IMA Journal of Applied Mathematics - current issue

The singular sources method for 3D inverse acoustic obstacle scattering problems

Ben Hassen, M. F., Erhard, K., Potthast, R. — Mon, 25 Jan 2010 09:44:40 PST

We investigate the ‘singular sources method’ (SSM) for the reconstruction of 3D scattering obstacles with sound-soft, sound-hard or impedance boundary condition. The task of this work is the proof of concept for the ‘numerical feasibility’ of the SSM for reconstructions of 3D objects in acoustics. To our knowledge, these are the first numerical results for the SSM in three dimensions. In particular, we will show reconstructions for different geometric objects for which the boundary condition is not needed to be known in the reconstruction scheme.

Solutions and symmetry reductions of the n-dimensional non-linear convection-diffusion equations

Ji, L., Qu, C., Ye, Y. — Mon, 25 Jan 2010 09:44:41 PST

This paper discusses a wide class of n-dimensional non-linear convection–diffusion equations with source term. It is shown that the radially symmetric equations admit certain types of conditional Lie–Bäcklund symmetries. As a result, exact solutions and symmetry reductions to 2D dynamical systems of the resulting equations are obtained. Those solutions extend the known ones such as self-similar solutions and instantaneous source-type solutions of the porous medium equation with absorption term. The behaviour of extinction and blow-up to many of the solutions are described.

Energy and helicity dissipation rates of the NS-alpha and NS-alpha-deconvolution models

Layton, W., Rebholz, L., Sussman, M. — Mon, 25 Jan 2010 09:44:41 PST

We consider the Navier-Stokes (NS)-alpha and the family of high-accuracy NS-alpha-deconvolution models of turbulence on = [0, L]³ subject to periodic boundary conditions. For body-force-driven turbulence, we prove directly from the model equations of motion the following bounds on the time-averaged modified energy dissipation rate, <_{, N}(w_{, N})>, and unmodified helicity dissipation rate, <(w_{, N})>, for the Nth model (N = 0, 1, 2, ...): Here, N is the degree of the approximate deconvolution operator, U_N and L_N are global velocity and length scales and C₁ and C₂ are constants that do not depend on U_N.

Qualitative dynamics of a vaccination model for HSV-2

Podder, C. N., Gumel, A. B. — Mon, 25 Jan 2010 09:44:41 PST

A new mathematical model for the transmission dynamics of herpes simplex virus type 2 (HSV-2), which takes into account disease transmission by infected individuals in the quiescent state and an imperfect HSV-2 vaccine, is designed and qualitatively analysed. In the absence of vaccination, it is shown that the model has a globally asymptotically stable (GAS) disease-free equilibrium (DFE) point whenever an epidemiological threshold, known as the ‘basic reproduction number’, is less than unity. Further, this model has a unique endemic equilibrium whenever the reproduction number exceeds unity. Using a non-linear Lyapunov function, it is shown that the unique endemic equilibrium is GAS (for a special case) when the associated reproduction threshold is greater than unity. On the other hand, the model with vaccination undergoes a vaccine-induced backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium when the reproduction threshold is less than unity. Threshold analysis of the vaccination model reveals that the use of an imperfect HSV-2 vaccine could have positive or negative population-level impact (in reducing disease burden). Simulations of the vaccination model show that an HSV-2 vaccine could lead to effective disease control or elimination if the vaccine efficacy and the fraction of susceptible individuals vaccinated at steady state are high enough (at least 80% each).

Linear wave forcing of an array of axisymmetric ice floes

Bennetts, L. G., Squire, V. A. — Mon, 25 Jan 2010 09:44:41 PST

Under linear and time-harmonic conditions, a set of periodic Green's functions is derived to combine the interactions of an infinite number of identical equispaced floating bodies. The bodies themselves are compliant thin elastic plates that can represent ice floes, and unlike previous studies, they are permitted to vary axisymmetrically in thickness through both their upper and their lower surfaces, with a realistic draught also admitted. Initially, the governing equations are simplified by means of an expansion of the vertical dependence of the unknown velocity potential combined with a variational principle, which reduces calculations to the horizontal plane alone. The unknowns of the resulting equations are written as an integral representation in the free-surface domain and as a Fourier expansion in the domain of the ice-covered fluid, and these are matched at their common boundary to complete the solution process. Our method is validated using numerical results for example problems. The effects of varying the distance between the floes, as well as the introduction of thickness variations and submergence, are also demonstrated.

Asymptotic behaviour of thin linearly elastic layers of oscillating thickness

El Jarroudi, M. — Mon, 25 Jan 2010 09:44:41 PST

We consider a 3D linearly elastic material whose thickness is a positive Lipschitz continuous function r(x₁, x₂) ≤ . We study the asymptotic behaviour, when tends to 0, of the associated sequence of rescaled energy functionals using -convergence methods. According to the bounds of the function r, we obtain two possible asymptotic behaviours: a membrane and a flexure one.