This paper deals with n-job, 2-machine flowshop/mean flowtime scheduling problems working under a “no-idle” constraint, that is, when machines work continuously without idle intervals. A branch and bound technique has been developed to solve the problem.

In two-dimensional bow-like flows past a semi-infinite body, one must in general expect a free-surface discontinuity, in the form of a splash or spray jet. However, there is numerical evidence that special body shapes do exist for which this splash is absent. In this study, we first establish conditions on the geometry of the bow in order that it should be splash-free at zero gravity, by solving the mathematical problem exactly. We then obtain solutions for finite non-zero gravity, by solving a non-linear integral equation numerically. A class of splashless body geometries with a downward directed segment at the extreme of the bow, to which the free surface attaches tangentially, is demonstrated in detail.

The acoustic response of a two-dimensional nearly-closed cavity to an excitation through a small opening is examined, using the method of matched asymptotic expansions. The Helmholtz mode of vibration is discussed using a low-frequency expansion of the velocity potential in the cavity interior. The variation in frequency and magnitude of the resonator response is explored, both for the Helmholtz and the natural-frequency modes.

Defining a spherical Struve function we show that the Struve transform of half integer order, or spherical Struve transform,

where n is a non-negative integer, may under suitable conditions be solved for f(t):

where is the sum of the first n + 1 terms in the asymptotic expansion of φn(x) as x → ∞. The coefficients in the asymptotic expansion are identified as

It is further shown that functions φn (x) which are representable as spherical Struve transforms satisfy n + 1 integral constraints, which in turn allow the construction of many equivalent inversion formulae.

The partially stiff system of ordinary differential equations

is studied by the methods developed in the earlier papers in this series. Here e is a small positive parameter, x and y are n- and m-vectors respectively, and A is nonsingular. A useful basis for the solution space of the homogeneous system is constructed and the method of variation of parameters is used to obtain useful representations of all solutions. Sufficient conditions are derived under which the formal approximation

is close to the actual solution. it is found that purely imaginary eigenvalues for A require more stringent requirements for the formal technique to be valid. A brief discussion of the case when A is singular shows that there are a great number of possibilities requiring consideration for a general theory. it is suggested that local computation of such cases is likely to be the most effective weapon for any specific system.

The principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

The method suggested earlier for solving the problems of optimal design from a limited set of elastic materials is generalized to a viscoelasticity case. The computational experiment for the problem of free oscillations of a spherical shell shows that characteristics of a viscoelastic layered structure may be improved due to peculiarities of wave propagation through the boundaries of layers made of different materials.

Solutions are found to two cusp-like free-surface flow problems involving the steady motion of an ideal fluid under the infinite-Froude-number approximation. The flow in each case is due to a submerged line source or sink, in the presence of a solid horizontal base.

In this paper we shall develop existence-uniqueness as well as constructive theory for the solutions of systems of nonlinear boundary value problems when only approximations of the fundamental matrix of the associated homogeneous linear differential systems are known. To make the analysis widely applicable, all the results are proved component-wise. An illustration which dwells upon the sharpness as well as the importance of the obtained results is also presented.

We consider the (n, p) boundary value problem

where λ > 0 and 0 ≤ p ≤ n - l is fixed. We characterize the values of λ such that the boundary value problem has a positive solution. For the special case λ = l, we also offer sufficient conditions for the existence of positive solutions of the boundary value problem.

A numerical solution of the RLW equation is presented using a cubic spline collocation method. Basic cubic spline relations are outlined and incorporated into the numerical solution procedure. Two test problems are studied to show the robustness of the proposed procedure.

An algorithm is given for transforming a polynomial with n coefficients to a continued fraction accurate to the same order. Only n numbers are held in storage at each stage. An extension to produce an inverse polynomial, also accurate to order n, is described.

A relation between positive commutators and absolutely continuous spectrum is obtained. If i[Y, Z] = 2Y holds on a core for Z and if Y is positive then we have a system of imprimitivity for the group on , from which it follows that Y has no singular continuous spectrum.

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.

An approximate nonlinear perturbation analysis for the re-entry roll resonance model is given. The results are used to identify the dynamic processes involved, as characterised by terms in the model equations, and to suggest a prudent management rule for this and similar transiently-resonant systems.

We consider a nonlinear second-order elliptic boundary value problem in a bounded domain Ω ⊂ RN with mixed boundary conditions. The solution is found via linearisation. We design a robust and efficient approximation scheme. Error estimates for the linearisation algorithm are derived in L2(Ω), H1(Ω) and L∞(Ω) spaces under the minimal regularity assumptions of the exact solution.

A theory is developed for the computer control of variable-structure systems, using periodic zero-order-hold sampling. A simple two-dimensional system is first analysed, and necessary and sufficient conditions for the occurrence of pseudo-sliding modes are discussed. The method is then applied to a discrete model of a cylindrical robot. The theoretical results are illustrated by computer simulations.

The paper presents new demonstrably convergent first-order iterative algorithms for unconstrained discrete-time optimal control problems. The algorithms, which solve the linear-quadratic problem in one iterative step, are particularly suited for solving nonlinear problems with linear constraints via penalty function methods. The proofs of the reduction of cost at each iteration and convergence of the algorithms are provided.