A thin, partially insulating circular rug is placed on a uniform half space up through which a steady heat flow passes. The corresponding dual integral equations are solved using Tranter's method, finite Legendre transforms and Mellin-Bames contour integrals. An untabulated Bessel (or Stieltjes) transform similar to the discontinuous WeberSchafheitlin integral is evaluated, and a simple expression derived for the rug's surface temperature.

Pointwise bounds are obtained for the solution of a Dirichlet problem involving the nonlinear Liouville equation in the plane, Illustrative calculations are performed for a square domain.

When material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant, the steady-state regime is governed by a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions. In this paper we consider questions of existence of solutions to these equations. It is shown that, with the exception of the special case in which the mass-transfer is uninhibited on the boundary, a solution always exists, whereas in this special case a solution exists only for sufficiently low values of the exothermicity. Bounds are established for the solutions and the occurrence of minimal and maximal solutions is shown for some cases. Finally the behaviour of the solution set with respect to one of the parameters is studied.

In an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.

This note examines maximum principles for systems of parabolic partial differential equations describing diffusion in the presence of three diffusion paths. The particular system under consideration arises from a random walk model. For a more general system constraints on the various constants are given which guarantee maximum principles. Remarkably, the physical system arising from the random walk model automatically satisfies these constraints.

In previous papers, three terms have been included in Ogden's stress-deformation function for incompressible isotropic elastic materials. The material constants have been calculated by elementary methods and the resulting fits to sets of experimental data have been moderately good.

The purpose of the present paper is to improve upon established correlation between theory and experiment by means of a systematic optimization procedure for calculating material constants. For purposes of illustration the Levenberg-Marquardt non-linear least squares optimization algorithm is adapted to determine the material constants in Ogden's stress-deformation function.

The use of this algorithm for three-term stress-deformation functions improves somewhat on previous results. Calculations are also carried out in respect of a four-term stress-deformation function and further improvement in the fit is achieved over a large range of deformation.

The asymptotic expansion for a spectral function of the Laplacian operator, involving geometrical properties of the domain, is demonstrated by direct calculation for the case of a doubly-connected region in the form of a narrow annular membrane. By utilizing a known formula for the zeros of the eigenvalue equation containing Bessel functions, the area, total perimeter and connectivity are all extracted explicitly.

Segal's unitarizing complex structure J is shown, in the Fermi-Dirac case, to be the orthogonal component in the polar decomposition of the real skew adjoint generator of classical dynamics. It is proven that in the Bose-Einstein case, the classical symplectic dynamics cannot be unitarized unless the generator is similar to a real skew adjoint operator.

With the classical Hamiltonian strictly positive, J is the pseudo-orthogonal component in the polar decomposition of the generator, using spectral theory in Krein space with indefinite metric. Thus, J can be expressed simply in terms of the projection E(0) onto the subspace of classical solutions with negative frequency. This complements the physicists' experience that conceptual difficulties arise when dynamically invariant separation of positive and negative frequency solutions is impossible.

Field equations for coupled gravitational and zero mass scalar fields in the presence of a point charge are obtained with the aid of a static spherically symmetric conformally flat metric. A closed from exact solution of the field equations is presented which may be considered as describing the field of a charged particle at the origin surrounded by the scalar meson field in a flat space-time.

The general solution of the rth inhomogeneous linear difference equation is given in the form

The coefficients , i = 2, …, r, and b(n−r)(n) can be evaluated from n values , k = 0, …, n − 1, which santisfy an rth order homogenous linear difference equation. In the rth order homogeneous case and if n ≥ 2r, the method requires the evaluation of r determinants of successive orders n − 2r + 1, n − 2r + 2, …, n − r. If r ≤ n ≤ 2r − 1, only n − r determinants are required, with orders varying from 1 to n − r. In the second order ihnomogenous case, can be evaluated from a continued fraction amd a simple product.

A model governing the combustion of a material is considered. The model consists of two non-linear coupled parabolic equations with initial and boundary conditions. An approximation for the rate of reactant consumption is made to enable the system to the treated by laplace transform. Three simple geometries are considered; namely, an infinite slab, an infinite circular and a sphere. The results obtained are then compared with numerical solutions for spme specific values of the parameters. There is good agreement over time duration for which numerical work was performed.

New polynomial solutions of the Navier-Stokes equations for steady uni-directional flow of viscous incompressible fluid, with a free surface, down inclined channels of specialized cross-section are considered. An inverse method is uded to obtain the geometrical shape of the channel by equating the polynomials solution to zero (i.e. the no-slip condition) and thence determining the boundary shape.