markov chain. Transition rates for such processes are evaluated. These results have application to the study of animal movements.

The effect of an enclosed air cavity on the natural vibration frequencies of a rectangular membrane is investigated. The modes specified by an even integer are not affected. For the odd-odd modes, the frequency equation is found via a Green's function formulation and is solved to first order in a parameter representing the effect of the cavity of the rectangular drum. The frequencies are raised, with the fundamental being most affected. In the case of degeneracies, each degenerate mode contributes to the frequency shift, but the degeneracy itself is not broken to first order.

This paper deals with a class of network optimization problems in which the flow is a function of time rather than static as in the classical network flow problem, and storage is permitted at the nodes. A solution method involving discretization will be presented as an application of the ASG algorithm. We furnish a proof that the discretized solution converges to the exact continuous solution. We also apply the method to a water distribution network where we minimize the cost of pumping water to meet supply and demand, subject to both linear and nonlinear constraints.

In this article, exact and approximate techniques are used to obtain parameters of interest for two problems involving differential equations of power-law type. The first problem is related to non-linear steady-state diffusion, and is investigated by means of a hodograph transformation and an approximation using a path-independent integral. The second problem involves Poiseuille flow of a pseudo-plasticfluid, and a path-independent integral is derived which yields an exact result for the geometry under consideration.

Explicit formulae are derived for the projected gradient vector and trial dual variables required in the application of Rosen's method [4] to the solution of a Minimum Cost Network problem.

The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region in R2 together with an impedance condition on its inner boundary and another impedance condition on its outer boundary, from the complete knowledge of the eigenvalues for the two-dimensional Laplacian using the asymptotic expansion of the spectral function for small positive t.

Consideration of functions whose second difference along the trajectories of a difference equation is positive gives a stability theorem for autonomous discrete-time systems. Such functions can be used to estimate domains of nonglobal stability.

A formal framework is constructed for the comparison of different stabilization techniques, such as Wiener filtering, regularization, Courant's method and Landweber–Strand iterations, for the solution of first kind integral equations. It is shown that, when they are applied to convolution equations, all these methods can be reinterpreted as Wiener filters. This equivalence is then used to derive some specific results about regularization, Courant's method and Landweber–Strand iteration.

In this paper, we consider convex programs with linear constraints where the objective function involves nested maxima of linear functions as well as a convex function. A dual program is constructed which has interpretational significance and may be easier to solve than the primal formulation. A numerical example is given to illustrate the method.

A differential game model of a technological service industry is reformulated as an equivalent game over a function space by direct substitution of the solutions of the state equations. For this game, Nash equilibria are shown to exist under certain mild assumptions. A generalization is considered in which each firm has a choice of three different objective functions, which may reflect distinct management options in a technological service industry. Nash equilibria for the generalized version exist under similar mild assumptions.

A method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.