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 displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms of a priori estimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.

We present a mathematical model built to describe the fluid dynamics for the heat transfer fluid in a parabolic trough power plant. Such a power plant consists of a network of tubes for the heat transport fluid. In view of optimisation tasks in the planning and in the operational phase, it is crucial to find a compromise between a very detailed description of many possible physical phenomena and a necessary simplicity needed for a fast and robust computational approach. We present the model, a numerical approach, simulation for single tubes and also for realistic network settings. In addition, we optimise the power output with respect to the operational parameters.

Not only are thin fluid films of enormous importance in numerous practical applications, including painting, the manufacture of foodstuffs, and coating processes for products ranging from semi-conductors and magnetic tape to television screens, but they are also of great fundamental interest to mathematicians, physicists and engineers. Thin fluid films can exhibit a wealth of fascinating behaviour, including wave propagation, rupture, and transition to quasi-periodic or chaotic structures. More details of various aspects of thin-film flow can be found in the recent review articles by Oron, Davis & Bankoff (1997) and Myers (1998), and in the volumes edited by Kistler & Schweizer (1997) and Batchelor, Moffatt & Worster (2000).

Consensus-based optimisation (CBO) is a versatile multi-particle metaheuristic optimisation method suitable for performing non-convex and non-smooth global optimisations in high dimensions. It has proven effective in various applications while at the same time being amenable to a theoretical convergence analysis. In this paper, we explore a variant of CBO, which incorporates truncated noise in order to enhance the well-behavedness of the statistics of the law of the dynamics. By introducing this additional truncation in the noise term of the CBO dynamics, we achieve that, in contrast to the original version, higher moments of the law of the particle system can be effectively bounded. As a result, our proposed variant exhibits enhanced convergence performance, allowing in particular for wider flexibility in choosing the noise parameter of the method as we confirm experimentally. By analysing the time evolution of the Wasserstein-$2$ distance between the empirical measure of the interacting particle system and the global minimiser of the objective function, we rigorously prove convergence in expectation of the proposed CBO variant requiring only minimal assumptions on the objective function and on the initialisation. Numerical evidences demonstrate the benefit of truncating the noise in CBO.

Phosphorus diffusion in silicon shows a number of anomalous effects, and we apply asymptotic methods to a model problem which includes most of these. Both constant surface concentration problems and the diffusion of implanted dopant are considered. An unusual feature of the model is the non-local dependence of the tail diffusivity on the peak concentration.

Nonintegrable differential-difference equations are constructed which support two-kink and two-soliton solutions. These equations are related to the discrete Burgers hierarchy and a discrete form of the Korteweg-de Vries equation. In particular, discretisations of equations related to the Fitzhugh-Nagumo-Kolmogorov-Petrovskii-Piskunov, Satsuma-Burgers-Huxley equations are derived. Methods presented here can also be used to derive non-integrable differential-difference equations describing the elastic collision of more than two kinks or solitary waves.

Linear Fredholm operators (J, k) of the form u(x, t) = (J, k) v (x, t) ≡ J(t) v (x, t)+ k(t, s) v (x, s)ds can be found which map solutions v of the linear matrix system of n partial differential equations ∂u/∂t = DLv + Bv, into solutions u of the like system ∂u/∂t = DLu + Au, when the diagonal matrix D with positive elements and the matrices A and B commute with the linear, scalar operator L. For solution sets in appropriate function spaces, this mapping (J, k) is unique, independent of L, and 1–1 onto if it preserves initial values so that u(x, 0) = v(x, 0). When the set of solutions is restricted to those with zero initial values, this uniqueness aspect of (j, k) breaks down, and there are many linear maps of this Fredholm form which preserve the zero initial conditions and map all such solutions of the first equation in the appropriate function space into solutions of the second. When L is an unbounded operator like the Laplacian ∇2, the initial value problems have many solutions depending on the values of u and u on the boundaries of the region of the solutions, as well as their values in the region at time t = 0. Danckwerts introduced the concept of a ‘constant preserving’ map of Volterra form mapping solutions of the scalar diffusion equation ∂u/∂t = D∇2u, which are initially zero, onto solutions of a scalar diffusion equation ∂u/∂t = D∇2u + Au, with a linear, homogeneous, constant coefficient source term. This concept of a ‘constant preserving’ map extends to the nth order matrix Fredholm maps described. A map (J, k∗) is said to be constant preserving if for all constant n x n matrices C, (J, k∗)C = C, and hence J(t) + k∗(t, s)ds = I, where I is the unit matrix. In the restricted solution spaces where u(x, 0) and v(x, 0) are zero, there is a unique 1–1 onto Danckwerts map of this type transforming solutions of the v equation into solutions of the u equation. In the cases where the coupling matrix A is constant and B is zero, the kernel k∗ of the Danckwerts map can be expressed in terms of the kernel k for the more general Fredholm map associated with the same equations, but mapping the larger solution sets containing elements which need not vanish at time zero. The Danckwerts mapping is used to establish a generalized Ussing flux-ratio theorem for some coupled diffusion problems involving several chemically interacting components.

The main result of this paper is a non-uniqueness theorem for the self-similar solutions of a model for phase transitions in binary alloys. The reason for this non-uniqueness is the discontinuity in the coefficients of heat conduction and diffusion at the inter-phase. Also the existence of a self-similar solution and the stability criterion are discussed.

The relationship between mathematical models of active parameters for transmission lines is studied. Treating transmission lines as a series of differential lumped circuits, we show that pairs of line parameters for inductance and capacitance per unit length must satisfy one of two constraints. One of these is a symmetry condition, which is satisfied by passive (i.e. constant) parameters. If a parameter pair satisfies either of these constraints, the energy per unit length and power loss in the line may be written as integrals of known functions, no matter what the pair's dependence on line current and voltage may be. Potential applications of these results to other subject areas are discussed.

Vibration energy harvesting aims to harness the energy of ambient random vibrations for power generation, particularly in small-scale devices. Typically, stochastic excitation driving the harvester is modelled as a Brownian process and the dynamics are studied in the equilibrium state. However, non-Brownian excitations are of interest, particularly in the nonequilibrium regime of the dynamics. In this work we study the nonequilibrium dynamics of a generic piezoelectric harvester driven by Brownian as well as (non-Brownian) Lévy flight excitation, both in the linear and the Duffing regimes. Both the monostable and the bistable cases of the Duffing regime are studied. The first set of results demonstrate that Lévy flight excitation results in higher expectation values of harvested power. In particular, it is shown that increasing the noise intensity leads to a significant increase in power output. It is also shown that a linearly coupled array of nonlinear harvesters yields improved power output for tailored values of coupling coefficients. The second set of results show that Lévy flight excitation fundamentally alters the bifurcation characteristics of the dynamics. Together, the results underscore the importance of non-Brownian excitation characterised by Lévy flight in vibration energy harvesting, both from a theoretical viewpoint and from the perspective of practical applications.

A mathematical model of the voltage drop which arises in on-chip power distribution networks is used to compare the maximum voltage drop in the case of different geometric arrangements of the pads supplying power to the chip. These include the square or Manhattan power pad arrangement, which currently predominates, as well as equilateral triangular and hexagonal arrangements. In agreement with the findings in the literature and with physical and SPICE models, the equilateral triangular power pad arrangement is found to minimize the maximum voltage drop. This headline finding is a consequence of relatively simple formulas for the voltage drop, with explicit error bounds, which are established using complex analysis techniques, and elliptic functions in particular.

Considering a coordinate-free formulation of helical symmetry rather than more traditional definitions based on coordinates, we discuss basic properties of helical vector fields and compare results from the literature obtained with other approaches. In particular, we discuss the role of the stream function for the topology of the streamline pattern in incompressible flows. On this basis, we perform a comprehensive study of the topology of the flow field generated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin (Hardin, J. C. 1982 Phys. Fluids25, 1949–1952) contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Okulov, V. L. 1995 Russ. J.Eng. Thermophys.5, 63–75) we obtain a closed-form approximation which is considerably easier to analyse. Critical points of the stream function can be found from the zeroes of a single real function of one variable, and we show that three different flow topologies can occur, depending on a single dimensionless parameter. By including the self-induced velocity on the vortex filament by a localised induction approximation, the stream function is slightly modified and an extra parameter is introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combination of parameters.

This special issue of the European Journal of Applied Mathematics is devoted to papers on free boundary problems, largely for Hele-Shaw or Stokes flows. The majority of the papers were presented at a meetingSupport from the London Mathematical Society is gratefully acknowledged. on these topics held in Oxford in August 1998, 100 years after H. S. Hele-Shaw first described his cell [2], and 40 years on from the classic experiment of Saffman & Taylor [4]. The conference delegates sent a message of good wishes to P. Ya Polubarinova-Kochina (below), whose paper in 1945 [3], together with that of Galin in the same year [1], may be said to have initiated the modern study of the Hele-Shaw free boundary problem. We learned with great sadness of Polubarinova-Kochina's death this year, at the age of 100; her active scientific career spanned some 75 years.

As part of the preparation for the conference, with the assistance of K. A. Gillow we have assembled a 600-paper bibliography on Hele-Shaw and Stokes flow; it can be found at www.maths.ox.ac.uk/˜howison/Hele-Shaw/. We hope it will be useful to the many researchers in these areas, which even after a century of investigation will clearly retain their mathematical and practical interest for many years to come.

A three-dimensional model of the varistor device is proposed. The thermal and electric conductivity of the material are taken to depend, in addition to the electric potential, on the temperature. Two theorems of existence and uniqueness of solutions for the boundary-value problem which determine the potential and the temperature inside the device are proposed. Levy–Caccioppoli global inversion theorem is used for the proof.

In this paper a four-dimensional macroeconomic model of a small open economy, describing the development of income, capital stock, interest rate and money stock, which was constructed in [5] (Makovínyiová, K. & Zimka, R. (2009) On stability in generalized Schinasi's macroeconomic model under fixed exchange rates. Tatra Mt. Math. Publ. 43, 115–122), is analysed. Sufficient conditions for the existence of one pair of purely imaginary eigenvalues and two eigenvalues with negative real parts in the linear approximation matrix of the model are found. Formulae for the calculation of the bifurcation coefficients of the model are derived. A statement about the existence of limit cycles is made. A numerical example is given illustrating the results.

The Polubarinova–Galin equation describes a parametric conformal map for the zero-surface-tension Hele-Shaw flow driven by injection or suction. In this paper, we classify degree three polynomial solutions to the Polubarinova–Galin equation into three categories: global solutions, solutions which can be continued after blow-up and solutions which cannot be continued after blow-up. The coefficient region of the initial functions in each category is obtained.

The competition between inertia and solidification for the high-Reynolds-number flow of molten aluminium across a cool solid aluminium surface is investigated. A two-dimensional molten aluminium droplet is of finite extent and is surrounded by a passive gas. The droplet initially freezes due to rapid thermal conduction into the solid. Depending on the initial velocity of the molten aluminium, one of two situations may develop: (i) If the molten aluminium has a non-decreasing initial velocity profile, solidification continues until the passing of the trailing edge of the liquid/gas interface or the flow is engulfed; (ii) If the molten aluminium has a decreasing initial velocity profile, the droplet narrows and thickens resulting in a reduction in the heat flux and in the rate of solidification; this will eventually lead to fluid clumping and shock formation. The rate of solidification may also be reduced by increasing the ambient temperature. The results are interpreted in terms of the recast observed during the solidification phase of laser percussion drilling.

In this paper, we consider rational or multi-cut solutions to Hele-Shaw flow driven by suction. A lower bound of the distance between the moving boundary and the suction point is given and this lower bound enables us to find necessary conditions for rational or multi-cut solutions to blow up due to the boundary touching the sink.