A new class of history-dependent variational–hemivariational inequalities was recently studied in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618). There, an existence and uniqueness result was proved and used in the study of a mathematical model which describes the contact between a viscoelastic body and an obstacle. The aim of this paper is to continue the analysis of the inequalities introduced in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618) and to provide their numerical analysis. We start with a continuous dependence result. Then we introduce numerical schemes to solve the inequalities and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modelled with a viscoelastic constitutive law, the contact is given in the form of normal compliance, and friction is described with a total slip-dependent version of Coulomb's law.

We show how to find all the integrating factors and corresponding first integrals for any system of Ordinary Differential Equations (ODEs). Integrating factors are shown to be all solutions of both the adjoint system of the linearised system of ordinary differential equations and a system that represents an extra adjoint-invariance condition. We present an explicit construction formula to find the resulting first integrals in terms of integrating factors, and discuss techniques for finding integrating factors. In particular, we show how to utilize known first integrals and symmetries to find new integrating factors. Illustrative examples are given.

We consider a hybrid system consisting of a cable linked at its end to a rigid body. It is proved that such a hybrid system can be asymptotically stabilized by means of dissipative boundary feedbacks. Uniform decay estimates of energy are also established.

Recent studies of liquid films driven by competing forces due to surface tension gradients and gravity reveal that undercompressive travelling waves play an important role in the dynamics when the competing forces are comparable. In this paper, we provide a theoretical framework for assessing the spectral stability of compressive and undercompressive travelling waves in thin film models. Associated with the linear stability problem is an Evans function which vanishes precisely at eigenvalues of the linearized operator. The structure of an index related to the Evans function explains computational results for stability of compressive waves. A new formula for the index in the undercompressive case yields results consistent with stability. In considering stability of undercompressive waves to transverse perturbations, there is an apparent inconsistency between long-wave asymptotics of the largest eigenvalue and its actual behaviour. We show that this paradox is due to the unusual structure of the eigenfunctions and we construct a revised long-wave asymptotics. We conclude with numerical computations of the largest eigenvalue, comparisons with the asymptotic results, and several open problems associated with our findings.

For a given scalar partial differential equation (PDE), a potential variable can be introduced through a conservation law. Such a conservation law yields an equivalent system (potential system) of PDEs with the given dependent variable and the potential variable as its dependent variables. Often there is also another equivalent scalar PDE (potential equation) with the potential variable as its dependent variable. The Nonclassical Method for obtaining solutions of PDEs is a generalization of the Classical Method for obtaining invariant solutions from point symmetries admitted by a given PDE. As a prototypical example, the nonlinear heat conduction equation is used to demonstrate that the Nonclassical Method applied to a potential equation can yield new solutions (nonclassical potential solutions) of a given PDE that are unobtainable as invariant solutions from admitted point symmetries of the given PDE, a related potential system or the potential equation, or from nonclassical solutions generated by applying the Nonclassical Method ($\tau{\equiv} 1$) to the given scalar PDE.

In this paper, we provide the mathematical basis for three different magneto-acoustic imaging approaches (vibration potential tomography, magneto-acoustic tomography with magnetic induction and magneto-acoustic current imaging) and propose new algorithms for solving the inverse problem for each of them.

The wave equation $(u_{t} + u u_{x})_{x} - u = 0$ is a model for shallow water waves with Coriolis force, sound waves in a bubbly liquid and more generally “is the canonical asymptotic equation for genuinely nonlinear waves that are nondispersive as their wavelength tends to zero” in the words of Hunter [13]. This Ostrovsky–Hunter equation has steadily-translating, spatially periodic solutions which exist only when $c \leq c_{limit}$. The limiting wave (‘parabolic wave’) is exactly given by a piecewise quadratic polynomial in $x$ with a discontinuous slope at the crest. We show that near the limit, the travelling waves (‘paraboloidal waves’) can be approximated by matched asymptotic expansions: the inner solution rounds off the point while the outer solution, valid over most of the spatial domain, is to lowest order just the parabolic wave. In the opposite limit of small amplitude, we derive a Fourier-and-powers-of-amplitude expansion (‘Stokes' series’). We show that this is remarkably accurate even very close to the limiting wave and converges to the limiting wave for unit amplitude. We demonstrate also that the Fourier pseudospectral method gives first order convergence even for the slope-discontinuous parabolic wave.

We present a two-dimensional micro-scale model for crystal dissolution and precipitation in a porous medium. The local geometry of the pore is represented as a thin strip and the model allows for changes in the pore volume. A formal limiting argument, for the limit of the width of the strip going to zero, leads to a system of one-dimensional effective upscaled equations. We show that the effective equations allow for travelling-wave solutions and prove the existence and uniqueness of these. Numerical solutions of the effective equations are compared with numerical solutions of the original equations on the thin strip and with analytical results. We also show that a model from the literature that does not allow changes in the pore volume can be obtained from the present model as a formal limit.

This paper deals with the mathematical characterization of microstructure in elastic solids. We formulate our ideas in terms of rank-one convexity and identify the set of probability measures for which Jensen's inequality for this type of functions holds. This is the set of laminates. We also introduce generalized convex hulls of sets of matrices and investigate their structure.

We discuss the application of complex variable methods to Hele-Shaw flows and two-dimensional Stokes flows, both with free boundaries. We outline the theory for the former, in the case where surface tension effects at the moving boundary are ignored. We review the application of complex variable methods to Stokes flows both with and without surface tension, and we explore the parallels between the two problems. We give a detailed discussion of conserved quantities for Stokes flows, and relate them to the Schwarz function of the moving boundary and to the Baiocchi transform of the Airy stress function. We compare the results with the corresponding results for Hele-Shaw flows, the principal consequence being that for Hele-Shaw flows the singularities of the Schwarz function are controlled in the physical plane, while for Stokes flow they are controlled in an auxiliary mapping plane. We illustrate the results with the explicit solutions to specific initial value problems. The results shed light on the construction of solutions to Stokes flows with more than one driving singularity, and on the closely related issue of momentum conservation, which is important in Stokes flows, although it does not arise in Hele-Shaw flows. We also discuss blow-up of zero-surface-tension Stokes flows, and consider a class of weak solutions, valid beyond blow-up, which are obtained as the zero-surface-tension limit of flows with positive surface tension.

We consider a nonlinear, strongly coupled, parabolic system arising in the modelling of burglary in residential areas. This model appeared in Pitcher (Eur. J. Appl. Math., 2010, Vol. 21, pp. 401–419), as a more realistic version of the Short et al. (Math. Models Methods Appl. Sci., 2008, Vol. 18, pp. 1249–1267) model. The system under consideration is of chemotaxis-type and involves a logarithmic sensitivity function and specific interaction and relaxation terms. Under suitable assumptions on the data of the problem, we give a rigorous proof of the existence of a global and bounded, classical solution, thereby solving a problem left open in previous work on this model. Our proofs are based on the construction of approximate entropies and on the use of various functional inequalities. We also provide explicit numerical conditions for global existence when the domain is a square, including concrete cases involving values of the parameters which are expected to be physically relevant.

Equilibrium solutions to the one-dimensional Gierer–Meinhardt model in the form of sequences of spikes of different heights are constructed asymptotically in the limit of small activator diffusivity ε. For a pattern with k spikes, the construction yields k1 spikes that have a common small amplitude and k2 = k− k1 spikes that have a common large amplitude. A k- spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. It is shown that such solutions exist when the inhibitor diffusivity D is less than some critical value Dm that depends upon k1, on k2, and on other parameters associated with the Gierer–Meinhardt model. It is also shown that these asymmetric k-spike solutions bifurcate from the symmetric solution branch sk, for which k spikes have equal height. These asymmetric solutions provide connections between the branch sk and the other symmetric branches sj , for j = 1,…, k− 1. The stability of the asymmetric k-spike patterns with respect to the large O(1) eigenvalues and the small O(ε2) eigenvalues is also analyzed. It is found that the asymmetric patterns are stable with respect to the large O(1) eigenvalues when D > De, where De depends on k1 and k2, on certain parameters in the model, and on the specific ordering of the small and large spikes within a given k-spike sequence. Numerical values for De are obtained from numerical solutions of a matrix eigenvalue problem. Another matrix eigenvalue problem that determines the small eigenvalues is derived. For the examples considered, it is shown that the bifurcating asymmetric branches are all unstable with respect to these small eigenvalues.

This paper focuses on the role of a government of a large population of interacting agents as a meanfield optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a Partial Differential Equation of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a Γ -convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle.

This paper examines the effect of anisotropic growth on the evolution of mechanical stresses in a linear-elastic model of a growing, avascular tumour. This represents an important improvement on previous linear-elastic models of tissue growth since it has been shown recently that spatially-varying isotropic growth of linear-elastic tissues does not afford the necessary stress-relaxation for a steady-state stress distribution upon reaching a nutrient-regulated equilibrium size. Time-dependent numerical solutions are developed using a Lax-Wendroff scheme, which show the evolution of the tissue stress distributions over a period of growth until a steady-state is reached. These results are compared with the steady-state solutions predicted by the model equations, and key parameters influencing these steady-state distributions are identified. Recommendations for further extensions and applications of this model are proposed.

We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface. The motion is driven by a constant surface tension acting at the free boundary so that, with the effects of gravity ignored, one expects the boundary to approach a circular form as time evolves. It is shown that, if at some initial instant the region occupied by the fluid is given by a rational conformal map of the unit disc, then it must retain this property as long as the region remains simply-connected. Moreover, its evolution may be described analytically; in simple cases this description is explicit, but in more complicated problems the numerical integration of a system of first order differential equations may be required.

For an external problem in IRd (d=2, 3) such that the unknown function satisfies the wave equation outside a finite domain, we generate artificial boundary conditions transparent to outgoing waves. These conditions permit an equivalent replacement of the original external problem by the problem inside the artificial boundary which is a circle (d=2) or a sphere (d=3): The questions of numerical implementation of the artificial conditions (that are non-local in both space and time) are considered. Special attention is paid to the reduction of necessary computational resources; in particular, a way of incorporating these conditions into numerical methods which makes the computational formulae local in time is suggested. The aspects of treating artificial boundaries of a non-spherical shape are discussed. Numerical examples of two- and three-dimensional scattering problems demonstrate the accuracy of proposed artificial boundary conditions.

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, the b-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations.

In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013J. Solid State Electr.17, 467–479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimension-two Turing--Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing–Hopf dynamics.

A variety of mass preserving moving boundary problems for the thin film equation, ut = −(unuxxx)x, are derived (by formal asymptotics) from a number of regularisations, the case in which the substrate is covered by a very thin pre-wetting film being discussed in most detail. Some of the properties of the solutions selected in this fashion are described, and the full range of possible mass preserving non-negative solutions is outlined.