We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite-volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

The behaviour of an axisymmetric bubble in a pure liquid forced by an acoustic pressure field is analysed. The bubble is assumed to have a sharp deformable interface, which is subject both to surface tension and to Rayleigh viscosity damping. Two modelling regimes are considered. The first is a linearized solution, based on the assumption of small axisymmetric deformations to an otherwise spherical bubble. The second involves a semi-numerical solution of the fully nonlinear problem, using a novel spectral method of high accuracy. For large-amplitude nonspherical bubble oscillations, the fully nonlinear solutions show that a complicated resonance structure is possible and that curvature singularities may occur at the interface, even in the presence of surface tension. Rayleigh viscosity at the interface prevents singularity formation, but eventually causes the bubble to become purely spherical unless shape-mode resonances occur. An extended analysis is also presented for purely spherical bubbles, which allows for a more detailed study of the effects of resonance and the Rayleigh viscosity at the bubble surface.

We propose a novel time-asymptotically stable, implicit–explicit, adaptive, time integration method (denoted by the $\theta $-method) for the solution of the fractional advection–diffusion-reaction (FADR) equations. The spectral analysis of the method (involving the group velocity and the phase speed) indicates a region of favourable dispersion for a limited range of Péclet number. The numerical inversion of the coefficient matrix is avoided by exploiting the sparse structure of the matrix in the iterative solver for the Poisson equation. The accuracy and the efficacy of the method is benchmarked using (a) the two-dimensional fractional diffusion equation, originally proposed by researchers earlier, and (b) the incompressible, subdiffusive dynamics of a planar viscoelastic channel flow of the Rouse chain melts (FADR equation with fractional time-derivative of order ) and the Zimm chain solution (). Numerical simulations of the viscoelastic channel flow effectively capture the nonhomogeneous regions of high viscosity at low fluid inertia (or the so-called “spatiotemporal macrostructures”), experimentally observed in the flow-instability transition of subdiffusive flows.

We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

We devise schemes for producing, in the least possible time, p identical objects with n agents that work at differing speeds. This involves halting the process to transfer production across agent types. For the case of two types of agent, we construct schemes based on the Euclidean algorithm that seeks to minimize the number of pauses in production.

For microscale heterogeneous partial differential equations (PDEs), this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. A key innovation herein is to analyse the ensemble of all phase-shifts of the heterogeneity. Dynamical systems theory then frames the homogenization as a slow manifold of the ensemble. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of an exact homogenization at finite scale separation, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology complements existing well-established results by providing a new rigorous and flexible approach to homogenization that potentially also provides correct macroscale initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.

Clustering is a method of allocating data points in various groups, known as clusters, based on similarity. The notion of expressing similarity mathematically and then maximizing it (minimize dissimilarity) can be formulated as an optimization problem. Spectral clustering is an example of such an approach to clustering, and it has been successfully applied to visualization of clustering and mapping of points into clusters in two and three dimensions. Higher dimension problems remained untouched due to complexity and, most importantly, lack of understanding what “similarity” means in higher dimensions. In this paper, we apply spectral clustering to long timeseries EEG (electroencephalogram) data. We developed several models, based on different similarity functions and different approaches for spectral clustering itself. The results of the numerical experiment demonstrate that the created models are accurate and can be used for timeseries classification.

We find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from periodic and localized initial conditions as separate cases. The particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist) at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localized initial condition, are shown to approach the appropriate particular solution.

Many industrial design problems are characterized by a lack of an analytical expression defining the relationship between design variables and chosen quality metrics. Evaluating the quality of new designs is therefore restricted to running a predetermined process such as physical testing of prototypes. When these processes carry a high cost, choosing how to gather further data can be very challenging, whether the end goal is to accurately predict the quality of future designs or to find an optimal design. In the multi-fidelity setting, one or more approximations of a design’s performance are available at varying costs and accuracies. Surrogate modelling methods have long been applied to problems of this type, combining data from multiple sources into a model which guides further sampling. Many challenges still exist; however, the foremost among them is choosing when and how to rely on available low-fidelity sources. This tutorial-style paper presents an introduction to the field of surrogate modelling for multi-fidelity expensive black-box problems, including classical approaches and open questions in the field. An illustrative example using Australian elevation data is provided to show the potential downfalls in blindly trusting or ignoring low-fidelity sources, a question that has recently gained much interest in the community.

Singularly perturbed ordinary differential equations often exhibit Stokes’ phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes lines. These curves originate at singular points in the leading-order solution to the differential equation. In many important problems, it is impossible to obtain a closed-form expression for these leading-order solutions, and it is therefore challenging to locate these singular points. We present evidence that the analytic leading-order solution of a linear differential equation can be replaced with a numerical rational approximation using the adaptive Antoulas–Anderson (AAA) method. Despite such an approximation having completely different singularity types and locations, we show that the subsequent exponential asymptotic analysis accurately predicts the exponentially small behaviour present in the solution. For sufficiently small values of the asymptotic parameter, this approach breaks down; however, the range of validity may be extended by increasing the number of poles in the rational approximation. We present a related nonlinear problem and discuss the challenges that arise due to nonlinear effects. Overall, our approach allows for the study of exponentially small asymptotic effects without requiring an exact analytic form for the leading-order solution; this permits exponential asymptotic methods to be used in a much wider range of applications.

We consider planar flow involving two viscous fluids in a porous medium. One fluid is injected through a line source at the origin and moves radially outwards, pushing the second, ambient fluid outwards. There is an interface between the two fluids and if the inner injected fluid is of lower viscosity, the interface is unstable to small disturbances and radially directed unstable Saffman–Taylor fingers are produced. A linearized theory is presented and is compared with nonlinear results obtained using a numerical spectral method. An additional theory is also discussed, in which the sharp interface is replaced with a narrow diffuse interfacial region. We show that the nonlinear results are in close agreement with the linearized theory for small-amplitude disturbances at early times, but that large-amplitude fingers develop at later times and can even detach completely from the initial injection region.

Mathematical modelling of microwaves travelling through bauxite ore provides a way to compute moisture content in the free space transmission method given data on signal attenuation, phase shift and variable bauxite depth. We extend a recently developed four-layer model that uses coupled ordinary differential wave equations for the electric field together with continuity boundary conditions at interfaces between ore, air and antenna to find a solution that incorporates multiple internal reflections in ore and air. The model provides good fits to data, depending on ore permittivity and conductivity.

Our extensions are to use effective medium models to obtain electromagnetic properties of the ore mixture from moisture content and to incorporate the damping effects of scattering from the ore surface. Our model leads to a formula for the received signal showing how signal strengths SS and phase shifts depend on the moisture content of the bauxite ore, through the effects of moisture on permittivity and conductivity. We show that SS may be noninvertible, indicating that attenuation data alone cannot be used to infer moisture content. Combining with phase data typically corrects the noninvertibility. Reducing the operating frequency dramatically improves the usefulness of signal strength data for inferring moisture content.

Tissues form from collections of cells that interact together mechanically via cell-to-cell adhesion, mediated by transmembrane cell adhesion molecules. Under a sufficiently large amount of induced stress, these tissues can undergo elastic deformation in the direction of tension, where they then elongate without any topological changes, and experience plastic deformation within the tissue. In this work, we present a novel mathematical model describing the deformation of cells, where tissues are elongated in a controlled manner. In doing so, the cells are able to undergo remodelling through elastic and then plastic deformation, in accordance with experimental observation. Our model describes bistable sizes of a cell that actively deform under stress to elongate the cell. In the absence of remodelling, the model reduces to the standard linear interaction model. In the presence of instant remodelling, we provide a bifurcation analysis to describe the existence of the bistable cell sizes. In the case of general remodelling, we show numerically that cells within a tissue may populate both the initial and elongated cell sizes, following a sufficiently large degree of stress.

The effects of apparatus-induced dispersion on nonuniform, density-dependent flow in a cylindrical soil column were investigated using a finite-element model. To validate the model, the results with an analytical solution and laboratory column test data were analysed. The model simulations confirmed that flow nonuniformities induced by the apparatus are dissipated within the column when the distance to the apparatus outlet exceeds $3R/2$, where R represents the radius of the cylindrical column. Furthermore, the simulations revealed that convergent flow in the vicinity of the outlet introduces additional hydrodynamic dispersion in the soil column apparatus. However, this effect is minimal in the region where the column height exceeds $3R/2$. Additionally, it is found that an increase in the solution density gradient during the solute breakthrough period led to a decrease in flow velocity, which stabilized the flow and ultimately reduced dispersive mixing. Overall, this study provides insights into the behaviour of apparatus-induced dispersion in nonuniform, density-dependent flow within a cylindrical soil column, shedding light on the dynamics and mitigation of flow nonuniformities and dispersive mixing phenomena.

Three typical elastic problems, including beam bending, truss extension and compression, and two-rings collision are simulated with smoothed particle hydrodynamics (SPH) using Lagrangian and Eulerian algorithms. A contact-force model for elastic collisions and equation of state for pressure arising in colliding elastic bodies are also analytically derived. Numerical validations, on using the corresponding theoretical models, are carried out for the beam bending, truss extension and compression simulations. Numerical instabilities caused by largely deformed particle configurations in finite/large elastic deformations are analysed. The numerical experiments show that the algorithms handle small deformations well, but only the Lagrangian algorithm can handle large elastic deformations. The numerical results obtained from the Lagrangian algorithm also show a good agreement with the theoretical values.

We provide an analytic solution of the Rössler equations based on the asymptotic limit $c\to \infty $ and we show in this limit that the solution takes the form of multiple pulses, similar to “burst” firing of neurons. We are able to derive an approximate Poincaré map for the solutions, which compares reasonably with a numerically derived map.

Mathematical modelling has been used to support the response to the COVID-19 pandemic in countries around the world including Australia and New Zealand. Both these countries have followed similar pandemic response strategies, using a combination of strict border measures and community interventions to minimize infection rates until high vaccine coverage was achieved. This required a different set of modelling tools to those used in countries that experienced much higher levels of prevalence throughout the pandemic.

In this article, we provide an overview of some of the mathematical modelling and data analytics work that has helped to inform the policy response to the pandemic in Australia and New Zealand. This is a reflection on our experiences working at the modelling–policy interface and the impact this has had on the pandemic response. We outline the various types of model outputs, from short-term forecasts to longer-term scenario models, that have been used in different contexts. We discuss issues relating to communication between mathematical modellers and stakeholders such as health officials and policymakers. We conclude with some future challenges and opportunities in this area.

Six patents were secured by E. H. Lanier from 1930 to 1933 for aeroplane designs that were intended to be exceptionally stable. A feature of five of these was a flow-induced “vacuum chamber” which was thought to provide superior stability and increased lift compared to typical wing designs. Initially, this chamber was in the fuselage, but later designs placed it in the wing by replacing a section of the upper skin of the wing with a series of angled slats. We report upon an investigation of the Lanier wing design using inviscid aerodynamic theory and viscous numerical simulations. This took place at the 2005 Australia–New Zealand Mathematics-in-Industry Study Group. The evidence from this investigation does not support the claims but, rather, suggests that any improvement in lift and/or stability seen in the few prototypes that were built was, most probably, due to thicker airfoils than were typical at the time.

Double-masking may be used to reduce the transmission of a virus. If additionally the masks are compressible, with different permeabilities and behaviour under compression, then it may be possible to design a mask that allows for easy breathing under normal breathing conditions, but is relatively impermeable under coughing or sneezing conditions. Such a mask could be both comfortable to wear and effective. We obtain analytical solutions for the steady-state flow-through behaviour of such a double mask under flow-out conditions. The results show that the reduction in permeability required to produce a relatively impermeable mask under high flux expulsion (sneezing) conditions could be achieved using either a single filter compressible mask or two filters with different poroelastic parameters. The parameters can be more easily adjusted using a double mask. For both single- and double-mask cases, there is an abrupt cut off, whereby through-flux levels reduce from a maximum value to zero as pressure drop levels increase beyond a critical value. Additionally, in the double-mask case, there exists a second steady-state solution for particular parameter ranges. This second solution is unlikely to occur under normal circumstances.

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, $\tau $, between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on $\tau $. A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive $\tau $.