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.

The minimization of signal distortion was one approach applied successfully in the theory of optimum signal detection for arrays [3]. The processors considered operated on the input as received.

In some applications it is desirable to clip the received signal before processing and the problem of optimum processing of such clipped signals then arises. Several approaches to this problem are being studied, but the present paper is concerned with that based on minimum signal distortion.

The interest in retrial queueing systems mainly lies in their application to telephone systems. This paper studies multiserver retrial queueing systems with n servers. The arrival process is a quite general point process. An arriving customer occupies one of the free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after a random time retries in order to occupy a server. The orbit has one waiting space only, and an arriving customer, who finds all servers busy and the waiting space occupied, is lost from the system. Time intervals between possible retrials are assumed to have arbitrary distribution (the retrial scheme is explained more precisely in the paper). The paper provides analysis of this system. Specifically the paper studies the optimal number of servers to decrease the loss proportion to a given value. The representation obtained for the loss proportion enables us to solve the problem numerically. The algorithm for numerical solution includes effective simulation, which meets the challenge of a rare events problem in simulation.

The aim of this article is to prove a symmetry result for several overdetermined boundary value problems. For the two first problems, our method combines the maximum principle with the monotonicity of the mean curvature. For the others, we use essentially the compatibility condition of the Neumann problem.

Elastic behaviour of a nonhomogeneous transversely isotropic half-space is studied under the action of a smooth rigid axisymmetric indentor. Hankel transforms of different orders have been used. It is observed that in contrast to a homogeneous medium, the pressure distribution in the contact region in a nonhomogeneous medium is not directly available, rather it is obtainable from the solution of a Fredholm integral equation. The integral equation is solved for a flat-ended punch and paraboloidal indentations for various values of the nonhomogeneity parameter, and the effects of nonhomogeneity in elastic behaviour on stresses have been shown graphically. The results of the associated homogeneous case are readily available from the results of the present study.

A Demianski-type metric is investigated in connection with the Einstein-Maxwell fields. Using complex vectorial formalism, some exact solutions of Einstein-Maxwell field equations for source-free electromagnetic fields plus pure radiation fields are obtained. The radiating Demianski solution, the Debney-Kerr-Schild solution and the Brill solution are derived as particular cases.

A problem in combustion theory with tenperature-dependent conductivity is considered. It is shown that information regarding criticality dependence on data and parameters can be obtained from a transformed equation in which the conductivity is constant, while the nonlinear source term is modified. Some previous work can then be used in such a study.

We study the existence of the invariant region for the equations of nonisentropic gas dynamics. We obtain the mean-integral of the conserved quantity after making an intensive study of the Riemann problem. Using the extremum principle and the Lagrangian multiplier method, we prove that the one-dimensional equations of nonisentropic gas dynamics for an ideal gas possess a unique invariant region. However, the invariant region is not bounded.

The large-amplitude oscillations and buckling of an anisotropic cylindrical shell subjected to the initial inplane biaxial normal stresses have been analysed. The concept of anisotropy used by Lekhnitsky has been introduced into the field equations for cylindrical shells of isotropic material deduced by Donnell. The method of Galerkin and the method of successive approximation have been used to obtain the desired approximate solution. The expression for the critical loads for the buckling of anisotropic cylindrical shells has been obtained during intermediate stages of analysis. Some relevant frequency response graphs of the obtained solution are also presented. The minimum critical loads for various classes of anisotropy have also been given at the end of the discussion, to exhibit the effects of large deflections and imperfections on elastic buckling.

We consider the pricing of European options under a modified Black–Scholes equation having fractional derivatives in the “spatial” (price) variable. To be specific, the underlying price is assumed to follow a geometric Koponen–Boyarchenko–Levendorski process. This pure jump Lévy process could better capture the real behaviour of market data. Despite many difficulties caused by the “globalness” of the fractional derivatives, we derive an explicit closed-form analytical solution by solving the fractional partial differential equation analytically, using the Fourier transform technique. Based on the newly derived formula, we also examine, in theory, many basic properties of the option price under the current model. On the other hand, for practical purposes, we impose a reliable implementation method for the current formula so that it can be easily used in the trading market. With the numerical results, the impact of different parameters on the option price are also investigated.

A simple model for the propagation of a combustion wave is proposed and the speed of propagation is predicted. It is assumed that the reactant ignites at a specified temperature and then burns until depleted with reaction rate dependent on temperature and reactant concentration. The exact solution and linear stability are determined in the case of constant heat generation and a numerical scheme is developed to generate traveling wave solutions in the more general case. This numerical method is applied to the case where the temperature dependence of the reaction rate is modeled by the Arrhenius function.

There are advantages in viewing orthogonal functions as functions generated by a random variable from a basis set of functions. Let Y be a random variable distributed uniformly on [0,1]. We give two ways of generating the Zernike radial polynomials with parameter l, {Zll+2n(x), n≥0}. The first is using the standard basis {xn,n≥0} and the random variable Y1/(l+1). The second is using the nonstandard basis {xl+2n,n≥0} and the random variable Y1/2. Zernike polynomials are important in the removal of lens aberrations, in characterizing video images with a small number of numbers, and in automatic aircraft identification.

Simple proofs are given of improved results of Brown and Shepp which are useful in calculations with fractal sets. A new inequality for convex functions is also obtained.

The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.

Mathematical modelling of Wairakei geothermal field is reviewed, both lumped-parameter and distributed-parameter models. In both cases it is found that reliable predictions require five to ten years of history for calibration. With such calibration distributed-parameter models are now used for field management. A prudent model of Wairakei, constructed without such historical data, would underestimate field capacity and provide only general projections of the type of changes in surface activity and subsidence.

Similarity solutions of the steady-state equation of transport for the distribution function F0 of cosmic rays in the interplanetary region are obtained by theuse of transformation groups. The solutions are derived in detail for a spherically-symmetric model of the interplanetary region with an effective radial diffusion coefficient κ = κ0(p)rb with r the heliocentric radial distance. p the particle momentum, κ0(p) an arbitary function of p, and the solar wind velocity is radial and of constant speed V. Solutions for which the similarity variable η is a function of r only are also derived; these are of particular impoartance when the F0 is specified on a boundary of given radius. Non spherically-symmetric solutions can also be obtained by group methods and examples of such solutions are listed, without derivation, for the equation of transport incorporating the effects of anisotropic diffusion (diffusion coefficient κ1 in the radial direction and κ2 normal to it). The solutions are the most extensive steady-state analytic solutions yet obtained, and contain previous analytic solutions as special cases.

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations.

Newton's method is applied to an operator that satisfies stronger conditions than those of Kantorovich. Convergence and error estimates are compared in the two situations. As an application, we obtain information on the existence and uniqueness of a solution for differential and integral equations.

We will study one-parameter families of differentiable optimal control problems given by:

Here, at given times t the inequality constraint functions are of semi-infinite nature, the objective functional may also be of max-type. For each s ∈ ℝ the problem is equivalent to a one-parameter family (Ps (t))t∈[a,b] of differentiable optimization problems. From these the consideration of generalized critical trajectories, such as a local minimum trajectory, comes into our investigation. According to a concept introduced by Hettich, Jongen and Stein in optimization, we distinguish eight types of generalized critical trajectories. Under suitable continuity, compactness and integrability assumptions, those problems, which exclusively have generalized critical points being of one of these eight types, are generic. We study normal forms and characteristic examples, locally around these trajectories.

Moreover, we indicate the related concept of structural stability of optimal control problems due to the topological behaviour of the lower level sets under small data perturbations. Finally, we discuss the numerical consequences of our investigations for pathfollowing techniques with jumps.

Thin spray-on liners (TSLs) have been found to be effective for structurally supporting the walls of mining tunnels and thus reducing the occurrence of rock bursts, an effect primarily due to the penetration of cracks by the liner. Surface tension effects are thus important. However, TSLs are also used to simply stabilize rock surfaces, for example, to prevent rock fall, and in this context crack penetration is desirable but not necessary, and the tensile and shearing strength and adhesive properties of the liner determine its effectiveness. We examine the effectiveness of nonpenetrating TSLs in a global lined tunnel and in a local rock support context. In the tunnel context, we examine the effect of the liner on the stress distribution in a tunnel subjected to a geological or mining event. We show that the liner has little effect on stresses in the surrounding rock and that tensile stresses in the rock surface are transmitted across the liner, so that failure is likely to be due to liner rupture or detachment from the surface. In the local rock support context, loose rock movements are shown to be better achieved using a liner with small Young’s modulus, but high rupture strength.