A steady two-dimensional jet of an inviscid incompressible fluid rising and falling under gravity is considered. The jet is aimed vertically upwards and the flow is assumed to be bounded entirely by two free surfaces. The problem is solved numerically by finite differences. Accurate results for the free surface profiles are presented.

Here we discuss the stability and convergence of a quadrature method for Symm's integral equation on an open smooth arc. The method is an adaptation of an approach considered by Sloan and Burn for closed curves. Before applying the quadrature scheme, we use a cosine substitution to remove the endpoint singularity of the solution. The family of methods includes schemes with any order O(hp) of convergence.

Recently in several papers the boundary element method has been applied to non-linear problems. In this paper we extend the analysis to strongly nonlinear boundary value problems. We shall prove the convergence and the stability of the Galerkin method in Lp spaces. Optimal order error estimates in Lp space then follow. We use the theory of A-proper mappings and monotone operators to prove convergence of the method. We note that the analysis includes the u4 -nonlinearity, which is encountered in heat radiation problems.

The well-known Wilkinson expressions for the first derivatives of (ordinary) eigen-values and eigenvectors of simple matrices, in terms of the set of eigenvalues and eigenvectors, are redifferentiated and combined to obtain partial differential equations for the eigenvalues. Analogous expressions are obtained for the first derivatives of generalised eigenvalues and eigenvectors of simple pairs of matrices (A, B), defined by . Again, redifferentiation and combination yields slightly more complicated partial differential equations for the generalised eigenvalues. When the matrices depend on a few parameters θ1, θ2, …, the resulting differential equations for the eigenvalues, with those parameters as independent variables, can easily be derived. These parametric equations are explicit representations of analytic perturbation results of Kato, expressed by him as rather abstract complex matrix integrals. Connections with bounds for eigenvalues derived by Stewart and Sun can also be made. Two applications are exhibited, the first being to a broken symmetry problem, the second being to working out the second-order perturbations for a classical problem in the theory of waves in cold plasmas.

This paper deals with the convergence aspect of diffusive delay Lotka-Volterra systems with infinite delays. It is well known that such a system has a globally asymptotically stable steady state if the negative feedbacks of the intraspecific competitions are dominant and instantaneous. It is shown here that such a globally asymptotically stable steady state continues to exist even if the instantaneous assumption is removed, provided that solutions of the system are eventually uniformly bounded and the delays involved in the intraspecific competitions are small. This work generalises several recent related ones.

Brockett has studied the isospectral flow Ḣ = [H, [H, N]], with [A, B] = AB ∔ BA, on spaces of real symmetric matrices. The flow diagonalises real symmetric matrices and can be used to solve linear programming problems with compact convex constraints. We show that the flow converges exponentially fast to the optimal solution of the programming problem and we give explicit estimates for the time needed by the flow to approach an ε-neighbourhood of the optimum. An interior point algorithm for the standard simplex is analysed in detail and a comparison is made with a continuous time version of Karmarkar algorithm.

In two dimensions it is found that the most general autonomous Hamiltonian possessing a Laplace-Runge-Lenz vector is The Poisson bracket of the two components of this vector leads to a third first-integral, cubic in the momenta. The Lie algebra of the three integrals under the operation of the Poisson bracket closes, and is shown to be so(3) for negative energy and so(2, 1) for positive energy. In the case of zero energy, the algebra is W(3, 1). The result does not have a three-dimensional analogue, apart from the usual Kepler problem.