Boundary value problems for linear transport equations sometimes require the explicit construction of solutions when boundary conditions are prescribed only on parts of the boundary and thus necessitate the construction of half-range expansions. In contrast to the standard eigenfunction expansion, a half-range expansion of a given function must be reconstructed just over half the domain, using just half of the eigenfunctions. The difficulty of such expansions arises because the eigenfunctions are not orthogonal, though they are complete, over half the domain, and there is no obvious method of obtaining the expansion coefficients. Here we use complex variable techniques to find explicit formulas for the coefficients of half-range expansions for regular, negative definite Sturm–Liouville operators. We prove that the half-range expansion formula is unique, and find the corresponding half-range Green functions.

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

We consider travelling wave solutions of a reaction–diffusion system arising in a model for infiltration with changing porosity due to reaction. We show that the travelling wave solution exists, and is unique modulo translations. A small parameter ε appears in this problem. The formal limit as ε → 0 is a free boundary problem. We show that the solution for ε > 0 tends, in a suitable norm, to the solution of the formal limit.

We propose and analyse an age-structured model for within-host HIV virus dynamics which is incorporated with both virus-to-cell and cell-to-cell infection routes, and proliferations of both uninfected and infected cells in the form of logistic growth. The model turns out to be a hybrid system with two differential-integral equations and one first-order partial differential equation. We perform some rigorous analyses for the considered model. Among the interesting dynamical behaviours of the model is the occurrence of backward bifurcation in terms of the basic reproduction number R0 at R0 = 1, which raises new challenges for effective infection control. We also discuss the cause of such a backward bifurcation, based on our analytical results.

An analysis is made of the damping of sound and structural vibrations by vorticity production in the apertures of a bias flow, perforated elastic plate. Unsteady motion causes vorticity to be generated at the aperture edges; the vorticity and its energy are swept away by the bias flow and result in a net loss of acoustic and vibrational energy. In this paper we investigate the interaction of an arbitrary fluid-structure disturbance with a small circular aperture in the presence of a high Reynolds number, low Mach number bias flow. By considering the limit in which the aperture is small compared to the length scale of the impinging disturbance, it is shown that the effect of the interaction can be represented by a concentrated source in the plate bending wave equation consisting of a delta function and two of its axisymmetric derivatives. A generalized bending wave equation is then formulated for a plate perforated with an homogeneous distribution of small, bias flow circular apertures. This equation is used to predict the attenuation of sound and resonant bending waves by vorticity production. Acoustic damping is found to be significant provided the fluid loading is sufficiently small for the plate to be regarded as rigid (e.g. for an aluminium plate in air when the frequency is not too small). On the other hand, a bending wave is effectively damped only when the fluid loading is large enough for the wave to produce a substantial pressure drop across the plate; when this occurs the predicted attenuations are comparable with those usually achieved by the application of elastomeric damping materials. Numerical predictions are presented for steel and aluminium plates in air and water.

Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$, and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$-elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments.

We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, e.g. smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and non-local drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimisation method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2–1 junction.

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by deriving partial energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution u(x,t) converges to a solution of an elliptic obstacle problem as t → +∞.

An eigenvalue problem is considered for a multi-structure consisting of a three-dimensional finite solid connected to an arbitrary smooth elastic thin shell of revolution. Two-sided estimates are obtained for the first six eigenfrequencies of the multi-structure. Explicit asymptotic formulae are given.

This paper provides an algorithm for identifying the multipoles of a meromorphic function from the boundary measurements, which can be applied to inverse source problems for a Poisson equation to reconstruct its source such as linear combination of mono and dipolar point sources. After establishing an algorithm that allows to identify the number, the locations and the moments of poles, we provide a stability estimate of the algorithm under the measurement error.

A leader is to be elected from n people using the following algorithm. Each person flips a coin. Those people who wind up with tails (which occurs with probability p, 0 < p < 1) move on to the next stage. Those with heads are eliminated. Let Hn denote the number of stages needed until there is a single winner. We analyze the moments and the probability distribution of Hn. In the symmetric model we have an unbiased coin with p = 1/2; in the asymmetric model p ≠ 1/2. We analyze these models asymptotically, for n → ∞, using a variety of analytical and numerical approaches. This leads to simple derivations of some existing results, as well as some new results for the asymmetric case. Our analysis makes some assumptions about the forms of various asymptotic expansions as well as their asymptotic matching.

In 1958 started the study of the families of algebraic limit cycles in the class of planar quadratic polynomial differential systems. In the present we known one family of algebraic limit cycles of degree 2 and four families of algebraic limit cycles of degree 4, and that there are no limit cycles of degree 3. All the families of algebraic limit cycles of degree 2 and 4 are known, this is not the case for the families of degree higher than 4. We also know that there exist two families of algebraic limit cycles of degree 5 and one family of degree 6, but we do not know if these families are all the families of degree 5 and 6. Until today it is an open problem to know if there are algebraic limit cycles of degree higher than 6 inside the class of quadratic polynomial differential systems. Here we investigate the birth and death of all the known families of algebraic limit cycles of quadratic polynomial differential systems.

This work models and analyses the dynamics of a general spring-mass-damper system that is in frictional contact with its support, taking into account frictional heat generation and a reactive obstacle. Friction, heat generation and contact are modelled with subdifferentials of, possibly non-convex, potential functions. The model consists of a non-linear system of first-order differential inclusions for the position, velocity and temperature of the mass. The existence of a global solution is established and additional assumptions yield its uniqueness. Nine examples of conditions arising in applications, for which the analysis results are valid, are presented.

This note deals with a new model of elastic–perfectly plastic materials in which the yield stress is regarded as a threshold above which plastic flow occurs rather than a constraint which cannot be violated. This modelling change allows us to treat a number of signalling and impact problems not solvable within the classic framework of elastic–perfectly plastic materials.

The aim of this paper is to study the dimension reduction analysis of an elastic plate with small thickness reinforced with increasing number of thin ribbons developing fractal geometry. We prove the $\Gamma $-convergence of the energy functionals to a two-dimensional effective energy including singular terms supported within the Sierpinski carpet.

We study a one-parameter class of examples of optimal transport problems between a two-dimensional source and a one-dimensional target. Our earlier work identified a nestedness condition on the surplus function and marginals, under which it is possible to solve the problem semi-explicitly. In the family of examples we consider, we classify the values of parameters which lead to nestedness. In those cases, we derive an almost explicit characterisation of the solution.

Previous study of contact of power-law graded materials concerned the contact of a rigid body (punch) with an elastic inhomogeneous foundation whose inhomogeneity is characterised by the Young modulus varying with depth as a power function. This paper models Hertzian and adhesive contact of two elastic inhomogeneous power-law graded bodies with different exponents. The problem is governed by an integral equation with two different power kernels. A nonstandard method of Gegenbauer orthogonal polynomials for its solution is proposed. It leads to an infinite system of linear algebraic equations of a special structure. The integral representations of the system coefficients are evaluated, and the properties of the system are studied. It is shown that if the exponents coincide, the infinite system admits a simple exact solution that corresponds to the case when the Young moduli are different but the exponents are the same. Formulas for the length of the contact zone, the pressure distribution and the surface normal displacements of the contacting bodies are obtained in the form convenient for computations. Effects of the mismatch in the Young moduli exponents are studied. A comparative analysis of the Hertzian and adhesive contact models clarifies the effects of the surface energy density on the contact pressure, the contact zone size and the profile of the contacting bodies outside the contact area.

Heat transport in granular and porous media occurs through conduction in the solid and radiation through the voids. By exploiting the separation of length scales between the small typical particles or voids and the large size of whole region, the method of multiple scales can be applied. For a purely diffusive system, this yields a problem on the macroscale with an effective conductivity, deduced by solving a ‘cell problem’ on the microscale. Here, we apply the method when radiation and conduction are both present; however, care must be taken to correctly handle the integral nature of the radiative boundary condition. Again, an effective conductivity is found by solving a ‘cell problem’ which, because of the non-linearity of radiative transfer, to be solved for each temperature value. We also incorporate modifications to the basic theory of multiple scales in order to deal with the non-local nature of the radiative boundary condition. We derive the multiple scales formulation of the problem and report on numerical comparisons between the homogenised problem and direct solution of the problem. We also compare the effective conductivity to that derived using Maxwell models and effective medium theory.

In this work, we carry out an analytical and numerical investigation of travelling waves representing arced vegetation patterns on sloped terrains. These patterns are reported to appear also in ecosystems which are not water deprived; therefore, we study the hypothesis that their appearance is due to plant–soil negative feedback, namely due to biomass-(auto)toxicity interactions.

To this aim, we introduce a reaction-diffusion-advection model describing the dynamics of vegetation biomass and toxicity which includes the effect of sloped terrains on the spatial distribution of these variables. Our analytical investigation shows the absence of Turing patterns, whereas travelling waves (moving uphill in the slope direction) emerge. Investigating the corresponding dispersion relation, we provide an analytic expression for the asymptotic speed of the wave. Numerical simulations not only just confirm this analytical quantity but also reveal the impact of toxicity on the structure of the emerging travelling pattern.

Our analysis represents a further step in understanding the mechanisms behind relevant plants‘ spatial distributions observed in real life. In particular, since vegetation patterns (both stationary and transient) are known to play a crucial role in determining the underlying ecosystems’ resilience, the framework presented here allows us to better understand the emergence of such structures to a larger variety of ecological scenarios and hence improve the relative strategies to ensure the ecosystems’ resilience.