A generalization of the Keller–Segel model for chemotactic systems is studied. In this model there are several populations interacting via several sensitivity agents in a two-dimensional domain. The dynamics of the population is determined by a Fokker–Planck system of equations, coupled with a system of diffusion equations for the chemical agents. Conditions for global existence of solutions and equilibria are discussed, as well as the possible existence of time-periodic attractors. The analysis is based on a variational functional associated with the system.

We study the problem of designing a periodic interface between two homogeneous materials with different impedance properties, in such a way that time-harmonic waves incident on the interface over a given range of angles have minimal total reflected energy. It is shown that the problem can be ‘relaxed’ to include continuously varying profiles. A simple gradient descent minimization scheme is proposed and examples from several numerical calculations are given.

We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

In the focusing problem we seek a solution to the porous medium equation whose initial distribution is in the exterior of some compact set (e.g. a ball). At a finite time T the gas will reach all points of the initially empty region R. We construct a selfsimilar solution of the radially symmetric focusing problem. This solution is an example of a selfsimilar solution of the second kind, i.e. one in which the similarity variable cannot be determined a priori from dimensional considerations. Our solution also shows that in more than one space dimension, the velocity of the gas is infinite at the centre of R at the focusing time T.

We consider a nonlinear diffusion equation describing the planar spreading of a viscous fluid injected between an elastic sheet and an underlying rigid plane. The dynamics depends sensitively on the physical conditions at the contact line where the sheet is lifted off the plane by the fluid. We explore two possibilities for these conditions (or “regularisations”): a pre-wetted film and a constant-pressure fluid lag (a gas-filled gap between the fluid edge and the contact line). For both flat and inclined planes, we compare numerical and asymptotic solutions, identifying the distinct stages of evolution and the corresponding characteristic rates of spreading.

The equation

is a possible dimensionless version of a model for the configuration of a very long strut resting on a nonlinear elastic foundation with axial loading P. By seeking to establish the existence of homoclinic orbits connecting the zero equilibrium of (*), now regarded as defining a four dimensional dynamical system, to itself one is pursuing the so-called ‘dynamical phase-space analogy’ for the spatial configuration suggested by the form of the equation. The existence of homoclinic solutions is then interpreted as indicating the presence of spatially localized buckling of the deformed strut.

In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation. These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.

In this paper we present a new formulation of a large class of phase-field models, which describe solidification of a pure material and allow for both surface energy and interface kinetic anisotropy, in terms of the Hoffman–Cahn ξ-vector. The ξ-vector has previously been used in the context of sharp interface models, where it provides an elegant tool for the representation and analysis of interfaces with anisotropic surface energy. We show that the usual gradient-energy formulations of anisotropic phase-field models are expressed in a natural way in terms of the ξ-vector when appropriately interpreted. We use this new formulation of the phase-field equations to provide a concise derivation of the Gibbs–Thomson–Herring equation in the sharp-interface limit in three dimensions.

This paper studies many fundamental aspects of a newly proposed multi-class traffic flow model. For the first time it presents a complete discussion on the hyperbolicity of the system; and based upon this, the admissible waves of the Riemann problem are deeply investigated. Many important conclusions are made in discussions, and the related physical meanings are interpreted. For the confirmation of main conclusions, numerical examples are given at the end.

We introduce several novel and computationally efficient methods for detecting “core–periphery structure” in networks. Core–periphery structure is a type of mesoscale structure that consists of densely connected core vertices and sparsely connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that that vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which we express as a perturbation of a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically generated networks and a variety of networks constructed from real-world data sets.

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this paper we present the two approaches and discuss their modelling assumptions, notions of optimality and different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection and discuss the scope of these methods with a data example from the computer experiments literature.

New Lie and conditional symmetries of nonlinear diffusion equations with convection terms are constructed. Examples of new ansätze and exact solutions for some nonlinear equations (in particular, for the Murray and the Newell–Whitehead equations) are presented.

In technical applications, uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of Wiener (Amer. J. Math. 60(4), 897–936, 1938) was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from academics and from industry. For each of the applications, chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical and experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.

This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

The classical time-dependent drift-diffusion model for semiconductors is considered for small scaled Debye length (which is a singular perturbation parameter). The corresponding limit is carried out on both the dielectric relaxation time scale and the diffusion time scale. The latter is a quasineutral limit, and the former can be interpreted as an initial time layer problem. The main mathematical tool for the analytically rigorous singular perturbation theory of this paper is the (physical) entropy of the system.

This paper discusses the initial value problem

where A, Bi and Ci are d × d complex matrices, pi, qi ∈ (0, 1), i = 1, 2, …, and y0 is a column vector in ℂd. By using ideas from the theory of ordinary differential equations and the theory of functional equations, we give a comprehensive analysis of the asymptotic behaviour of analytic solutions of this initial value problem.

This paper is concerned with two frequency-dependent susceptible–infected–susceptible epidemic reaction–diffusion models in heterogeneous environment, with a cross-diffusion term modelling the effect that susceptible individuals tend to move away from higher concentration of infected individuals. It is first shown that the corresponding Neumann initial-boundary value problem in an n-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly in-time bounded regardless of the strength of the cross-diffusion and the spatial dimension n. It is further shown that, even in the presence of cross-diffusion, the models still admit threshold-type dynamics in terms of the basic reproduction number $\mathcal {R}_0$ – i.e. the unique disease-free equilibrium is globally stable if $\mathcal {R}_0\lt1$, while if $\mathcal {R}_0\gt1$, the disease is uniformly persistent and there is an endemic equilibrium (EE), which is globally stable in some special cases with weak chemotactic sensitivity. Our results on the asymptotic profiles of EE illustrate that restricting the motility of susceptible population may eliminate the infectious disease entirely for the first model with constant total population but fails for the second model with varying total population. In particular, this implies that such cross-diffusion does not contribute to the elimination of the infectious disease modelled by the second one.

We consider nonnegative steady-state solutions of the evolution equation

formula here

Our class of coefficients f, g allows degeneracies at h = 0, such as f(0) = 0, as well as divergences like g(0) = ±∞. We first construct steady states and study their regularity. For f, g > 0 we construct positive periodic steady states, and non-negative steady states with either zero or nonzero contact angles. For f > 0 and g < 0, we prove there are no non-constant positive periodic steady states or steady states with zero contact angle, but we do construct non-negative steady states with nonzero contact angle. In considering the volume, length (or period) and contact angle of the steady states, we find a rescaling identity that enables us to answer questions such as whether a steady state is uniquely determined by its volume and contact angle. Our tools include an improved monotonicity result for the period function of the nonlinear oscillator. We also relate the steady states and their scaling properties to a recent blow-up conjecture of Bertozzi and Pugh.