We consider and analyse sampling theories in the reproducing kernel Hilbert space (RKHS) in this paper. The reconstruction of a function in an RKHS from a given set of sampling points and the reproducing kernel of the RKHS is discussed. Firstly, we analyse and give the optimal approximation of any function belonging to the RKHS in detail. Then, a necessary and sufficient condition to perfectly reconstruct the function in the corresponding RKHS of complex-valued functions is investigated. Based on the derived results, another proof of the sampling theorem in the linear canonical transform (LCT) domain is given. Finally, the optimal approximation of any band-limited function in the LCT domain from infinite sampling points is also analysed and discussed.

In this paper we describe solutions of a one-dimensional model of inviscid fluid jets that develop finite time singularities in a fractal-like manner. We also discuss the extreme sensitivity of the solutions of this problem with respect to small perturbations of the initial data.

We introduce a differential model to study damage accumulation processes in the presence of chemical reactions. The influence of micro-structure leads to a nonlinear parabolic system characterized by the presence of a characteristic length. Here, we first present an analytical description of the qualitative behaviour of solutions which blow-up in finite time. Numerical simulations are given to describe the shape of solutions near the rupture time and the influence of the chemical reagents. As in the non-reactive model, the failure of the material occurs in a region of positive measure, due to the diffusive effects of the micro-structure, although some localization phenomena are observed. Moreover, if we increase the chemical concentration beyond a given threshold, which depends upon the specific conditions of the material, we observe a strong acceleration in the damage process.

In this paper we investigate the population dynamics of a species with age structure in the case where the diffusion and death rates of the matured population are both age-dependent. We develop a new application of the age-structure technique in terms of an integral equation. For unbounded spatial domains, we study the existence of travelling waves, whilst in bounded domains, we investigate the existence of positive steady-state solutions and their stability.

Exact analytical representations are obtained describing self-similar unsteady flows of multi-phase immiscible fluids in the vicinity of non-circular, but constant strength, fronts. It is assumed that Darcy's law holds for each phase and that the mobilities are known functions of the saturations. Equivalent representations are obtained for Hele-Shaw cell flows that are produced when a viscous fluid is injected into a region containing some other viscous fluid. The fluids may be Newtonian fluids or non-Newtonian fluids for which the coefficients of viscosity depend on the shear stress. Even though the flows are unsteady and two dimensional, the representations are obtained by using hodograph techniques.

The main objective of this work is to present, for practical use, some original results about several qualitative properties of the solutions of a large class of degenerate diffusion-convection equations arising from fluid mechanics. Current interest in models of the simultaneous motion of two immiscible incompressible liquids results from its significance for many applied fields such as, for instance, the theoretical modelling of oil reservoirs where the pores of a threedimensional porous medium contain some hydrocarbon component (oil). In secondary recovery, a second inexpensive fluid (water) is injected into the porous medium in order to push the oil towards the producing wells.

Global smooth solutions are shown to exist for the system governing magneto-thermoviscoelastic phenomena in an electrically and thermally conducting isotropic solid immersed in an electromagnetic field. It is assumed that displacement currents are negligible, and that all field quantities depend on one space variable only; Joule heating is included.

The subject of the present paper is the derivation and analysis of mathematical models for the formation of a mushy region during calcium carbonate corrosion. More specifically there is emphasis on the variation of the overall diffusion resulting from the changing shape of a single pore due to corrosion process and on the resulting volume expansion of the material as the outcome of the transformation of calcium carbonate to gypsum. These models are derived by averaging, with the use of the multiple scales method applied on microscopic moving-boundary problems. The latter problems describe the transformation of calcium carbonate into gypsum in the microscopic scale. The derived macroscopic models are solved numerically with the use of an implicit in time, finite element method. The results of the simulations for various microstructure geometries in the micro-scale and a discussion are also presented.

In this paper, the problem of restoration of cloud contaminated optical images is studied in the case when we have no information about brightness of such images in the damage region. We propose a new variational approach for exact restoration of optical multi-band images utilising Synthetic Aperture Radar (EOS – Spatial Data Analytics, GIS Software, Satellite Imagery – is a cloud-based platform to derive remote sensing data and analyse satellite imagery for business and science purposes) images of the same regions. We prove existence of solutions, propose an alternating minimisation method for computing them, prove convergence of this method to weak solutions of the original problem and derive optimality conditions.

In this study, we examine a steady two-dimensional slow flow past a rigid cylinder coated with a thin layer of immiscible fluid. The Reynolds number for the external bulk flow is assumed small and flow within the film is driven by the action of the bulk fluid’s tangential viscous stress acting at the interface. Using double asymptotic expansions based on the bulk fluid’s Reynolds number and the aspect ratio of the film thickness to the cylinder’s radius, we derive the leading- and first-order equations governing the steady-state film dynamics, and obtain analytical solutions, in terms of the film thickness, for the bulk flow. We solve the governing film equations, finding that solutions feature a drained region. We briefly discuss the influence of the Capillary number and fluid viscosities, and conclude by showing how the presence of the film affects the drag on the film-coated cylinder.

Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearisation (Frechet derivative) equation holding on the space of solutions to the PDE, and they are well-known to comprise a linear space having the structure of a Lie algebra. Solutions of the adjoint linearisation equation holding on the space of solutions to the PDE are called adjoint-symmetries. Their algebraic structure for general PDE systems is studied herein. This is motivated by the correspondence between variational symmetries and conservation laws arising from Noether’s theorem, which has a modern generalisation to non-variational PDEs, where infinitesimal symmetries are replaced by adjoint-symmetries, and variational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certain Euler-Lagrange condition). Several main results are obtained. Symmetries are shown to have three different linear actions on the linear space of adjoint-symmetries. These linear actions are used to construct bilinear adjoint-symmetry brackets, one of which is a pull-back of the symmetry commutator bracket and has the properties of a Lie bracket. The brackets do not use or require the existence of any local variational structure (Hamiltonian or Lagrangian) and thus apply to general PDE systems. One of the symmetry actions is shown to encode a pre-symplectic (Noether) operator, which leads to the construction of symplectic 2-form and Poisson bracket for evolution systems. The generalised KdV equation in potential form is used to illustrate all of the results.

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

Steady two-dimensional fluid flow over an obstacle is solved using complex variable methods. We consider the cases of rectangular obstacles, such as large boulders, submerged in a potential flow. These may arise in geophysics, marine and civil engineering. Our models are applicable to initiation of motion that may result in subsequent transport. The local flow depends on the obstacle shape, slowing down in confining corners and speeding up in expanding corners. The flow generates hydrodynamic forces, drag and lift, and their associated moments, which differ around each face. Our model replaces the need for ill-defined drag and lift coefficients with geometry-dependent functions. We predict smaller flow velocities to initiate motion. We show how a joint-bound boulder can be transported against gravity, and analyse the influence of a wake region behind an isolated boulder.

The limit Wigner measure of a WKB function satisfies a simple transport equation in phase-space and is well suited for capturing oscillations at scale of order $O(\epsilon)$, but it fails, for instance, to provide the correct amplitude on caustics where different scales appear. We define the semi-classical Wigner function of an $N$-dimensional WKB function, as a suitable formal approximation of its scaled Wigner function. The semi-classical Wigner function is an oscillatory integral that provides an $\epsilon$-dependent regularization of the limit Wigner measure, it obeys a transport-dispersive evolution law in phase space, and it is well defined even at simple caustics.

The time-varying flow in which fluid is withdrawn from or added to a reservoir of infinite or arbitrary finite depth through a point sink or source of variable strength beneath a free surface is considered. Backed up by some analytic work, a numerical method is used, and the results are compared with previous work on steady and unsteady flows. In the case of withdrawal for an impulsively started flow, it is found that the critical flow rate increases with reservoir depth, although it changes little as the depth increases beyond double the sink submergence depth. The largest flow rate at which steady solutions can evolve in source flows follows a similar pattern although at a considerably higher value. Simulations indicate that some of the previously calculated steady state solutions at higher flow rates may be unstable, if they exist at all.

The bifurcation of symmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg–Landau equations by the methods of formal asymptotics. The behaviour of the bifurcating branch depends upon the parameters d, the size of the superconducting slab, and κ, the Ginzburg–Landau parameter. It was found numerically by Aftalion & Troy [1] that there are three distinct regions of the (κ, d) plane, labelled S1, S2 and S3, in which there are at most one, two and three symmetric solutions of the Ginzburg–Landau system, respectively. The curve in the (κ, d) plane across which the bifurcation switches from being subcritical to supercritical is identified, which is the boundary between S2 and S1∪S3, and the bifurcation diagram is analysed in its vicinity. The triple point, corresponding to the point at which S1, S2 and S3 meet, is determined, and the bifurcation diagram and the boundaries of S1, S2 and S3 are analysed in its vicinity. The results provide formal evidence for the resolution of some of the conjectures of Aftalion & Troy [1].

Volume 9 (1998), pp. 485–506

One of the author's names was misprinted in this article. EPIFIANO G. VIRGA should read EPIFANIO G. VIRGA.

In addition, on page 502, line 9, the argument leading to evaluate g(∞) := limb→+∞g(b) is not conclusive. The passage

‘On the other hand, if it were g(∞)>0, taking the limit as b→+∞ in (9.9), we would get A=πh2, which by (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that g(∞)=0.’

should be changed as follows:

‘On the other hand, taking the limit as b→+∞ in (9.9), we would get A=πh2, by which (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that, in the limit as b→+∞, the equilibrium configuration must be singular whenever A>πh2, and so g(∞)=0 because, otherwise, by (9.7) there would be asymptotic regular configurations, as soon as h/2kc(τ0+w) is sufficiently small.’

On page 502, line −7, ‘below’ should be replaced with ‘above’ and ‘regular’ with ‘singular’.

We apologise for the errors in the above paper and hope that no inconvenience has been caused to readers.

In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.

The present paper investigates the mechanism of interface closure in the root region of the solutions for steady deep-cellular growth. This phenomenon is determined by a transcendentally small factor beyond all orders. It is found that the root region comprises three inner-inner regions; the inner system in the root region has a simple turning point, whose presence generates the so-called trapped-waves mechanism, which is responsible for the interface closure at the bottom of root. The quantization condition derived from the trapped-waves mechanism yields the eigenvalue that determines the location of interface closure and its dependence on the interfacial energy and other physical parameters.