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.

This paper treats the multi-peg generalization of the Tower of Hanoi problem with n(≥ 1) disks and p(≥ 3) pegs, P1, P2,…, Pp. Denoting by M(n, p) the presumed minimum number of moves required to transfer the tower of n disks from the source peg, P1, to the destination peg, Pp, under the condition that each move transfers the topmost disk from one peg to another such that no disk is ever placed on top of a smaller one, the Dynamic Programming technique has been employed to find the optimality equation satisfied by M(n, p). Though an explicit expression for M(n, p) is given, no explicit expressions for the partition numbers (at which M(n, p) is attained) are available in the literature for p ≥ 5. The values of the partition numbers have been given in this paper.

Utilising Jones' method associated with the Wiener-Hopf technique, explicit solutions are obtained for the temperature distributions on the surface of a cylindrical rod without an insulated core as well as that inside a cylindrical rod with an insulated inner core when the rod, in either of the two cases, is allowed to enter, with a uniform speed, into two different layers of fluid with different cooling abilities. Simple expressions are derived for the values of the sputtering temperatures of the rod at the points of entry into the respective layers, assuming the upper layer of the fluid to be of finite depth and the lower of infinite extent. Both the problems are solved through a three-part Wiener-Hopf problem of special type and the numerical results under certain special circumstances are obtained and presented in tabular forms.

A spectral method is used to consider the porous medium combustion in an infinite slab. The infinite system of ordinary differential equations for the amplitude functions is truncated and comparisons are made for different numbers of modes included in the numerical computation. It is shown that the qualitative behaviour of the solution is captured by the first eigenmode. Dependence of the solution on initial data and a parameter is also considered.

This paper derives key equations for the determination of optimal control strategies in an important engineering application. A train travels from one station to the next along a track with continuously varying gradient. The journey must be completed within a given time and it is desirable to minimise fuel consumption. We assume that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. This assumption is based on the control mechanism for a typical diesel-electric locomotive. For each setting the power developed by the locomotive is directly proportional to the rate of fuel supply. We assume a single level of braking acceleration. For each fixed finite sequence of control settings we show that fuel consumption is minimised only if the settings are changed when certain key equations are satisfied. The strategy determined by these equations is called a strategy of optimal type. We show that the equations can be derived using an intuitive limit procedure applied to corresponding equations obtained by Howlett [9, 10] in the case of a piecewise constant gradient. The equations will also be derived directly by extending the methods used by Howlett. We discuss a basic solution procedure for the key equations and apply the procedure to find a strategy of optimal type in appropriate specific examples.

The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms.

Taylor's model of dispersion simply describes the long-term spread of material along a pipe, channel or river. However, often we need multi-mode models to resolve finer details in space and time. Here we construct zonal models of dispersion via the new principle of matching their long-term evolution with that of the original problem. Using centre manifold techniques this is done straightforwardly and systematically. Furthermore, this approach provides correct initial and boundary conditions for the zonal models. We expect the proposed principle of matched centre manifold evolution to be useful in a wide range of modelling problems.

Members of an hierarchy of integrable nonlinear evolution equations, related to the well-known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of the concentration, are adapted to derive a new integrable nonlinear equation which models the surface evolution of an arbitrarily-oriented theoretical anisotropic material by the concomitant action of evaporation-condensation and surface diffusion. The constitutive relations are explicitly formulated and these show that the theoretical anisotropic material behaves like a liquid crystal. The integrable nonlinear equation may be used to advantage as test cases for numerical schemes. Its form has many attributes of the nonlinear governing equation for an isotropic material. Closed-form solutions are constructed for the evolution of a ramped surface by concomitant evaporation-condensation and surface diffusion.

The generalised elliptic-type integral Rμ(k, α, γ)

where 0 ≤ k < 1, Re(γ) > Re(α) > 0, Re(μ) ≥ −0.5, is represented in terms of the Gauss hypergeometric function by Kalla, Conde and Hubbell [8]. In 1987, Kalla, Lubner and Hubbell derived a simple-structured single-term approximation for this function in the neighbourhood of k2 = 1 in some range of the parameters α, γ and μ. Another formula which complements the parameter range was recently derived by the author. In this paper a novel technique is used in deriving multiple-term efficient approximations (in the neighbourhood of k2 = 1) which may be considered as a generalisation to the concept of the single-term approximations mentioned above. Two non-overlapping expressions which almost cover the entire range of parameters (α, γ, μ) are derived. Closed-form solutions are obtained for single- and double-term approximations (in the neighbourhood of k2 = 1). Results show that the proposed technique is superior to existing approximations for the same number of terms. Our formulation has potential application for a wide class of special functions.

Numerical solutions for travelling combustion waves of a solid material are sought. The algorithm of computation is based on a two-sided shooting method. It is found that there is a lower bound of the wave speed c, say c*, such that for c < c* no numerical solution can be constructed. This c* is a function of the activation energy of the medium.

The completion-time variance (CTV) and the waiting-time variance (WTV) are two performance measures which are commonly used in optimization of single-machine scheduling systems. This paper shows that when the number of jobs is large the two measures are nearly equivalent in a probabilistic environment.

In this paper, we discuss MDP-the moment optimal problem for the first-passage model. A policy improvement iteration algorithm is given for finding the k-moment optimal stationary policy.

Long periodic waves propagating in a closed channel are considered. The fluid consists of two layers of constant densities separated by a layer in which the density varies continuously. The numerical results of Vanden-Broeck and Turner [8] are extended. It is shown that their solutions are particular members of a family of solutions. Solutions are selected by requiring that the streamfunction takes values on the upper and lower walls which are consistent with a uniform stream far upstream. The new solutions are qualitatively similar to those of Vanden-Broeck and Turner [8]. In particular, there are periodic waves characterized by a train of ripples at their troughs. It is shown numerically that these waves approach solitary waves with oscillatory tails as their wavelength increases. Moreover special solutions for which the amplitude of the ripples is almost zero are identified. Such solutions without ripples were previously found for solitary waves with surface tension.

The Huygens' property is exploited to study propagation relations for solutions of certain types of linear higher order Cauchy problems. Motivated by the solution properties of the abstract wave problem, addition formulas are developed for the solution operators of these problems. The application of these alternative forms of the solution operators to data leads to connecting operator relations between distinct solutions of the problems at different times. We examine this solution behaviour for both analytic and abstract Cauchy problems. A basic algorithm for constructing addition formulas for solutions of ordinary differential equations is included.

We extend the concept of V-pseudo-invexity and V-quasi-invexity of multi-objective programming to the case of nonsmooth multi-objective programming problems. The generalised subgradient Kuhn-Tucker conditions are shown to be sufficient for a weak minimum of a multi-objective programming problem under certain assumptions. Duality results are also obtained.

The flow induced when fluid is withdrawn through a line sink from a layered fluid in a homogeneous, vertically confined porous medium is studied. A nonlinear integral equation is derived and solved numerically. For a given sink location, the shape of the interface can be determined for various values of the flow rate. The results are compared with exact solutions obtained using hodograph methods in a special case. It is found that the cusped and coning shapes of the interface can be accurately obtained for the sink situated at different depths in the fluid and the volume of flow into the sink per unit of time.

In this paper we have proved that the generalized incomplete gamma functions and their extensions are mutually related through integral and differential representations.