In this paper we consider an optimal control problem governed by a system of nonlinear hyperbolic partial differential equations with deviating argument, Darboux-type boundary conditions and terminal state inequality constraints. The control variables are assumed to be measurable and the state variables are assumed to belong to a Sobolev space. We derive an integral representation of the increments of the functionals involved, and using separation theorems of functional analysis, obtain necessary conditions for optimality in the form of a Pontryagin maximum principle. The approach presented here applies equally well to other nonlinear constrained distributed parameters with deviating argument.

This paper establishes an estimate for the asymptotic behaviour of the spectrum of a direct strain feedback (DSF) control system. The results show that the system operator corresponding to the closed loop system cannot have an analytic extension and that the decay rate for the system energy is not proportional to the feedback constant.

Sufficient conditions are derived for the relative controllability of nonlinear neutral Volterra integrodifferential systems with distributed delays in the control variables. The results are a generalization of previous results and are obtained by using Schauder's fixed-point theorem.

Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

The temporal instability of a developing swirling incompressible jet is considered. The jet development (in the streamwise direction) is modelled by combining a near-field and far-field approximation to the jet velocity profile into a one parameter family of basic velocity fields. The single parameter in the jet velocity field then allows us to model the radial spreading of the jet and the decay of swirl observed experimentally. Two distinct modes of instability of this model profile are found. The first is that found from a stability analysis of a fully developed swirling jet in the far field whilst the second is relevant to a “top-hat” jet with an imposed rigid body rotation. We demonstrate that the effect of azimuthal swirl is to destabilise both modes of instability. Additionally our results suggest that the near-nozzle modes of instability will dominate; indeed the growth rates of these modes are significantly larger than those found from previous studies of a fully developed jet in the far-field region.

This paper is concerned with a reinvestigation of the problem of water wave scattering by a wall with multiple gaps by using the solution of a singular integral equation with a combination of logarithmic and power (Cauchy-type) kernels in disjoint multiple intervals. Use of Havelock's expansion of water wave potential reduces the problem to such an integral equation in the horizontal velocity across the gaps. The solution of the integral equation is obtained by utilizing the solutions of Cauchy-type integral equations in (0,∞) and also in multiple disjoint intervals. An explicit expression for the reflection coefficient is obtained for a wall with n gaps and supplemented by numerical results for up to three gaps.

This work deals with low-frequency asymptotic solutions using the method of matched asymptotic expansions. It is based on two papers by Buchwald [3] and Buchwald and Tran Cong [4] who studied the diffraction of elastic waves by a small circular cavity and a small elliptic cavity, respectively, in an otherwise unbounded domain. Here we clarify and systematize some aspects of their work and extend it to the diffraction of elastic waves by a small cylindrical cavity with a hypotrochoidal boundary. Results for the case of an incident P-wave are compared, in the special case of an elliptic boundary, with the results from the numerical solution of the boundary integral equation method.

We prove a continuous version of a relaxation theorem for the nonconvex Darboux problem xlt ε F(t, τ, x, xt, xτ). This result allows us to use Warga's open mapping theorem for deriving necessary conditions in the form of a maximum principle for optimization problems with endpoint constraints. Neither constraint qualification nor regularity assumption is supposed.

In this paper, we use an ordinary differential equation approach to study the existence of similarity solutions for the equation u1 = Δ(uα) + θu–β in Rn × (0, ∞) where β > 0, θ ∈ [0, 1}, and n ≥ 1. This includes the slow diffusion equation when α > = 1, and the standard heat equation when α = 1, and the fast diffusion equation when 0 < α < 1. We prove that there are forward self-similar solutions for this equation with initial data of the form c|x|p, where p = 2/(α + β) if θ = 1; p ≥ 0 and 2 + (1 – α)p > 0 if θ = 0, for some positive constant c.

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 paper we have introduced extensions νυ(α, x; b) and Γν(α, x; b) of the generalized Gamma functions γ(α x; b) and Γ(α, x; b) considered recently by Chaudhry and Zubair. These extensions are found useful in the representations of the Laplace and K-transforms of a class of functions. We have also defined a generalization of the inverse Gaussian distribution. The cumulative and the reliability functions of the generalized inverse Gaussian distribution are expressed in terms of these functions. Some useful properties of the functions are also discussed.

The eddy currents induced in a thin sheet of variable conductivity by a sinusoidally varying primary magnetic field are investigated in the low frequency limit when the depth of penetration of the primary field is much greater than the thickness of the sheet. The problem is formulated in terms of a set of integro-differential equations. The method of solution is applicable to bodies with arbitrary planar shape and the result is particularly useful in inverse problems involving bodies with conductivity inhomogeneities.

Withdrawal flow through a point sink on the bottom of a fluid of finite depth is considered. The fluid is at rest at infinity, and a stagnation point is present at the free surface, directly above the point sink. Numerical solutions are computed by means of the method of fundamental solutions, and it is observed that flows of this type are apparently possible for Froude number less than about 1.5. Relationships to previous work are discussed.

In this paper we propose a P1 finite element preconditioning using the so-called ‘hat-function’, to a collocation scheme constructed by quadratic splines for a 2nd-order separable elliptic operator and we show that the resulting preconditioning system of equations is well conditioned with the condition number independent of the number of unknowns.

The problem of reflection of water waves by a nearly vertical porous wall has been investigated. A perturbational analysis has been applied for the first order correction to be employed to the corresponding vertical wall problem. The Green's function technique has been used to obtain the solution of the boundary value problem at hand, after utilising a mixed Fourier transform together with an idea involving the regularity of the transformed function along the real axis. The cases of fluid of finite as well as infinite depth have been taken into consideration. Particular shapes of the wall have been considered and numerical results are also discussed.