A matrix solution and a determinantal solution are obtained for a general linear recurrence relation.

Solutions of a homogeneous (r + 1)-term linear difference equation are given in two different forms. One involves the elements of a certain matrix, while the other is in terms of certain lower Hessenberg determinants. The results generalize some earlier results of Brown [1] for the solution of a 3-term linear difference equation.

We are concerned with the solution of the second kind Fredholm equation (and eigenvalue problem) by a projection method, where the projection is either an orthogonal projection on a set of piecewise polynomials or an interpolatory projection at the Gauss points of subintervals.

We study these cases of superconvergence of the Sloan iterated solution: global superconvergence for a smooth kernel, and superconvergence at the partition points for a kernel of “Green's function” type. The mathematical analysis applies for the solution of the inhomogeneous equation as well as for an eigenvector.

The combustion of a material can be modelled by two coupled parabolic partial differential equations for the temperature and concentration of the material. This paper deals with properties of the solution of these equations inside a cylinder or a sphere and under given initial conditions. Bounds for the variation of the temperature with the initial conditions are first established by considering a decoupled form of the equations. Then the coupled system is used to obtain approximate expressions for the temporal evolution of temperature and concentration.

Equations are derived which describe the evolution of the mean flow generated by a progressing water wave packet. The effect of friction is included, and so the equations are subject to the boundary conditions first derived by Longuet-Higgins [10]. Solutions of the equations are obtained for a wave packet of finite length, and also for a uniform wave train. The latter solution is compared with experiments.

Para-Bose coherent states, defined as “displaced” ground states, are obtained using the differential operator representation for the annihilation operator.

A deferred correction procedure for the approximate solution of the second-kind equation is introduced, compared with an extrapolation procedure, and illustrated for integral and differential equations.

Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals, high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. In pourous media such as rocks, fissures provide a system of high-flow paths. Recently, based on a continuum approach, these phenomena have been modelled, resulting in coupled systems of partial differential equations of parabolic type for the concentrations in bulk and in the high-diffusivity paths. This theory assumes that each point of the medium is simultaneously occupied by more than one diffusion or flow path. Here a simple discrete random walk model of diffusion in a medium with double diffusivity is given.

An algorithm is given for calculating the asymptotic behaviour of the temperature of the fluid in an adiabatic calorimeter, and used to derive the asymptote for a finite cylinder.

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.

A method is derived for the solution of boundary value problems governed by a second-order elliptic partial differential equation with variable coefficients. The method is obtained by expressing the solution to a particular problem in terms of an integral taken round the boundary of the region under consideration.

Under the appropriate physical hypotheses, the problem of determining the pressure distribution in a gas-filled slider bearing becomes a singular perturbation problem as Λ, the bearing number, tends to infinity. This paper extends the results of an earlier one by the author to consider the case where the film profile has jump discontinuities in slope at points interior to the bearing. Application of the methods of general singular perturbation theory establishes the appropriate existence-uniqueness results for this problem, and a means is devised by which uniformly valid asymptotic approximations to the pressure distribution may be obtained for large values of Λ.

In a paper published in 1949, E. R. Love [10] found an integral relation between a separated solution (in oblate spheroidal coordinates) to a particular mixed boundary-value problem and a solution to the same problem using an integral representation. This note examines further representations of the same type which occur in some simple two and three-dimensional potential problems.

This paper considers the improvement of approximate eigenvalues and eigenfunctions of integral equations using the method of deferred correction. A convergence theorem is proved and a numerical example illustrating the theory is given.

Here we consider a particular class of stochastic geometric programs in which the randomness occurs in the decision variables. Specifically we analyse a program in which we specify a joint normal probability for the dicision variables and require the constraint set to be satisfied in the chance constrained mode. A numerical example is given to illustrate the approach.

In a recent paper estimates of the solutions of two nonlinear differential equations were made by use of the hypercircie method. Here exact solutions are given which are compared with those estimates.

Slow catalyst poisoning can result in the sudden failure of a chemical reactor operating isothermally with substrate-inhibited kinetics. At failure, a satisfactory steady state is exchanged for one of low conversion. The method of matched asymptotic expansions is used to give a detailed description of the exchange process in the phase plane. The structure of the jump is ascertained by separate asymptotic expansions across two adjoining transition regions in which the independent variables contain unknown shifts.

Certain definite integrals involving spherical Bessel functions are treated by relating them to Fourier integrals of the point multipoles of potential theory. The main result (apparently new) concerns

where l1, l2 and N are non-negative integers, and r1 and r2 are real; it is interpreted as a generalized function derived by differential operations from the delta function δ(r1 − r2). An ancillary theorem is presented which expresses the gradient ∇2nYlm(∇) of a spherical harmonic function g(r)YLM(Ω) in a form that separates angular and radial variables. A simple means of translating such a function is also derived.

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.

A theory is provided for the natural seiching frequencies and radiative decay rates for a shallow-water basin whose connection to open water is restricted by a submerged wall or reef. The transition from an essentially-open basin to a closed basin, as the aperture reduces to zero, is discussed using a matching procedure. Graphs of frequencies and damping factors as functions of aperture size are obtained for idealized two-dimensional shelf configurations, involving a constant-depth shallow basin connected to constant-depth, but not necessarily shallow, open water.