In this paper, we are interested in the problem of determining the shape of sound-soft or impedance obstacles in the acoustic wave scattering with an anisotropic inhomogeneous medium. The main theme is to extend the factorization method to this setting. Precisely, we can reconstruct the obstacle by the eigenvalues and eigenfunctions of the far-field operator. We also provide some numerical simulations

We perform a linear stability analysis of three-layer radial porous media and Hele-Shaw flows with variable viscosity in the middle layer. A nonlinear change of variables results in an eigenvalue problem that has time-dependent coefficients and eigenvalue-dependent boundary conditions. We study this eigenvalue problem and find upper bounds on the spectrum. We also give a characterization of the eigenvalues

The modelling of linear and nonlinear reaction–subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time fractional derivative. It is known that the precise form depends on the interaction of dispersal and reaction, and leads to qualitative differences. We refine these results

AbstractThe ability of small-cell wireless networks to self-organize is crucial for improving capacity and performance in modern communication networks. This paper considers one of the most basic questions: what is the expected distance to a cell’s nearest neighbour in a spatially distributed network? We analyse a model problem in the asymptotic limit of large total received signal and compare the

In this work, we systematically derive the governing equations of supersonic conical flow by projecting the 3D Euler equations onto the unit sphere. These equations result from taking the assumption of conical invariance on the 3D flow field. Under this assumption, the compressible Euler equations reduce to a system defined on the surface of the unit sphere. This compressible flow problem has been

We derive the formulas for the resonance frequencies and their sensitivity when the nano-slit structures are used in the detection of thin layers. For a thin layer with a thickness of |$H$| deposited over the nanostructure, we show quantitatively that for both single and periodic slit structures with slit aperture size |$\delta $|⁠, the sensitivity of resonance frequency reduces as |$H$| increases

AbstractWe derive and analyse a new way to calculate the shear modulus of an inhomogeneous elastic material from time-dependent magnetic resonance elastography (MRE) measurements of its interior displacement. Even with such a rich data source, this is a challenging inverse problem because the coefficient of the shear modulus in the governing equations can be small (or potentially zero). Our approach

We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order

We study the long-time dynamics of 2D linear Fokker–Planck equations driven by a drift that can be decomposed in the sum of a large shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential

We investigate the reciprocity gap functional method, which has been developed in the inverse scattering theory, in the context of electrical impedance tomography. In particular, we aim to reconstruct an inclusion contained in a body, whose conductivity is different from the conductivity of the surrounding material. Numerical examples are given, showing the performance of our algorithm.

In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier

The application of Markov chains to modelling refugee crises is explored, focusing on local migration of individuals at the level of cities and days. As an explicit example, we apply the Markov chains migration model developed here to United Nations High Commissioner for Refugees data on the Burundi refugee crisis. We compare our method to a state-of-the-art ‘agent-based’ model of Burundi refugee movements

We present a simplified model of data flow on processors in a high-performance computing framework involving computations necessitating inter-processor communications. From this ordinary differential model, we take its asymptotic limit, resulting in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid governed by a conservation law. We derive a Hamilton–Jacobi

In this paper, we perform a careful analysis of the forced PP04 model for climate change, in particular the behaviour of the ice ages. This system models the transition from a glacial to an inter-glacial state through a sudden release of oceanic carbon dioxide into the atmosphere. This process can be cast in terms of a Filippov dynamical system, with a discontinuous change in its dynamics related to

AbstractWe examine the convective and absolute instability of a 2D axisymmetric viscoelastic liquid jet falling vertically in a medium of an inviscid gas under the influence of gravity. We use the upper-convected Maxwell model to describe the viscoelastic liquid jet and together with an asymptotic approach, based on the slenderness of the jet, we obtain steady-state solutions. By considering travelling

This paper concerns constructing a semi-analytic solution procedure for integrating the fully non-linear Serre equations (or 1D Green–Naghdi equations for constant water depth). The validity of the solution procedure is checked by investigating a moving solitary wave for which the analytical solution is known. The semi-analytic procedure constructed in this study is confirmed to be good at observing

AbstractThis paper is concerned with the inverse scattering problem of time-harmonic elastic waves by a mixed-type scatterer, which is given as the union of an impenetrable obstacle and a crack. We develop the modified factorization method to determine the shape of the mixed-type scatterer from the far field data. However, the factorization of the far field operator $F$ is related to the boundary integral

AbstractResults are presented for nonlinear equilibrium solutions of the Navier–Stokes equations in the boundary layer set up by a flat plate started impulsively from rest. The solutions take the form of a wave–roll–streak interaction, which takes place in a layer located at the edge of the boundary layer. This extends previous results for similar nonlinear equilibrium solutions in steady 2D boundary

Welcome to this special issue of the IMA Journal of Applied Mathematics dedicated to the winner and finalists of the IMA Lighthill–Thwaites Prize 2019. This biennial prize was established by the IMA in recognition of the achievement of its first two presidents—Professors Sir James Lighthill and Sir Bryan Thwaites—in laying the foundations of what has become a major scientific institute. It was their

Cancer invasion and metastatic spread to secondary sites in the body are facilitated by a complex interplay between cancer cells of different phenotypes and their microenvironment. A trade-off between the cancer cells’ ability to invade the tissue and to metastasize, and their ability to proliferate has been observed. This gives rise to the classification of cancer cells into those of mesenchymal and

We study the dynamics of detachment in 2D capillary adhesion by considering a plate that is initially attached to a flat, rigid substrate via the surface tension of a bridging liquid droplet. In particular, we focus on the effect of allowing the plate to tilt freely during its subsequent motion. A linear stability analysis shows that small perturbations from equilibrium decouple into two modes: one

In this paper, we consider a lumped-parameter model to predict renal pressures and flow rate during a minimally invasive surgery for kidney stone removal, ureterorenoscopy. A ureteroscope is an endoscope designed to work within the ureter and the kidney and consists of a long shaft containing a narrow, cylindrical pipe, called the working channel. Fluid flows through the working channel into the kidney

This paper considers the use of compliant boundary conditions to provide a homogenized model of a finite array of collinear plates, modelling a perforated screen or grating. While the perforated screen formally has a mix of Dirichlet and Neumann boundary conditions, the homogenized model has Robin boundary conditions. Perforated screens form a canonical model in scattering theory, with applications

Optical tomography is a typical non-invasive medical imaging technique, which aims to reconstruct geometric and physical properties of tissues by passing near infrared light through tissues for obtaining the intensity measurements. Other than optical properties of tissues, we are interested in finding locations of small inclusions inside the object from boundary measurements, based on the time-dependent

We establish a framework to investigate the global synchronization of coupled reaction–diffusion neural networks with time delays. The coupled networks under consideration can incorporate both the internal delays in each individual network and the transmission delays across different networks. The coupling scheme for the coupled networks is rather general, and its performance is not adversely affected

Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1D Gierer–Meinhardt (GM) model, we show that a non-constant precursor gradient in the decay rate

Delayed acceleration feedback (DAF) is known to have a positive impact on the stability properties of dynamical models in several applications. Motivated by this, we study the impact of DAF on the classical car-following model (CCFM). First, we show that DAF shrinks the locally stable region. We then show that the resulting model, similar to the CCFM, loses local stability via a Hopf bifurcation. However

A filter comprises porous material that traps contaminants when fluid passes through under an applied pressure difference. One side effect of this applied pressure, however, is that it compresses the filter. This changes the permeability, which may affect its performance. As the applied pressure increases, the flux of fluid processed by the filter will also increase but the permeability will decrease

A new approach is proposed for obtaining the dynamic elastic response of a multilayered elastic solid caused by axisymmetric, time-harmonic elastic singularities. The method for obtaining the elastodynamic Green’s functions of the point force, double forces and center of dilatation is presented. For this purpose, the boundary conditions in an infinite solid at the plane passing through the singularity

This paper is concerned with uniqueness in inverse electromagnetic scattering with phaseless far-field pattern at a fixed frequency. In our previous work (2018,SIAM J. Appl. Math. 78, 3024–3039), by adding a known reference ball into the acoustic scattering system, it was proved that the impenetrable obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the acoustic

Mixing is an omnipresent process in a wide range of industrial applications, which supports scientific efforts to devise techniques for optimizing mixing processes under time and energy constraints. In this endeavour, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the

This article describes a reduction of a non-autonomous Brusselator reaction–diffusion system of partial differential equations on a spherical cap with time-dependent curvature using the method of centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case

Fluid droplets can be induced to move over rigid or flexible surfaces under external or body forces. We describe the effect of variations in material properties of a flexible substrate as a mechanism for motion. In this paper, we consider a droplet placed on a substrate with either a stiffness or surface energy gradient and consider its potential for motion via coupling to elastic deformations of the

We construct a dynamical system based on the Källén–Crafoord–Ghil conceptual climate model which includes the ice–albedo and precipitation–temperature feedbacks. Further, we classify the stability of various critical points of the system and identify a parameter which change generates a Hopf bifurcation. This gives rise to a stable limit cycle around a physically interesting critical point. Moreover

The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is different for acoustic and electromagnetic waves for example

This paper investigates the different possible behaviours of a recent asymptotic model for oscillation-mark formation in the continuous casting of steel, with particular focus on how the results obtained vary when the heat transfer coefficient (⁠|$m$|⁠), the thermal resistance (⁠|$R_{mf}$|⁠) and the dependence of the viscosity of the flux powder as a function of temperature, |$\mu _{f}\left ( T\right

This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form |$h\approx{{e}^{i(kx-\omega t)}}$|⁠, where |$h$| is a response to a disturbance, |$k$| is a real wavenumber and |$\omega (k)$| is a wavelength-dependent complex frequency. In a previous paper, King et al. (2016, Stability of algebraically

AbstractThis paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form $h\approx{{e}^{i(kx-\omega t)}}$, where $h$ is a response to a disturbance, $k$ is a real wavenumber and $\omega (k)$ is a wavelength-dependent complex frequency. In a previous paper, King et al. (2016, Stability of algebraically

We consider a fractional phase-field crystal (FPFC) model in which the classical Swift–Hohenberg equation (SHE) is replaced by a fractional order Swift–Hohenberg equation (FSHE) that reduces to the classical case when the fractional order |$\beta =1$|⁠. It is found that choosing the value of |$\beta $| appropriately leads to FSHE giving a markedly superior fit to experimental measurements of the structure

We study the stochastic susceptible-infected-susceptible model of epidemic processes on finite directed and weighted networks with arbitrary structure. We present a new lower bound on the exponential rate at which the probabilities of nodes being infected decay over time. This bound is directly related to the leading eigenvalue of a matrix that depends on the non-backtracking and incidence matrices

The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. The numerical results suggest that there exist three

In this short communication, exact solutions are obtained for a range of non-Newtonian flows between stationary parallel plates. The pressure-driven flow of fluids with a variational viscosity that adheres to the Carreau governing relationship are considered. Solutions are obtained for both shear-thinning (viscosity decreasing with increasing shear-rate) and shear-thickening (viscosity increasing with

Professor John Blake spent a considerable part of his scientific career on studying bubble dynamics and acoustic cavitation. As Blake was a mathematician, we will be focusing on the theoretical and numerical studies (and much less on experimental results). Rather than repeating what is essentially already known, we will try to present the results from a different perspective as much as possible. This

Transport in biological systems often occurs in complex spatial environments involving random structures. Motivated by such applications, we investigate an idealized model for solute transport past an array of point sinks, randomly distributed along a line, which remove solute via first-order kinetics. Random sink locations give rise to long-range spatial correlations in the solute field and influence

This paper is concerned with nonlinear interactions of fundamental equatorial modes. In order to understand the mechanism of large-scale atmospheric motions in the near equator regime—especially the observed wavenumber-frequency spectrum—we develop novel models describing interactions among Kelvin, Yanai and Poincaré waves. Based on the methods of multiple scales and Galerkin projection, the primitive

We prove the existence of a family of immersed obstacles that have zero wave resistance in the context of the 2D Neumann–Kelvin problem. We first build a waveless potential by superposing a source and a sink in a uniform flow for an appropriate choice of parameters. The obstacle is obtained by a combination of streamlines of the waveless potential. Numerical simulations show that the construction is

A geometrically invariant concept of fast–slow vector fields perturbed by transport terms describing molecular diffusion is proposed in this paper. It is an extension of our concept of singularly perturbed vector fields for ODEs to reaction–diffusion systems with chemical reactions having wide range of characteristic time scales, while transport processes remain comparatively slow. Under this assumption

The present work is based on a mixture theory of poroelastic media which is consistent with the classical Darcy’s law and uplift force in soil mechanics. In addition, it also results in having an inertial effect on the motion of solid constituent as commonly expected, in contrast to Biot’s theory, where relative acceleration is introduced as an interactive force between solid and fluid constituents

A new method is introduced for the simulation of multiple impacts in granular media using the Kuwabara-Kono (KK) contact model, a nonsmooth (not Lipschitz continuous) extension of Hertz contact that accounts for viscoelastic damping. We use the technique of modified equations to construct time-discretizations of the nondissipative Hertz law matching numerical dissipation with KK dissipation at different

This work is a natural continuation of our recent study devoted to the scattering of a plane incident wave by a semi-infinite impedance sector. We develop an approach that enables us to compute different components in the far-field asymptotics. The method is based on the Sommerfeld integral representation of the scattered wave field, on the careful study of singularities of the integrand and on the

The modelling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coefficient $\alpha $ and compact terms. The coefficient $\alpha $ changes sign. The region where $\{\alpha>0\}$ is propagative, and the region where $\{\alpha <0\}$ is non propagative and elliptic. The two models are coupled through the line $\varSigma =\{\alpha =0\}$. Generically

We derive a second-order correction to an existing leading-order model for surface waves in linear elasticity. The same hyperbolic–elliptic equation form is obtained with a correction term added to the surface boundary condition. The validity of the correction term is shown by re-examining problems which the leading-order model has been applied to previously, namely a harmonic forcing, a moving point

We develop a fluid-structure interaction (FSI) model of the mitral valve (MV) that uses an anatomically and physiologically realistic description of the MV leaflets and chordae tendineae. Three different chordae models-complex, 'pseudo-fibre' and simplified chordae-are compared to determine how different chordae representations affect the dynamics of the MV. The leaflets and chordae are modelled as

New mathematics has often been inspired by new insights into the natural world. Here we describe some ongoing and possible future interactions among the massive data sets being collected in neuroscience, methods for their analysis and mathematical models of the underlying, still largely uncharted neural substrates that generate these data. We start by recalling events that occurred in turbulence modelling