Gauss' Theorema Egregium produces a partial differential equation which relates the Gaussian curvature K to components of the metric tensor and its derivatives. Well-known partial differential equations (PDEs) such as the Schrödinger equation and the sine-Gordon equation can be derived from Gauss' equation for specific choices of K and coördinate systems. In this paper we consider a class of Bäcklund Transformations which corresponds to coördinate transformations on surfaces with a given Gaussian curvature. These Bäcklund Transformations lead to the construction of solutions to certain classes of non-linear second order PDEs of hyperbolic type by identifying these PDEs as the Gauss equation in some coördinate system. The possibility of solving the Cauchy Problem has also been explored for these classes of equations.

Analytical, approximate and numerical methods are used to study the Neumann boundary value problem

− uxx + q2u = u2(1 + sin x), for 0 < x < π,

subject to ux(0) = 0, ux(π) = 0,

for q2 ∈ (0,∞). Asymptotic approximations to (1) are found for q2 small and q2 large. In the case where q2 is large u(x) ≈ 3qδ(x − π/2). When q2 = 0 we show that the only possible solution is u ≡ 0. However, there exist non-zero solutions for q2 > 0 as well as the trivial solution u ≡ 0. To O(q4) in the q2 small case u(x) = q2π(π + 2)−1, so that bifurcation occurs about the trivial solution branch u ≡ 0 at the first eigenvalue λ0 = 0 and in the direction of the first eigenfunction ξ0 = constant.

We obtain a bifurcation diagram for (1), which confirms that there exists a positive solution for q2 ∈ (0, 10). Symmetry-breaking bifurcations and blow-up behaviour occur on certain regions of the diagram. We show that all non-trival solutions to the problem must be positive.

The formal outer solution u = q2û appears to satisfy û = û2(1 + sin x), so that û ≡ 0 and û = (1 + sin x)−1 are possible limit solutions. However, in the non-trivial case ûx(0) = −1 and ûx(π) = 1; this means that û does not satisfy the boundary conditions required for a solution of (1). This behaviour usually implies that for q2 large a boundary layer exists near x = 0 (and one near x = π), which corrects the slope. However, we find no evidence for such a solution structure, and only find perturbations in the direction of a delta function about u ≡ 0. We show using the monotone convergence theorem for quadratic forms that the inverse of the operator on the left-hand side of (1) is strongly convergent as q2 → ∞. We show that strong convergence of the operator is sufficient to stop outer-layer behaviour occurring.

By means of piecewise continuous vector functions, which are analogues of the classical Lyapunov functions and via the comparison method, sufficient conditions are found for conditional, stability of the zero solution of a system of impulsive differential-difference equations.

The free surface flow of a finite depth fluid past a semi-infinite body is considered. The fluid is assumed to have constant vorticity throughout and the free surface is assumed to attach smoothly to the front face of the body. Numerical solutions are found using a boundary integral method in the physical plane and it is shown that solutions exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a comer is also considered. Vorticity is included in the flow and it is shown that the behaviour of the solutions is qualitatively the same as that found in the problem described above.

Newton's method is applied to an operator that satisfies stronger conditions than those of Kantorovich. Convergence and error estimates are compared in the two situations. As an application, we obtain information on the existence and uniqueness of a solution for differential and integral equations.

Recently Dragomir and Goh have produced some interesting new bounds relating to entropy measures in information theory. We establish several refinements of their results.

Lotka-Volterra systems have been used extensively in modelling population dynamics. In this paper, it is shown that chaotic behaviour in the sense of Smale can exist in timeperiodically perturbed systems of Lotka-Volterra equations. First, a slowly varying threedimensional perturbed Lotka-Volterra system is considered and the corresponding unperturbed system is shown to possess a heteroclinic cycle. By using Melnikov's method, sufficient conditions are obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra system is examined, leading finally to an application with a model for the self-organisation of macromolecules.

In this paper we prove the existence of asymptotic expansions of the error of the spline collocation method applied to Fredholm integral equations of the first kind with logarithmic kernels. These expansions justify the use of Richardson extrapolation for the acceleration of convergence of the method. The results are stated and proven for a single equation, corresponding to the parameterization of a boundary integral equation on a smooth closed curve. As a byproduct we obtain the nodal superconvergence of the scheme. These results are then extended to smooth open arcs and to systems of integral equations. Finally we prove that such expansions also exist for the Sloan iteration of the numerical solution.

We consider an ordinary differential equation arising in the study of the Ricci flow on R2. The existence and uniqueness of solutions of this equation are derived. We then study the asymptotic behaviour of these solutions at ±∞.

The paper presents some additional solutions of the diffusion equation

for the case s = 2, m = −1, a case that was left open in a previous discussion. These solutions behave in a manner that is physically acceptable as the time, t, increases and as the radial coordinate, r, tends to infinity.