An important test of the quality of numerical methods developed to track the interface between two fluids is their ability to reproduce test cases or benchmarks. However, benchmark solutions are scarce and virtually nonexistent for complex geometries. We propose a simple method to generate benchmark solutions in the context of the two-layer flow problem, a classical multiphase flow problem. The solutions are obtained by considering the inverse problem of finding the required channel geometry to obtain a prescribed interface profile. This viewpoint shift transforms the problem from that of having to solve a complex differential equation to the much easier one of finding the roots of a quartic polynomial.

We discuss some propositions of Holmes and Manning relating to the evolution of price in a cobweb market approaching equilibrium. We find in particular that the detailed behaviour of the linear model is quite typical of nonlinear cobweb models.

The present work deals with the problem of recovering a local image from localised projections using the concept of approximation identity. It is based on the observation that the Hilbert transform of an approximation identity taken from a certain class of compactly supported functions with sufficiently many zero moments has no significant spread of support. The associated algorithm uses data pertaining to the local region along with a small amount of data from its vicinity. The main features of the algorithm are simplicity and similarity with standard filtered back projection (FBP) along with the economic use of data.

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.

If a finite segment of a spectrum is known, the determination of the finite object function in image space (or the full spectrum in frequency space) is a fundamental problem in image analysis. Gerchberg's method, which solves this problem, can be formulated as a fixed point iteration. This and other related algorithms are shown to be equivalent to a steepest descent method applied to the minimization of an appropriate functional for the Fourier Inversion Problem. Optimal steepest descent and conjugate gradient methods are derived. Numerical results from the application of these techniques are presented. The regularization of the problem and control of noise growth in the iteration are also discussed.

Some recent results on the optimal choice of quadrature rules for the finite element solution of eigenvalue problems are discussed in the light of some results of the author and J. Paine.

This work is devoted to numerical studies of nearly optimal controls of systems driven by singularly perturbed Markov chains. Our approach is based on the ideas of hierarchical controls applicable to many large-scale systems. A discrete-time linear quadratic control problem is examined. Its corresponding limit system is derived. The associated asymptotic properties and near optimality are demonstrated by numerical examples. Numerical experiments for a continuous-time hybrid linear quadratic regulator with Gaussian disturbances and a discrete-time Markov decision process are also presented. The numerical results have not only supported our theoretical findings but also provided insights for further applications.

The scaled boundary finite element method (SBFEM) is a semi-analytical computational method initially developed in the 1990s. It has been widely applied in the fields of solid mechanics, oceanic, geotechnical, hydraulic, electromagnetic and acoustic engineering problems. Most of the published work on SBFEM has focused on its theoretical development and practical applications, but, so far, no explicit discussion on the numerical stability and accuracy of its solution has been systematically documented. However, for a reliable engineering application, the inherent numerical problems associated with SBFEM solution procedures require thorough analysis in terms of its causes and the corresponding remedies. This study investigates the numerical performance of SBFEM with respect to matrix manipulation techniques and their properties. Some illustrative examples are given to identify reasons for possible numerical difficulties, and corresponding solution schemes are proposed to overcome these problems.

In this paper we describe an elementary method for calculating the matrix exponential on an arbitrary time scale. An example is also given to illustrate the result.

A new implementation of the BFGS algorithm for unconstrained optimisation is reported which utilises a conjugate factorisation of the approximating Hessian matrix. The implementation is especially useful when gradient information is estimated by finite difference formulae and it is well suited to machines which are able to exploit parallel processing.

The problem of an infinitely long rigid punch of uniform cross-section moving across a viscoelastic half-space at constant velocity, large enough so that inertial effects cannot be neglected, is examined and solved in various approximations. Frictional shear is assumed to exist between the punch and the half-space. The method, which is an extension of that developed in previous papers [6, 7], is applicable for any form of viscoelastic behaviour in the half-space. For the special case of discrete spectrum behaviour the method is described in detail. For the case where the punch is cylindrical and viscoelastic effects are small compared with elastic effects, explicit expressions are given for all quantities of interest, in particular the coefficient of hysteretic friction. A general Hilbert transform formula is derived in the appendix.

In two-dimensional flow past a body close to a free surface, the upwardly diverted portion may separate to form a splash. We model the nose of such a body by a semi-infinite obstacle of finite draft with a smoothly curved front face. This problem leads to a nonlinear integral equation with a side condition, a separation condition and an integral constraint requiring the far-upstream free surface to be asymptotically plane. The integral equation, called Villat's equation, connects a natural parametrisation of the curved front face with the parametrisation by the velocity potential near the body. The side condition fixes the position of the separation point, whereas the separation condition, known as the Brillouin-Villat condition, imposes a continuity relation to be satisfied at separation. For the described flow we derive the Brillouin-Villat condition in integral form and give a numerical solution to the problem using a polygonal approximation to the front face.

This paper deals with a minimax control problem for semilinear elliptic variational inequalities associated with bilateral constraints. The control domain is not necessarily convex. The cost functional, which is to be minimised, is the sup norm of some function of the state and the control. The major novelty of such a problem lies in the simultaneous presence of the nonsmooth state equation (variational inequality) and the nonsmooth cost functional (the sup norm). In this paper, the existence conditions and the Pontryagin-type necessary conditions for optimal controls are established.

Para-Bose coherent states, defined as “displaced” ground states, are obtained using the differential operator representation for the annihilation operator.

We consider symmetry-breaking bifurcation points which arise in parameter-dependent nonlinear equations of the form f(x, λ) = 0. These types of bifurcation points are connected to pitchfork bifurcation points. A direct method is used to compute such points. Multiple shooting is used to discretise the two-point boundary-value problems to obtain a finite-dimensional problem.

A simple model for underground mineral leaching is considered, in which liquor is injected into the rock at one point and retrieved from the rock by being pumped out at another point. In its passage through the rock, the liquor dissolves some of the ore of interest, and this is therefore recovered in solution. When the injection and recovery points lie on a vertical line, the region of wetted rock forms an axi-symmetric plume, the surface of which is a free boundary. We present an accurate numerical method for the solution of the problem, and obtain estimates for the maximum possible recovery rate of the liquor, as a fraction of the injected flow rate. Limiting cases are discussed, and other geometries for fluid recovery are considered.

We consider a distributed optimization problem over a multi-agent network, in which the sum of several local convex objective functions is minimized subject to global convex inequality constraints. We first transform the constrained optimization problem to an unconstrained one, using the exact penalty function method. Our transformed problem has a smaller number of variables and a simpler structure than the existing distributed primal–dual subgradient methods for constrained distributed optimization problems. Using the special structure of this problem, we then propose a distributed proximal-gradient algorithm over a time-changing connectivity network, and establish a convergence rate depending on the number of iterations, the network topology and the number of agents. Although the transformed problem is nonsmooth by nature, our method can still achieve a convergence rate, ${\mathcal{O}}(1/k)$, after $k$ iterations, which is faster than the rate, ${\mathcal{O}}(1/\sqrt{k})$, of existing distributed subgradient-based methods. Simulation experiments on a distributed state estimation problem illustrate the excellent performance of our proposed method.

Halley's method is a famous iteration for solving nonlinear equations. Some Kantorovich-like theorems have been given. The purpose of this note is to relax the region conditions and give another Kantorovich-like theorem for operator equations.

This paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous Lotka-Volterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well as the uniqueness of a positive solution. In order to obtain these results we employ monotonicity methods, singular spectral theory and a fixed point index.

In this paper we study a variational inequality in which the principal operator is a generalised Laplacian with fast growth at infinity and slow growth at 0. By defining appropriate sub-and super-solutions, we show the existence of solutions and extremal solutions of this inequality above the subsolutions or between the sub- and super-solutions.