ALBERT

Description: The class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo, has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl. Math. , 48 , 279–319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58 , 309–317; Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond. A , 463 , 357–367). The new classes include fully implicit constitutive relations for the stress and the deformation gradient, and the interesting sub-class wherein the Cauchy–Green tensor or the linearized strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the Green–St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A , 463 , 357–367), other possible implicit relationships between other stress and kinematic measures have not been investigated. This paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic bodies, when they are expressed through implicit relations for different stress and stretch/strain measures.

Description: This work investigates the existence and non-existence of travelling wave solutions for Kolmogorov-type delayed lattice reaction–diffusion systems. Employing the cross iterative technique coupled with the explicit construction of upper and lower solutions in the theory of quasimonotone dynamical systems, we can find two threshold speeds $c^* $ and $c_* $ with $c^* \geq c_* 〉 0.$ If the wave speed is greater than $c^* ,$ then we establish the existence of travelling wave solutions connecting two different equilibria. On the other hand, if the wave speed is smaller than $c_* ,$ we further prove the non-existence result of travelling wave solutions. Finally, several ecological examples including one-species, two-species and three-species models with various functional responses and time delays are presented to illustrate the analytical results.

Description: In this paper, sufficient criteria for global asymptotic stability of a general stochastic non-autonomous Lotka–Volterra system with infinite delays are established. Then these criteria are applied to a competitive system, a predator–prey system and a cooperative system to demonstrate their applicability and effectiveness. Some recent results are extended and improved.

Description: In the paper, the problem of existence of almost periodic solutions to non-linear impulsive systems of Caputo fractional order is addressed. Sufficient conditions for stability of almost periodic solutions are also established. The investigations are carried out by using of a fractional comparison principle and the Lyapunov's function method. We apply our results to an impulsive cellular neural network model of fractional order.

Description: In this paper we propose a limited memory method, called the Broyden rank reduction (BRR) method, to efficiently compute fixed points of high-dimensional non-linear maps. The method is based on the singular value decomposition and has good convergence properties. We discuss some basic properties of the BRR method and prove convergence for linear systems. The method is developed for a concrete problem in engineering. In chemical engineering, periodically forced processes in packed bed reactors are generally simulated using pseudo-homogeneous one-dimensional models on a coarse grid discretization. In these simulations, one typically uses Broyden's method because of its simplicity and the fact that appropriate initial conditions can be chosen in a natural way. A disadvantage of Broyden's method is the memory usage to store the Broyden matrix. The BRR method resolves this issue and allows us to consider more complicated and accurate models. We show that the BRR method developed to simulate periodically forced processes is in certain situations a good option to compute fixed points of high-dimensional non-linear maps and illustrate this using examples from integral equations and parabolic partial differential equations. To benchmark our method, we conclude the paper with a comparison of the BRR method with the generally preferred Newton–GMRES( m ) method.

Description: In this paper, the relationship between representation formulas for unique regularized solutions of inverse source, as well as backward problems with final overdetermination for evolution equations and singular value decomposition (SVD) of corresponding input–output operators, is analysed. For each considered inverse problem this representation formula is derived via the solution of appropriate adjoint problems and the parameter of regularization. The Lagrange multiplier method is then used to show that each regularized inverse problem is equivalent to a coupled system of two (direct and adjoint) parabolic (or hyperbolic) problems, and the Lagrange multiplier is the weak solution of the corresponding adjoint problem. In the constant coefficient linear parabolic and hyperbolic equations cases, it is proved that the coupled problem is equivalent to the normal equation corresponding to each inverse/backward problem. As a result, an equivalence of the representation formula for unique regularized solution and the SVD of the input–output operator is obtained. This approach, in particular, allows one to construct regularizing filters for a considered class of inverse problems. Some numerical examples are presented to illustrate the usefulness and effectiveness of the proposed approach.

Description: This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier–Stokes equations, complexified in time. Specifically, for periodic boundary conditions on [0, L ] 2 , and a force g D( A (α–1)/2 ), we show that there is a fixed strip about the real-time axis on which a uniform bound $|A^{\alpha}u| 〈 m_{\alpha}\nu\kappa_{0}^{\alpha}$ holds for each $\alpha \in {\mathbb {N}}$ . Here is viscosity, $\kappa_{0}=2\pi/L$ , and m α is explicitly given in terms of g and α. We show that if any element in A is in $\mathcal{D}(A^{\alpha})$ , then all of A is in $\mathcal{D}(A^{\alpha})$ , and likewise with $\mathcal{D}(A^{\alpha})$ replaced by $C^{\infty}(\Omega)$ . We demonstrate the universality of this ‘all for one, one for all’ law is also shown to hold for the union of a hierarchical set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a non-zero force by showing that if this is so, the force must be in a particular function class.

Description: An exact solution is obtained for the steady plane magnetohydrodynamics oblique stagnation-point flow of a homogeneous, incompressible, electrically conducting micropolar fluid over a rigid uncharged dielectric at rest. The space is permeated by a non-uniform external magnetic field and by a suitable uniform external electric field. The total magnetic field in the fluid is parallel to the velocity at infinity. The problem is reduced to a non-linear ordinary differential boundary value problem by using similarity transformations. The effects of the magnetic field on the velocity and on the microrotation profiles are presented graphically and discussed.

Description: In this paper, we analyse the connection between approximate non-linear self-adjointness of perturbed partial differential equations (PDEs) and non-linear self-adjointness of the corresponding unperturbed PDEs, and consequently provide a simple approach to discriminate approximate non-linear self-adjointness of perturbed PDEs. Moreover, a succinct approximate conservation law formula by virtue of the known conservation laws of the corresponding unperturbed PDEs is given in an explicit form. As an application, we classify a class of perturbed wave equations to be approximately non-linearly self-adjoint and construct its general approximate conservation law formula. The specific examples demonstrate that approximate non-linear self-adjointness can be used to construct new approximate conservation laws of perturbed PDEs.

Description: Practical physical applications of mathematical nature are frequently met in engineering technology and involve the low-Reynolds number flow of micropolar conducting fluids under the effect of magnetic fields. Here, we consider the three-dimensional (3D) creeping motion (Stokes flow), in steady state, of a non-conductive colloidal suspension of ferromagnetic material embedded within an electrically conductive, viscous and incompressible, carrier liquid. In such cases, the ferromagnetic particles behave as rigid magnetic dipoles and react in the presence of an externally applied magnetic field, which is of general form and arbitrarily orientated in the 3D space. Therein, an induced magnetic field of minor significance is created, while the effective viscosity of the fluid is increasing and an additional magnetic pressure is appeared. The consistency of the governing set of partial differential equations with the principles of both ferrohydrodynamics and magnetohydrodynamics is established by taking into account magnetization and electrical conductivity of the fluid, respectively. Our main intension is to use the potential representation theory to improve previous models and construct a new complete and unique integro-differential representation of the magnetic Stokes flow of conducting liquids, valid for any non-axisymmetric geometry, which provides the velocity and total pressure fields in a closed form and in terms of easy-to-find potentials, via a semi-analytical formalism. To demonstrate the usefulness of our analytical approach, we assume a degenerate case of the aforementioned method to simulate the creeping flow of a micropolar fluid with conductive properties inside a circular duct.

Description: Written to coincide with the anniversary of the launch of our journal, this survey reviews some of the highlights from the first fifty years of the IMA Journal of Applied Mathematics and its predecessor the Journal of the IMA. Rather than an exhaustive survey, we have chosen some of our personal favourite articles from the last fifty years. For each, we have given only the briefest of summaries—readers can download each of these papers for themselves, as they have each been put on open access to coincide with this Issue. We also attempt to put the work in context. More importantly, these papers were chosen as they have had, and continue to have, a considerable influence. For each, we try to spell out the impact of the work, and by implication, the impact of our journal.

Description: Social media data tend to cluster around events and themes. Local newsworthy events, sports team victories or defeats, abnormal weather patterns and globally trending topics all influence the content of online discussion. The automated discovery of these underlying themes from corpora of text is of interest to numerous academic fields as well as to law enforcement organizations and commercial users. One useful class of tools to deal with such problems are topic models, which attempt to recover latent groups of word associations from the text. However, it is clear that these topics may also exhibit patterns in both time and space. The recovery of such patterns complements the analysis of the text itself and in many cases provides additional context. In this work we describe two methods for mining interesting spatio-temporal dynamics and relations among topics, one that compares the topic distributions as histograms in space and time and another that models topics over time as temporal or spatio-temporal Hawkes process with exponential trigger functions. Both methods may be used to discover topics with abnormal distributions in space and time. The second method also allows for self-exciting topics and can recover intertopic relationships (excitation or inhibition) in both time and space. We apply these methods to a geo-tagged Twitter dataset and provide analysis and discussion of the results.

Description: In 1952, Lighthill introduced the simplest possible model of a swimming micro-organism of finite size, intended as a model of a single-celled protozoan covered in beating cilia. The model consisted of a sphere, on the surface of which material points undergo small-amplitude oscillations. In 1971, Lighthill’s student, John Blake, completed the calculations and applied the theory to various species of ciliate. Subsequently we, and many others, have used the even simpler (though less realistic) model of a steady squirmer, a sphere whose surface moves tangentially with time-independent velocity. In this article we survey: -low-Reynolds-number locomotion, nutrient uptake and optimisation of individual squirmers; -hydrodynamic interactions between pairs of steady squirmers and their influence on clustering, self-diffusion and rheology in suspensions of squirmers, including the effect of being bottom-heavy; -measurements and modelling of metachronal waves in Volvox , the only truly sphericalmulti-celled organism, culminating in predictions of the mean swimming speed and angular velocity of free-swimming Volvox . The predictions are compared with experiment.

Description: 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 when substantial space-time velocity field measurements and numerical simulations allowed a new perspective on the governing equations of fluid mechanics. While no analogous global mathematical model of neural processes exists, we argue that big data may enable validation or at least rejection of models at cellular to brain area scales and may illuminate connections among models. We give examples of such models and survey some relatively new experimental technologies, including optogenetics and functional imaging, that can report neural activity in live animals performing complex tasks. The search for analytical techniques for these data is already yielding new mathematics, and we believe their multi-scale nature may help relate well-established models, such as the Hodgkin–Huxley equations for single neurons, to more abstract models of neural circuits, brain areas and larger networks within the brain. In brief, we envisage a closer liaison, if not a marriage, between neuroscience and mathematics.

Description: The properties of spatially localized structures in systems with a conservation law are reviewed and related to the properties of fronts in such systems. The theory is illustrated using the conserved Swift–Hohenberg equation, and the insights gained used to shed light on the process of crystallization from a melt and on two problems arising in hydrodynamics: convection in a rotating layer and convection in an imposed magnetic field. Spatially localized structures in systems with time-periodic forcing are also considered. Several open problems are highlighted.

Description: Thixotropic yield stress fluids have a yield stress and viscosity which change slowly over time as the fluid undergoes structural changes. This article reviews recent work which shows how such behaviour can arise as a limit of viscoelastic flow when a relaxation time becomes large. The large relaxation time introduces a small parameter which can be used as a basis for singular perturbation studies. A simple prototype for a viscoelastic model will be used to illustrate the analysis. We review the behaviour in startup of shear flow, oscillatory shear flow and elongational flow and identify potential directions for future research.

Description: The 120-year-old so-called Painlevé paradox involves the loss of determinism in models of planar rigid bodies in point contact with a rigid surface, subject to Coulomb-like dry friction. The phenomenon occurs due to coupling between normal and rotational degrees-of-freedom such that the effective normal force becomes attractive rather than repulsive. Despite a rich literature, the forward evolution problem remains unsolved other than in certain restricted cases in 2D with a single contact point. Various practical consequences of the theory are revisited, including models for robotic manipulators and the strange behaviour of chalk when pushed rather than dragged across a blackboard. Reviewing recent theory, a general formulation is proposed, including Poisson or energetic impact laws. The general problem in 2D with a single point of contact is discussed and cases of inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularization are discussed from a dynamical systems point of view. By passing to the infinite stiffness limit and allowing impact without collision, inconsistent and indeterminate cases are shown to be resolvable for all open sets of conditions. However, two unavoidable ambiguities that can be reached in finite time are discussed in detail, so called dynamic jam and reverse chatter. A partial review is given of 2D cases with two points of contact showing how a greater complexity of inconsistency and indeterminacy can arise. Extension to fully 3D analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the Painlevé paradox is highlighted and open problems are discussed.

Description: This article surveys the important role, in a variety of applied mathematical contexts, played by the so-called Schottky–Klein (S–K) prime function. While it is a classical special function, introduced by 19th century investigators, its theoretical significance for applications has only been realized in the last decade or so, especially with respect to solving problems defined in multiply connected, or ‘holey’, domains. It is shown here that, in terms of it, many well-known results pertaining only to the simply connected case (no holes) can be generalized, in a natural way, to the multiply connected case, thereby contextualizing those well-known results within a more general framework of much broader applicability. Given the wide-ranging usefulness of the S–K prime function it is important to be able to compute it efficiently. Here we introduce both a new theoretical formulation for its computation, as well as two distinct numerical methods to implement the construction. The combination of these theoretical and computational developments renders the S–K prime function a powerful new tool in applied mathematics.

Description: We discuss phase resetting behaviour when non-small, near instantaneous, perturbations are applied to a dynamical system containing a strong attractor that is equivalent to a winding map on an $n$ -dimensional torus. Almost all of the literature to date has focused on the application of phase transition curves derivable for systems with attracting limit cycles, when $n = 1$ , and the consequent possibilities for entrainment of the system subject to such periodic largish perturbations. In higher dimensions, the familiar tongues in the amplitude versus period plane, describing solutions entrained with periodic fast perturbations, have subtle structure (since only locally two-dimensional subsets of the $n$ -dimensional torus may admit such solutions) whilst the topological class of the $n$ -dimensional phase transition mapping is represented by a matrix of winding numbers. In turn this governs the existence of asymptotic solutions in the limits of small perturbations. This paper is an attempt to survey the entrainment behaviour for the full set of alternative (near rigid) phase transition mappings definable on the 2-torus. We also discuss higher dimensional effects and we suggest how this might be important for non-linear neural circuits including delays that routinely exhibit such attractors and may drive one another in cascades of periodic behaviour.

Description: This article considers the theoretical modelling of a novel electrostatic transducer in which the backplate consists of many spherical resonators. Three analytical models are considered, each of which produce impedance profiles of the device, in addition to transmission voltage responses and reception force responses, all of which closely agree. Design parameters are then varied to investigate their influence on the resonant frequencies and other model outputs.

Description: A class of backward doubly stochastic differential equations (BDSDEs), which has a more general form of the forward Ito integrals (compared with the ones which have been studied until now), is observed. The existence of the solution for this class of BDSDEs with continuous coefficients is given, a comparison theorem for this class of BDSDE is proved and the existence of minimal and maximal solution is derived. At the end, Kneser-type theorem for a larger class of BDSDEs is obtained. It is shown that for the elements of this class of equations, there is either unique or there are uncountable solutions.

Description: In this paper, three dynamic linear models of the main span of suspension bridges are studied. The models describe vertical and torsional oscillations of the deck under the action of lateral wind. The problems correspond to the situations when both the midspan cable bands are loosened, both the midspan bands are fixed, and one midspan band is fixed and the other is loosened. Three corresponding eigenvalue and eigenvector problems are formulated and analysed. The continuous dependence of eigenvalues and eigenvectors on data are proved. The analysis of the three eigenvalue and eigenvector problems against flutter is carried out, which reveals possible reasons of the collapse of the original Tacoma suspension bridge.

Description: We present a complete description on the spectrum and eigenfunctions of the following two-point boundary value problem (0.1) \begin{equation} \begin{cases} (p(x)f')'-(q(x)- \lambda r(x))f=0,\quad 0 〈 x 〈 L \\ f'(0)=(\alpha_{1} \lambda +\alpha_{2})f(0)\\ f'(L)=(\beta_{1}\lambda -\beta_{2})f(L) \end{cases} \end{equation} where $\lambda $ and $\alpha _{i},\ \beta _{i}$ are spectral and physical parameters, respectively, and $p(x),\ q(x),\ r(x)$ are positive and regular functions in $[0,L]$ . Our survey is focused mainly on the case $\alpha _{1} 〉 0$ and $\beta _{1} 〈 0$ , where neither self-adjoint operator theorems on Hilbert spaces nor Sturm's comparison results can be used directly. We describe the spectrum and the oscillatory results of the eigenfunctions from a geometrical approach, using a function related to the Prüfer angle. The proofs of the asymptotic results of the eigenvalues and separation theorem of the eigenfunctions are developed through classical second-order differential equation tools. Finally, the results on the spectrum of ( 0.1 ) are used to study the linear stability of three-layer Hele–Shaw flows, which is a simple model for the fingering phenomenon on the flooding oil recovery process.

Description: As known from the theory of centre manifolds, it is important to approximate invariant manifolds when discussing high order terms of the reduced systems in the investigation of bifurcations. In this paper, we give a method to approximate invariant manifolds which are obtained under pseudo-hyperbolicity. This method enables us to avoid the trouble of increasing dimension of the so-called approximate inertial manifold for smaller thickness of attractive neighbourhood. Unlike centre manifolds, exponential expansion and contraction may exist on such manifolds. Consequently, the technique of the contraction principle cannot be used directly. We not only overcome the difficulty so as to approximate such $C^{1}$ manifolds at a possibly higher order but also estimate, for non-local bifurcations, the lower bound of their locality radius such that the approximation is valid. Our approximation is demonstrated by examples of a wave equation and an ordinary differential equation.

Description: The Riemann problem for the pressureless Euler system with the Coulomb-like friction term is solved constructively, whose solutions involve delta shock wave and vacuum state. The position, propagation speed and strength of the delta shock wave are calculated explicitly from the generalized Rankine–Hugoniot conditions of the delta shock wave which are derived in detail, such that the influence of the Coulomb-like friction term on the Riemann solutions can be seen clearly. Furthermore, the generalized Riemann problem with measure-valued initial data is also considered and four kinds of different structures are displayed for the Riemann solutions. It is shown that the Riemann solutions converge to the corresponding ones of the pressureless Euler system when the friction term vanishes. In addition, the balance relations associated with the mass and momentum transportation are established for the delta shock wave solution.

Description: This paper is devoted to the attraction–repulsion chemotaxis system with a logistic source: \[ \begin{cases} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\mu\nabla\cdot(u\nabla w)+\mathcal{R}(u),\quad x\in\Omega,\ t 〉 0,\\ \varrho v_t=\Delta v-\alpha_1 v+\beta_1 u,\quad x\in\Omega,\ t 〉 0,\\ \varrho w_t=\Delta w-\alpha_2 w+\beta_2 u,\quad x\in\Omega,\ t 〉 0, \end{cases} \] where $\Omega \subset \mathbb {R}^N(N\geqslant 1)$ is a bounded domain with smooth boundary and $\mathcal {R}(s)\leqslant a-bs^\tau $ . For the case $\varrho =0$ , we show that when the repulsion prevails over the attraction in the sense that $\mu \beta _2-\chi \beta _1 〉 0$ , there exist global bounded classical solutions for any logistic damping $\tau \geqslant 1$ . When the attraction dominates the repulsion in the sense that $\mu \beta _2-\chi \beta _1 〈 0$ , the classical solutions are still global and bounded provided that the logistic damping is strong. For the case $\varrho 〉 0$ , we will investigate the similar problem for $N=1$ and $N=2$ . We will also study the regularity of stationary solutions.

Description: This paper presents a new approach for proving that the presence of a bounded defect in a uniform open waveguide cannot produce trapped modes, contrary to the case of a closed waveguide. The originality of the proof lies in the fact that it relies on a modal decomposition. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components of the field. The case of the 3D scalar wave equation is considered.

Description: This paper is concerned with travelling wave solutions of periodic monostable differential equations on a two-dimensional lattice. The existence, stability and uniqueness of supercritical travelling wave solutions are obtained by comparison principle.

Description: This paper considers a stochastic delay Lotka–Volterra model whose coefficients are dependent on expectations. By applying the M -matrix technique, the existence and uniqueness of the global positive solution and its exponential extinction are studied. An example is provided to illustrate the effectiveness of our results.

Description: We consider the selection and stability of periodic wave trains that form behind invasion fronts in predator–prey reaction–diffusion models where the prey compete in a non-local manner. We modify the selection criterion developed in earlier work, also applying it to the non-local system, and study how the properties of selected wave trains vary with the standard deviation of the non-local prey competition kernel. We find that the wavelength of selected wave trains decreases with the standard deviation of the non-local kernel and also that unstable wave trains are selected for a larger parameter range, suggesting that spatiotemporal chaos may be more common in highly non-local systems.

Description: We investigate the different wave-modes generated by a wave-maker in compressible flow. In addition to the propagating and evanescent waves found in the incompressible case, new radially propagating acoustic-gravity modes appear. We discuss the asymptotic behaviour of these waves, and give an example for a simple line wave-maker configuration.

Description: This paper concerns the derivation and preliminary analysis of a model for judder in devices such as brakes and clutches. The model is based on the competition between wear and thermal expansion which, in the multiple-pin configuration proposed by Barber (1969, Thermoelastic instabilities in the sliding of conforming solids. Proc. Roy. Soc. (London) , A312 , 381–394), leads to a novel, non-local dynamic contact problem.

Description: An averaged model of a coupled-inductor boost converter using the piecewise complementarity model of the converter under sliding motions is obtained. The model takes into account the idealized voltage–current characteristic of passive switches (diodes) present in the converter. Because of its lower complexity, the averaged model is more suitable for control design purposes when compared with the linear complementarity systems (LCS) model of the converter. The dynamic performance of the LCS model and the averaged models of the converter are validated through computer simulations using Matlab.

Description: A spherically symmetric isolated point defect in a 3D uniaxial nematic liquid crystal sample is often called a radial hedgehog . We use topological methods to describe local configurations of uniaxial and biaxial states into which a hedgehog naturally deforms under small perturbations: these include the biaxial torus and split core configurations studied in the literature using analytical and numerical methods. The topological results here take no account of the governing physical laws but provide a library of options from which the physics must make a choice.

Description: In this paper, the concept of discontinuity geometry of rigid impact oscillators is extended to soft impact oscillators and the mechanisms of grazing bifurcations of an impact oscillator with one-sided elastic constraint are studied by the geometric and dynamical systems methods. The existence conditions of periodic solutions are given by the discontinuity geometry objects and then are used to derive the discontinuity curves. Several bifurcation scenarios near grazing bifurcation are studied to examine the evolution of the system dynamics from a geometrical point of view. The geometrical conditions are obtained for the existence of periodic orbits with one impact, grazing and saddle-node bifurcations. Some geometrical insight is gained into a question whether there is a discontinuous jump or a continuous transition from a non-impacting period-1 to an impacting period-1 attractor at a grazing bifurcation.

Description: The border collision normal form is a family of continuous 2D piecewise smooth maps describing dynamics close to a critical parameter at which a fixed point intersects the switching surface. It is well known that if the fixed point is stable on one side of the bifurcation point, then after the bifurcation the system may have stable periodic orbits and/or chaotic attractors with a quasi-1D structure (robust chaos). We show that it is also possible to have a robust transition from a stable fixed point to an attractor with topological dimension two, i.e. the highest dimension possible in the phase space.

Description: Combining wave field decomposition, invariant imbedding and general phase theory can reformulate the ill-posed elliptic wave propagation problems into the exact, well-posed, one-way problems. As a main focus in the direct and inverse analysis of this approach, the singularity structure (turning points, focal points, cusps, kinks and poles) of the scattering and Dirichlet-to-Neumann operator symbols are presented and analysed in the transversely homogeneous limit. A specific 2D example is provided with numerical results.

Description: We mainly study the Cauchy problem of modified Camassa–Holm and Degasperis–Procesi equations. First, we establish the local well-posedness for the equation in Besov space. Secondly, we derive the conservation laws and a precise blow-up scenario. Moreover, we prove the existence of blow-up solutions and obtain its blow-up rate, provided the initial data satisfy certain conditions. Finally, we present the persistence properties of the equation.

Description: Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Description: We rigorously derive the leading-order terms in asymptotic expansions for the scattered electric and magnetic fields in the presence of a small object at distances that are large compared with its size. Our expansions hold for fixed wavenumber when the scatterer is a (lossy) homogeneous dielectric object with constant material parameters or a perfect conductor. We also derive the corresponding leading-order terms in expansions for the fields for a low-frequency problem when the scatterer is a non-lossy homogeneous dielectric object with constant material parameters or a perfect conductor. In each case, we express our results in terms of polarization tensors.

Description: The permafrost methane emission problem is the focus of attention on different climate models. Here, we present a mathematical model for permafrost lake methane emission and its influence on the climate system. We model this process using the theory of non-linear phase transitions. Further, we find that a climate catastrophe possibility depends on a value of feedback connecting the methane concentration in the atmosphere and temperature, and on the tundra permafrost methane pool. We note that the permafrost lake model that we developed for the methane emission positive feedback loop problem is a conceptual climate model.

Description: This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact dispersion property is adopted as the governing equation for unidirectional wave packet evolution. Following the idea from Zakharov's seminal paper (Zakharov, V. E. (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. , 9 , 190–194), the equation is transformed from the spatial–temporal domain to the wavenumber–temporal domain. The solution of the transformed equation is sought using the perturbation theory, for which the ansatz is expressed in the form of a regular expansion in the increasing order of a small parameter. After implementing the naïve perturbation method, due to non-linear mode generation and particular combinations of wavenumbers, the third-order solution contains spurious secular growth terms that appear as a consequence of resonant interaction and non-linear mode generation. These spurious secular growth terms can be prevented by implementing the method of strained parameters for which non-linear dispersion relation terms are produced for particular combination of wavenumbers.

Description: The evolution equations for a small spherical low-Reynolds-number swimmer actuated by means of an imposed treadmilling action on its surface are derived by means of an asymptotic analysis. The analysis rests on the assumption that the swimmer radius is small compared with its distance from the wall. It generalizes, to the more realistic 3D case, recent work on 2D treadmilling swimmers near a no-slip wall. The study is motivated by a recent work on the dynamics of Volvox algae near solid surfaces.

Description: We propose an approach to extend the isogeometric analysis (IGA) method to solve material interface problems. The development is carried out through incorporating the advantages of the extended finite element method into the standard IGA approach for solving problems with discontinuities. By applying both the XIGA and IGA methods to solve Poisson's equation problem containing weak discontinuities, we demonstrate that the XIGA achieves the optimal convergence rate, whereas the IGA only converges suboptimally. The proposed method is then successfully applied to solve bimaterial and curved material interface problems.

Description: In this work, we consider epitaxial growth of thin crystalline films. Thereby, we propose a new diffuse-interface approximation of a semi-continuous model resolving atomic distances in the growth direction but being coarse-grained in the lateral directions. Mathematically, this leads to a free boundary problem proposed by Burton, Cabrera and Frank for steps separating terraces of different atomic heights. The evolution of the steps is coupled to a diffusion equation for the adatom (adsorbed atom) concentration fulfilling Robin-type boundary conditions at the steps. Our approach allows to incorporate an Ehrlich–Schwoebel barrier as well as diffusion along step edges into a diffuse-interface model. This model results in a Cahn–Hilliard equation with a degenerate mobility coupled to diffusion equations on the terraces with a diffuse-interface description of the boundary conditions at the steps. We provide a justification by matched asymptotic expansions formally showing the convergence of the diffuse-interface model towards the sharp-interface model as the interface width shrinks to zero. The results of the asymptotic analysis are numerically reproduced by a finite element discretization.

Description: In this paper, we study the quasi-neutral limit of a 3D viscous capillary model for plasmas and semiconductors. Guided by the convergence-stability principle previously formulated by the second author for singular limit problems, we show that when the Debye length is sufficiently small, the initial value problem of the model has a unique smooth solution in the time interval where the corresponding incompressible Navier–Stokes equation has a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Debye length goes to zero. Moreover, we obtain the convergence of smooth solutions for the model towards those for the incompressible Navier–Stokes equation in the quasi-neutral limit, with a sharp convergence rate.

Description: This paper investigates the stabilization problem of neural networks with unbounded continuously distributed delays via impulsive control. By establishing an impulsive infinite delay differential inequality from the impulsive control point of view, some sufficient conditions ensuring the existence and global exponential stabilization of the unique equilibrium point are derived. Our result shows that impulses can be used to globally and exponentially stabilize the neural networks with unbounded continuously distributed delays if they are not originally stable or exponentially stable. Finally, two examples with their simulations are given to show the effectiveness of the proposed method.

Description: We formulate a mathematical model to study the in vivo keratin organization in terms of repartition of the keratin material between three different structural states: soluble proteins, filament precursors and filamentous state. The model describes a cycle of assembly and disassembly of keratin material. A generic nucleation function and structural stability of filament precursors are assumed. It is established that the three structural states always coexist and that the repartition of keratin material never exhibits any cyclic behaviour. Under some conditions, it is shown that the choice of the nucleation function does not affect the qualitative behaviour of the system. However, it might change the stable repartition of the keratin material in the cell.

Description: We present an extension of the numerical method of Loeper and Rapetti (2005, Numerical solution of the Monge–Ampère equation by a Newton's algorithm. C.R. Acad. Sci. Paris , I , 319–324) for the Monge–Ampère equation to non-uniform target densities and adopt it to solve the optimal transport problem with quadratic cost. The method employs a damped Newton algorithm to solve the Monge–Ampère equation. We show that the algorithm converges for sufficiently large damping coefficients, for the case where the source and target densities are sufficiently smooth, periodic and bounded away from zero. At each Newton iteration, we solve a non-constant coefficient linear partial differential equation. To improve the efficiency of the procedure, we use an analytically preconditioned fast Fourier transform method coupled with GMRES (Strain, J. (1994) Fast spectrally-accurate solution of variable-coefficients elliptic problems. Proc. Amer. Math. Sci. , 122 , 843–850) to solve this equation, as opposed to a more straightforward approach based on a second-order finite-difference discretization combined with biconjugate gradient used in the original LOEPER and RAPETTI paper. Finally, we present some numerical experiments in image processing to demonstrate the efficiency of the method.

Description: Herein we consider Rayleigh waves propagating in a given direction along the traction-free boundary of an anisotropic and prestressed elastic half-space, the effective incremental elasticity tensor and mass density of which are smooth functions of depth from the traction-free surface. We derive a high-frequency asymptotic formula which, for a large wave number, expresses the phase velocity of the Rayleigh waves in question in terms of the values of the effective incremental elasticity tensor, the mass density, and their first and higher-order normal derivatives at the traction-free boundary. We develop a procedure which, through an asymptotic expansion of the surface impedance matrix, can deliver an expression for each term of the asymptotic expansion of the Rayleigh-wave velocity. Our asymptotic formula thus gives a characterization of the frequency dependence of Rayleigh waves as caused by the vertical inhomogeneity of the medium, thereby providing a mathematical foundation for work on non-destructive evaluation of the depth dependence of near-surface stress and/or material parameters through their effect on the dispersion of Rayleigh waves.

Description: The compressible flow between a slider and a moving rough surface is examined asymptotically and numerically in the limit of very small gap height. The roughness is assumed to consist of a well-defined periodic pattern of grooves whose amplitude and wavelength are of the order of the gap height. A two-scale homogenization analysis is employed to determine a non-linear elliptic partial differential equation governing the leading order pressure in the gap on the scale of the slider. The equation involves coefficient functions that are determined numerically by averaging Stokes flows on the scale of the patterned roughness. Numerical results are obtained for the cases of transverse and longitudinal roughness for both no-slip and slip-flow boundary conditions, and for the limiting case of large bearing number. Comments and a brief analysis are given on the reduction of the governing equation for pressure in the conventional limit of long wavelength of the surface roughness.

Description: Inspired by the paper (Gong, W., Li, R., Yan, N. N. & Zhao, W. B. (2008) An improved error analysis for finite element approximation of bioluminescence tomography. J. Comput. Math. , 26 , 1–13), we consider an optimal finite element error estimate for an inverse problem in multispectral bioluminescence tomography. Different from achromatic or monochromatic measurements, hyperspectral or multispectral data can reduce the ill-posedness of the inverse problem and yield improved depth reconstruction. Compared with Gong et al. (2008, An improved error analysis for finite element approximation of bioluminescence tomography. J. Comput. Math. , 26 , 1–13), which just improved the error order of piecewise constant light source function, error order in this paper is optimal and all error estimates here are valid for a general smooth domain rather than a polyhedral/polygonal one. Moreover, under a boundedness assumption for the admissible source set, the constants in our error estimates do not depend on the regularization parameter, and therefore are bounded.

Description: In this paper, we prove that the parameter-to-solution map associated to the inverse problem of determining the diffusion coefficient in a parabolic partial differential equation satisfies the local tangential cone condition . In particular, we show stability and convergence of the regularized solutions by means of Landweber iteration. Our result has an immediate application to the local volatility calibration problem of the Black–Scholes model for European call options. We present a numerical validation based on simulated data to this calibration problem and discuss the results. We also prove convergence and stability of Kaczmarz-type strategies of the local volatility calibration problem by transforming the problem into a system of non-linear ill-posed equations.

Description: This paper presents a new class of eighth-order canonical polynomials in R n . Based on the sequential canonical dual transformation, all extrema are obtained. The method can be used to solve the associated seventh-order non-linear algebraic equations. Optimality conditions are provided to find both global minimizers and local extrema. Applications are illustrated by examples.

Description: We consider the transmission eigenvalue problem for an impenetrable obstacle with Dirichlet boundary condition surrounded by a thin layer of non-absorbing inhomogeneous material. We derive a rigorous asymptotic expansion for the first transmission eigenvalue with respect to the thickness of the thin layer. Our convergence analysis is based on a Max–Min principle and an iterative approach which involves estimates on the corresponding eigenfunctions. We provide explicit expressions for the terms in the asymptotic expansion up to order 3.

Description: We compute the natural frequencies for the oscillation of the free boundary of a perfect incompressible fluid in the presence of capillary forces and partially in contact with a solid container. First, we study the case when the fluid occupies the domain { ( x , y ): y ≤ h ( x , t )} with h ( x , t )=0 for | x | 〉 a and h ( x , t ) the free boundary for | x | ≤ a . We deduce an integrodifferential evolutionary equation for the linearized free boundary and impose two different boundary conditions: the condition that the contact line between free boundary and the solid is pinned and the condition that the contact line can move vertically with a contact angle /2. For both cases, we compute the natural oscillation frequencies for the free surface and compare the results with the frequencies of oscillation in the absence of solid walls. Secondly, we study the effect of having two parallel solid walls at | x | = b + a on the natural frequencies of oscillation of the free boundary.

Description: This paper is concerned with the inverse scattering problem of time-harmonic acoustic plane waves by a crack buried in a piecewise homogeneous medium. We first show the well-posedness of the solution to the direct problem by using the boundary integral method. Then a uniqueness result for the inverse problem is obtained, that is, the shape of the crack and the surface parameter can be uniquely determined by the far-field pattern. The proof of the uniqueness result is based on a generalization of the mixed reciprocity relation. In particular, a mathematical basis is given to retrieve the shape of the crack by using the linear sampling method and difficulties arise from the fact that the crack has an empty interior.

Description: We investigate a representation of piecewise linear function containing a complementarity condition, and call it a linear complementarity representation. This representation is also known as ‘Bokh1’ or ‘state-variable representation’ in literature. We discuss two spacial types of representation, called P-representation and ULT-representation, and give an elementary proof to show that either one of these two types of representation completely characterizes any piecewise linear function. We can also demonstrate that a P-representation can be transformed into a ULT-representation, and show how it can be implemented in an algorithmic manner. Furthermore, as an application of linear complementarity representation, we will present a formula of linear complementarity representation of the Choquet integral model over a finite set, which is a model used in the area of decision-making as an aggregation function.

Description: The purpose of this paper is to give a survey of the progress, advantages and limitations of various locating methods of complex matrices' eigenvalues. Some new methods of locating complex matrices’ eigenvalues as well as new applications of locating methods will be exhibited. And theoretical research and practical application of locating methods of eigenvalues will be promoted.

Description: Identification of a highly conductive orebody buried in Earth using an equivalent, perfectly conducting, triaxial model ellipsoid is investigated. The real data available (three-component magnetic fields collected along a borehole due to a single-frequency current loop at the Earth surface) are simulated via a low frequency, closed-form power series expansion of the electromagnetic fields scattered off an equivalent ellipsoid within a homogeneous, conductive medium, the source itself being idealized as a vertical magnetic dipole nearby. The approach provides formulations amenable to fast yet accurate computations, most of the work being in the construction of the formulations themselves, not in the numerical computations. The inversion scheme is described, which sees the iterative minimization of the least-square discrepancy between the fields due to a given ellipsoid and the data available. Unknowns are semi-axis lengths, angular orientations and coordinates of its centre. Numerical simulations illustrate the approach, before considering experimental single-well log data in a surface-to-borehole configuration at a mining site.

Description: A mathematical model is developed of a spring-loaded pressure relief valve connected to a reservoir of compressible fluid via a single, straight pipe. The valve is modelled using Newtonian mechanics, under assumptions that the reservoir pressure is sufficient to ensure choked flow conditions. The usual assumptions of ideal gas theory lead to a system of first-order partial differential equations for the motion, energy and momentum of the fluid in the pipe. A reduced-order model is derived using a collocation method under the assumption that the dominant pipe dynamics corresponds to a standing quarter wave. The model comprises just five non-linear ordinary differential equations, representing the position and velocity of the valve body, the pressure in the tank and the velocity and pressure amplitudes of the pipe quarter wave. Through comparison with simulations of the full model using a Lax–Wendroff method, it is shown that the reduced model is quantitatively accurate and is able to predict the onset of an oscillatory valve-chatter instability. The basic trends of this instability are shown to be robust to the inclusion of pipe friction and convective effects.

Description: In this paper, the valuation problem of a European call option in the presence of both stochastic volatility and transaction costs is considered. In the limit of small transaction costs and fast mean reversion, an asymptotic expression for the option price is obtained. While the dominant term in the expansion is shown to be the classical Black and Scholes solution, the correction terms appear at O ( 1/2 ) and O ( ). The optimal hedging strategy is then explicitly obtained for Scott's model.

Description: This paper is concerned with the spatial dynamics of a time-delayed reaction and diffusion malaria model. We first analyse the well-posedness of the initial-boundary value problem of the model. Then, we study the global stability of the disease-free or endemic steady state for the system by structuring two Lyapunov functionals. Moreover, by applying Schauder fixed-point theorem, we establish the existence of travelling wave solutions connecting the two steady states: the disease-free steady state and the endemic steady state if the basic reproduction ratio $\mathcal {R}_0 〉 1$ , and show the non-existence of travelling wave solutions connecting the disease-free steady state to itself if $\mathcal {R}_0 〈 1$ . Our analytic results show that $\mathcal {R}_0$ is a threshold value for global dynamics and the existence of travelling wave solutions to the malaria model.

Description: This paper presents a network disease model that incorporates balanced birth and death event and has infectious force in the infected state and carrier state, such as hepatitis B virus (HBV). By investigating the local stability of the disease-free equilibrium, the basic reproduction number R 0 is derived and established as a sharp threshold. In particular, by using suitable Lyapunov functions and graph-theoretic results based on Kirchhoff's Matrix Tree Theorem, it is proved that if R 0 〈1, then the disease-free equilibrium is globally asymptotically stable; whereas if R 0 〉1, there exists a unique endemic equilibrium, which is globally asymptotically stable. When birth and death event are ignored, the final size formula is determined. Moreover, the effects of various immunization strategies are investigated and compared by numerical simulations. The results obtained are informative for us to further understand the disease propagation and devise some effective interventions to combat the diseases.

Description: We investigate the potential and limitations of the wave generation by disturbances moving at the bottom. More precisely, we assume that the wavemaker is composed of an underwater object of a given shape which can be displaced according to a prescribed trajectory. We address the practical question of computing the wavemaker shape and trajectory generating a wave with prescribed characteristics. For the sake of simplicity, we model the hydrodynamics by a generalized forced Benjamin–Bona–Mahony equation. This practical problem is reformulated as a constrained non-linear optimization problem. Additional constraints are imposed in order to fulfil various practical design requirements. Finally, we present some numerical results in order to demonstrate the feasibility and performance of the proposed methodology.

Description: We investigate the numerical performance of the regularized deconvolution closure introduced recently by the authors. The purpose of the closure is to furnish constitutive equations for Irving–Kirkwood–Noll procedure, a well-known method for deriving continuum balance equations from the Newton's equations of particle dynamics. A version of this procedure used in the paper relies on spatial averaging developed by Hardy, and independently by Murdoch and Bedeaux. The constitutive equations for the stress are given as a sum of non-linear operator terms acting on the mesoscale average density and velocity. Each term is a ‘convolution sandwich’ containing the deconvolution operator, a composition or a product operator and the convolution (averaging) operator. Deconvolution is constructed using filtered regularization methods from the theory of ill-posed problems. The purpose of regularization is to ensure numerical stability. The particular technique used for numerical experiments is truncated singular value decomposition (SVD). The accuracy of the constitutive equations depends on several parameters: the choice of the averaging window function, the value of the mesoscale resolution parameter, scale separation, the level of truncation of singular values and the level of spectral filtering of the averages. We conduct numerical experiments to determine the effect of each parameter on the accuracy and efficiency of the method. Partial error estimates are also obtained.

Description: A stochastic variational inequality is proposed to model an elasto-plastic oscillator excited by a filtered white noise. We prove the ergodic property of the process and characterize the corresponding invariant measure. This extends Bensoussan–Turi's method (2008, Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Appl. Math. Optim. , 58 , 1–27) with a significant additional difficulty of increasing the dimension. Two points boundary value problem in dimension 1 is replaced by elliptic equations in dimension 2. In the present context, Khasminskii's method (1980, Stochastic Stability of Differential Equations . The Netherlands: Sijthoff and Noordhof) leads to the study of degenerate Dirichlet problems with partial differential equations and non-local boundary conditions.

Description: The eigenfunctions of the axisymmetric Stokes flow operator $E^{2}$ in the inverted prolate spheroidal system of coordinates are derived analytically, while an analytical method for the calculation of the eigenfunctions of $E^{4}$ in the inverted prolate spheroidal system of coordinates is presented. The stream function $\psi$ satisfies the Stokes flow equation $E^{4}\psi =0$ while the equation $E^{2}\psi =0$ expresses an irrotational flow. In the prolate spheroidal coordinates, the solution space $\ker \{E^2\}$ is decomposed in separable eigenfunctions and the solution space $\ker \{E^4\}$ enjoys a spectral decomposition in semiseparable eigenfunctions. In the inverse prolate spheroidal system, we show that the partial differential operator $E^{2}$ admits $R$ -separable solutions with $R$ being a function of the Euclidean distance, $r$ , while the operator $E^{4}$ admits $R$ -semiseparable solutions with $R$ being a function of the Euclidean distance on the third, $r^3$ . We derive the 0-eigenspace of $E^{2}$ and we prove that the $\ker \{E^{4}\}$ consists of both: the eigenfunctions and the generalized eigenfunctions of $\ker \{E^{2}\}$ . Furthermore, by employing the Kelvin transformation, we show that the obtained expressions satisfy the Kelvin theorems as these are applied to the Stokes flow. The obtained eigenfunctions may be used in the mathematical treatment of medical problems, such as the blood plasma flow around erythrocytes.

Description: Many practical approximations in science and engineering invoke a relatively long physical domain with a relatively thin cross-section. In this scenario, we typically expect the system to have structures that vary slowly in the long dimension. Extant approximation methodologies are typically either self-consistency or limits as the aspect ratio becomes unphysically infinite. The proposed new approach is to analyse the dynamics based at each cross-section in a rigorous Taylor polynomial. Centre manifold theory supports the local modelling of the system's dynamics with coupling to neighbouring locales treated as a non-autonomous forcing. The union over all cross-sections then provides powerful new support for the existence and emergence of a centre manifold model global in the long domain, albeit of finite size. Our resolution of the coupling between neighbouring locales leads to novel quantitative estimates of the error induced by long slow space variations. Two examples help us develop and illustrate the approach and results. The approach developed here may be used to quantify the accuracy of known approximations, to extend such approximations to mixed order modelling and to open previously intractable modelling issues to new tools and insights.

Description: This paper considers the initial-boundary value problem for the heat equation with a dynamic-type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition.

Description: We propose a model for acid-mediated tumour invasion involving two different scales: the microscopic one, for the dynamics of intracellular protons and their exchange with their extracellular counterparts, and the macroscopic scale of interactions between tumour cell and normal cell populations, along with the evolution of extracellular protons. We also account for the tactic behaviour of cancer cells, the latter being assumed to biase their motion according to a gradient of extracellular protons (following Bartel et al. (2012) pH-taxis: directional tumor cell migration along pH-gradients. Acta Physiol. , 204 , 113; Paradise et al. (2013) Directional cell migration in an extracellular pH gradient: a model study with an engineered cell line and primary microvascular endothelial cells. Exp. Cell Res. , 319 , 487–497; we call this pH taxis). A time dependent (and also time delayed) carrying capacity for the tumour cells in response to the effects of acidity is considered as well. The global well posedness of the resulting multiscale model is proved with a regularization and fixed point argument. Numerical simulations are performed to illustrate the behaviour of the model.

Description: An incompressible fluid flow model for a thin-film thrust bearing with slip flow is derived, leading to a modified Reynolds equation for a highly rotating rotor that incorporates a slip length shear condition on the bearing faces, extending previous bearing studies for new bearing applications associated with decreasing film thickness. Mathematical and numerical modelling is applied to the coupled process of the pressurized fluid flow through the bearing, with a Navier slip condition replacing a no-slip condition, and the axial motion of the rotor and stator. The derived modified Reynolds equation is coupled with the dynamic motion of the stator through the pressure exerted by the fluid film, with explicit analytical expressions for the pressure and force determined and the equation for the bearing gap reduced to a non-linear second-order non-autonomous ordinary differential equation. A mapping solver is used to investigate the time-dependent bearing gap for prescribed periodic motion of the rotor. A parametric study focuses on bearing operation under close contact motion to examine the minimum film thickness and possibility of bearing face contact.

Description: In this paper, the Wiener–Hopf factorization problem is presented in a unified framework with the Riemann–Hilbert factorization. This allows to establish the exact relationship between the two types of factorization. In particular, in the Wiener–Hopf problem one assumes more regularity than for the Riemann–Hilbert problem. It is shown that Wiener–Hopf factorization can be obtained using Riemann–Hilbert factorization on certain lines.

Description: In this paper, the singularities of solutions of plate-bending problems at corner points along the plate boundary are studied. The mathematical problem, formulated on the basis of the Reissner–Mindlin plate theory, involves a parameter $\varepsilon$ which measures the effect of shear deformation. As $\varepsilon$ tends to zero the governing equations can be shown to reduce to the biharmonic equation in the classical plate theory. Explicit solutions of the equations for general $\varepsilon 〉 0$ enable the analysis of the singularity at a corner point as the radial distance $r$ from the corner point tends to zero. Our analysis reveals why the singularities in the Reissner–Mindlin plate theory ( $r\rightarrow 0$ and $\varepsilon 〉 0$ fixed) differ from those in the classical plate theory ( $\varepsilon \rightarrow 0$ followed by $r\rightarrow 0$ ).

Description: Ignoring inertia, a deformable interface separating two fluid films is considered, subject to non-uniform tension driven by the solutal Marangoni effect in the presence of a scalar concentration field. Detailed description of adsorption kinetics is abrogated by a simple ansatz directly relating interfacial tension and bulk solute concentration. Consequently, the formal mathematical treatment and some of the results share features in common with the Rayleigh–Bénard–Marangoni thermocapillary problem. Normal mode perturbation analysis in the limit of small interface deformations establishes the existence of an unstable response for low wavenumber excitation. In the classification of Cross & Hohenberg (1993, Pattern formation outside of equilibrium. Rev. Mod. Phys. , 65 , 851–1112), both type I and type II behaviour are observed. By considering the zero wavenumber situation exactly, it is proved that all eigenvalues are purely imaginary with non-positive imaginary part; hence, a type III instability is not possible. For characteristic timescales of mass diffusion much shorter than the relaxation time of interfacial fluctuations (infinite crispation number): the response growth rate is obtained explicitly; only a single excitation mode is available, and a complete stability diagram is constructed in terms of the relevant control parameters. Otherwise, from a quiescent base state, an infinite discrete spectrum of modes is observed that exhibit avoided crossing and switching phenomena, as well as exceptional points where stationary state pairs coalesce into a single oscillatory standing wave pattern. A base state plane Poiseuille flow, driven by an external pressure gradient, generally exaggerates the response: growth rates of instabilities are enhanced, and stable decay is further suppressed with increasing base flow speed, but the inherent symmetry breaking destroys stationary and standing wave response. Results are obtained in this most general situation by implementing a numerical Chebyshev collocation scheme. The model was motivated by hydrodynamic processes supposed to be involved in gastric digestion of humans.

Description: In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction–diffusion predator–prey system with mutual interference described by the Crowley–Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction–diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction–diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

Description: We investigate the effect of turning delays on the behaviour of groups of differential wheeled robots and show that the group-level behaviour can be described by a transport equation with a suitably incorporated delay. The results of our mathematical analysis are supported by numerical simulations and experiments with E-Puck robots. The experimental quantity we compare with our revised model is the mean time for robots to find the target area in an unknown environment. The transport equation with delay better predicts the mean time to find the target than the standard transport equation without delay.

Description: The central aim of this partly expository paper is pressing the point that a certain transcendental equation for the final size of an susceptible-infected-removed epidemic model has an explicit solution in terms of the Lambert W -function, an elementary function. Known branch point and continued fraction approximations yield simple and highly accurate numerical approximations which are considerably better than the traditional Kermack–McKendrick approximation. A Karamata expansion easily yields refined forms of the second threshold theorem.

Description: Piezoelectric ultrasonic transducers usually employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwidth device it would seem natural therefore to utilize resonators that span a range of length scales. In this article, we derive a mathematical model to predict the dynamics of a fractal ultrasound transducer; the fractal in this case being the Sierpinski gasket. Expressions for the electrical and mechanical fields that are contained within this structure are expressed in terms of a finite element basis. The propagation of an ultrasonic wave in this transducer is then analysed and used to derive expressions for the non-dimensionalized electrical impedance and the transmission and reception sensitivities as a function of the driving frequency. Comparing these key performance measures to an equivalent standard (Euclidean) design shows some benefits of these fractal designs.

Description: A class of autonomous, even-order ordinary differential equations is discussed from the point of view of Lie symmetries. It is shown that for a certain power non-linearity, the Noether symmetry group coincides with the Lie point symmetry group. First integrals are established and exact solutions are found. Furthermore, this paper complements, for the one-dimensional case, some results in the literature of Lie group analysis of poliharmonic equations and Noether symmetries obtained in the last 20 years. In particular, it is shown that the exceptional negative power discovered in Bokhari et al. (2010, Symmetries and integrability of a fourth-order Euler–Bernoulli beam equation. J. Math. Phys. , 51 , 053517) is a member of a one-parameter family of exceptional powers in which the Lie symmetry group coincides with the Noether symmetry group.

Description: An optimal control approach to a simplified reaction–diffusion system describing cardiac defibrillation is proposed that allows for joint optimization of shape and duration of defibrillation pulses. Within the framework, optimized multi-phasic pulses with low energy, short duration and/or low amplitude can be designed according to specific needs. The approach is based on a novel time optimal control formulation for the monodomain model, which takes into consideration the dynamical system properties of the uncontrolled equation. The highly complex dynamics requires a consistent discretization of first- and second-order information to guarantee effective optimization schemes leading to successful defibrillation. Numerical examples underline the efficiency of the proposed method.

Description: The main goal of this paper is to introduce the so-called service cost savings games based on cooperative R $\& $ D situation. For these specific cooperative games, on one hand, we determine the Shapley value allocation, which is equivalent to a cooperative mechanism in R $\& $ D problems, for these service cost savings games through a decomposition method for games into one additive game and one Sharing car pooling cost game, exploiting the linearity of the Shapley value. On the other hand, we determine the nucleolus allocation as well, by exploiting fully the so-called 1-convexity property for these service cost savings games. Moreover, this significant property illustrates the possibility of the collaboration of the innovative firms.

Description: The initial boundary value problem for a non-linear wave equation of Kirchhoff type with damping and memory terms in the bounded domain is considered. The local, global existence and exponential decay result are discussed under certain conditions. We also prove that, under certain conditions, the energy will grow up at least as an exponential function of time when the weak damping term is non-linear and will blow up when the weak damping term is linear.

Description: Isostatic mounts are used in applications like telescopes and robotics to move and hold part of a structure in a desired pose relative to the rest, by driving some controls rather than driving the subsystem directly. To achieve this successfully requires an understanding of the structure of the coupled space of configurations and controls, and of the singularities of the mapping from the coupled space to the space of controls. It is crucial to avoid such singularities because generically they lead to large constraint forces and internal stresses which can cause distortion. In this paper we outline design principles for isostatic mount systems for dynamic structures, with particular emphasis on robots.

Description: This paper focuses on the existence of periodic solutions for coupled systems on networks with time-delay and linear coupling (CSNDLC). Based on graph theory, coincidence degree theory and the Lyapunov method, a systematic approach for the existence of periodic solutions to CSNDLC is developed. And this approach is applied to the linear coupled oscillators with time-delay on a network. In addition, global asymptotic stability criterion for its periodic solution is obtained. Finally, a numerical example is provided to illustrate the effectiveness of the results developed.

Description: A general reaction–diffusion system modelling glycolysis is investigated. The parameter regions for the stability and instability of the unique constant steady-state solution is derived, and the existence of time-periodic orbits and non-constant steady-state solutions are proved by the bifurcation method and Leray–Schauder degree theory. The effect of various parameters on the existence and non-existence of spatiotemporal patterns is analysed.

Description: Interface dynamics of post-contact states in regularized models of electrostatic–elastic interactions are analysed. A canonical setting for our investigations is the field of Micro-Electromechanical Systems (MEMS) in which flexible elastic structures may come into physical contact due to applied Coulomb forces. We study the dynamic features of a recently derived regularized model (Lindsay, A. E. et al. 2014 Regularized model of post-touchdown configurations in electrostatic MEMS: equilibrium analysis. Physica D ), which describes the system beyond the quenching singularity associated with touchdown, that is after the components of the device have come together. We build on our previous investigations of steady-state solutions by describing how the system relaxes towards these equilibria. This is accomplished by deriving a reduced dynamical system that governs the evolution of the contact set, thereby providing a detailed description of the intermediary dynamics associated with this bistable system. The analysis yields important practical information on the timescales of equilibration.

Description: A liquid stretched between two solid boundaries forms a liquid bridge. The static stability (‘shape stability’) of the liquid bridge to disturbances whose contact-line is (i) pinned or (ii) moves with fixed contact-angle is studied. The liquid/gas interface is idealized as a mathematical surface of constant surface tension. Elementary results from the calculus-of-variations are utilized to derive a sufficient condition for stability without explicitly solving the boundary value problem associated with the second variation. The focus is generating ‘quick’ checks on stability. The utility of the method is illustrated by limiting cases of the liquid bridge between parallel plates; the liquid cylinder and catenoid. Our stability criteria gives sharp bounds in some cases, recovering previously reported results and yields new predictions for mobile contact-line disturbances. We conclude with remarks concerning the effect of support geometry.

Description: A new general transform method for the solution of mixed boundary value problems (BVPs) for Laplace's equation in multiply connected circular domains is formulated. Circular domains are those with boundaries that are a union of circular arcs and $/$ or straight line segments. Constructive schemes based on the method are showcased by application to some concrete problems in fluid mechanics. In particular, a BVP of topical interest in the study of superhydrophobic surfaces is solved and compared with an existing numerical solution.

Description: The aim of this paper is to show that, for the eddy current model, the leading order term for the perturbation in the magnetic field, due to the presence of a small conducting magnetic inclusion, can be expressed in terms of a symmetric rank 2 polarization tensor. This tensor contains information about the shape and material properties of the object and is independent of position. We apply a recently derived asymptotic formula for the perturbed magnetic field, due to the presence of a conducting inclusion, which is expressed in terms of a new class of rank 4 polarization tensors (Ammari, H., Chen, J., Chen, Z., Garnier, J. & Volkov, D. (2014) Target detection and characterization from electromagnetic induction data. J. Math. Pures Appl. , 101 , 54–75.) and show that their result can be written in an alternative form involving a symmetric rank 2 tensor involving 6 instead of 81 complex components in an orthonormal coordinate frame. For objects with rotational and mirror symmetries we show that the number of coefficients is still smaller. We include numerical examples to demonstrate that the new polarization tensors can be accurately computed by solving a vector-valued transmission problem by $hp$ -finite elements and include examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical predictions.

Description: In the current work, we advance the integral equation solution of Guo & Meroney (2013, Theoretical solution for laminar flow in partially-filled pipes. J. Hydraul. Res. , 51 , 408–416) for the velocity distribution of a Newtonian fluid flowing under the effect of gravity in a partially filled pipe. Unlike in previous work, our new solution is analytical and is in series form. The solution is able to evaluate the velocity profile in a partially filled pipe for any fill depth. Additionally, the well-known velocity dip phenomenon is observed, and, using our new solution, it is predicted that the dip will begin once the pipe is filled above 86% of the channel depth. Excellent agreement is found between the analytical and a computed numerical solution to the flow problem.

Description: We investigate blow-up criteria for the local in time classical solution of the 3D incompressible nematic liquid crystal flows via the pressure and the orientation field. More precisely, we prove that $0 〈 T_{*} 〈 +\infty $ is the maximal time interval if and only if \[ \int_{0}^{T_{*}}\|P\|_{L^{\alpha}}^{\beta}+\|\nabla d\|_{L^{4}}^{8}\,{\rm d}t=\infty, \quad \hbox{ with } \frac{3}{\alpha}+\frac{2}{\beta}\leq 2, \quad \hbox{ and }\quad \frac{3}{2} 〈 \alpha \leq \infty \] and \[ \int_{0}^{T_{*}}\|\nabla P\|_{L^{\alpha}}^{\beta}+\|\nabla d\|_{L^{4}}^{8}\,{\rm d}t=\infty, \quad \hbox{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq 3,\quad \hbox{ and }\quad 1 〈 \alpha \leq \infty. \]

Description: We consider small-amplitude deformations of a thin-walled elastic tube, which initially has a uniform elliptical cross-section and is subject to a large axial pre-stress. We derive a boundary-layer model for the deformations near an end of such a tube that is pinned to a rigid elliptical support. The model is appropriate in the limit in which $\mathscr {F} \equiv d^2 F/[24{\pi } aK (1-\nu ^2)] \ll 1$ , where $d$ is the wall thickness, $F$ is the axial tension that gives rise to the pre-stress, $2\pi a$ is the tube circumference, $K$ is the bending stiffness of the tube wall and $\nu $ is its Poisson ratio. In particular, the model takes into account in-plane shear forces arising because of geometrical constraints. These forces are asymptotically small outside the boundary layer, and so were not present in the previous tube-law model of Whittaker et al. (2010a, A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Math. 63 , 465–492). Deformation profiles from the boundary-layer model are matched to solutions for the interior arising from the tube-law model of Whittaker et al. (2010a, A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Math. 63 , 465–492). The net effect is to modify the previous tube-end boundary condition on the interior solution, from zero normal displacement to a Robin-type condition. The predictions from the matched models compare favourably with full numerical simulations of the tube wall deformations. While the additional shear forces are only important in the boundary layer near the end, they can have a significant effect on the global solution when $\mathscr {F} \ll 1$ .

Description: Consider the generalized Kuramoto–Sivashinsky (gKS) equation. It is a model prototype for a wide variety of physical systems, from flame-front propagation, and more general front propagation in reaction–diffusion systems, to interface motion of viscous film flows. Our aim is to develop a systematic and rigorous low-dimensional representation of the gKS equation. For this purpose, we approximate it by a renormalization group equation which is qualitatively characterized by rigorous error bounds. This formulation allows for a new stochastic mode reduction guaranteeing optimality in the sense of maximal information entropy. Herewith, noise is systematically added to the reduced gKS equation and gives a rigorous and analytical explanation for its origin. These new results would allow one to reliably perform low-dimensional numerical computations by accounting for the neglected degrees of freedom in a systematic way. Moreover, the presented reduction strategy might also be useful in other applications where classical mode reduction approaches fail or are too complicated to be implemented.

Description: In this paper, we study the finite time blowup and global existence of the interfaces of the inhomogeneous non-Newtonian polytropic filtration equation with convection where ( x ) tends to zero as . In particular, we give the critical characterization of the left and right interfaces for the typical model case and , respectively, which extends the corresponding result for the porous medium equations by Ferreira et al. (2006, The interfaces of an inhomogeneous porous medium equation with convection. Comm. Partial Differential Equations , 31 , 497–514).

Description: The mixed convection boundary-layer flow near the lower stagnation point of a horizontal circular cylinder with a second-order slip velocity model and a constant surface heat flux is investigated. By choosing appropriate similarity variables, the partial differential equations and boundary conditions governing the flow are transformed into a similarity system. This is solved numerically for different values of the governing parameters. These solutions can have two branches, an upper and a lower branch, over certain ranges of the mixed convection parameter. These numerical studies are complemented by the derivation of the asymptotic limits of the system parameters. A stability analysis is performed, showing that the upper branch is stable and physically realizable, whereas the lower branch solutions are unstable.

Description: We are presenting a certain simplification of the well-known PL-continuation method, applied to continuous curve tracking, for zero set of the map The presented algorithm builds a family of regular ( k +1)-dimensional simplices following the set of zeros of the map F . The values of this map are calculated in the vertices of each simplex, then a decision is made as to which of the k -dimensional faces are to be taken as the faces intersected by the zero set F –1 (0). The method is based on sign changes of coordinate functions of the map F . The computed values of F are not used in further computations. This allows for limitation of the accuracy required to calculate values of F , i.e. the calculations may be discontinued when the signs of F values may be identified. The algorithm is less restrictive than the classical PL-continuation method and its modifications like the integer labelling. As a consequence it may be observed that in many cases the algorithm is able to follow multiple branches starting from the bifurcation point. The method is applied to the solution of practical 3D problems regarding identification of complex dispersion characteristics of microwave transmission lines. While the function F is being traced, a big computational effort is required for its value at every point of the domain. Therefore every method which can accelerate this process seems to be valuable.

Description: This paper examines the SGP4 orbit propagation equations for the specific case of an initially circular satellite orbit, with the aim of understanding the along-trajectory errors that are observed in practice. In particular, the SGP4 equations are compared with analytic expressions obtained by means of a regular perturbation expansion and two discrepancies have been found in the along-track angle u , both of order J 2 (the primary earth oblateness term): A slightly different value for the linear trend, governing the secular behaviour, A missing sinusoidal term, with period equal to the orbit period. It is suspected that the effects of the former discrepancy are masked in practice, so far as possible, by a suitable choice of the B * drag parameter.