A three-phase ensemble-averaged model is developed for the flow of water and air through a deformable porous matrix. The model predicts a separation of the flow into saturated and unsaturated regions. The model is closed by proposing an experimentally-motivated heuristic elastic law which allows large-strain nonlinear behaviour to be treated in a relatively straightforward manner. The equations are applied to flow in the ‘nip’ area of a roll press machine whose function is to squeeze water out of wet paper as part of the manufacturing process. By exploiting the thin-layer limit suggested by the geometry of the nip, the problem is reduced to a nonlinear convection-diffusion equation with one free boundary. A numerical method is proposed for determining the flow and sample simulations are presented.

Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang & Shen's [12] recent inpainting schemes based on the bounded variation and the elastica [11] image models. In this paper, we propose two novel inpainting models based on the Mumford–Shah image model [41], and its high order correction – the Mumford–Shah–Euler image model. We also present their efficient numerical realization based on the Γ-convergence approximations of Ambrosio & Tortorelli [2, 3] and De Giorgi [21].

The paper compares computational aspects of four approaches to compute conservation laws of single Differential Equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints for the conservation laws can be specified. Examples include new conservation laws that are non-polynomial in the functions, that have an explicit variable dependence and families of conservation laws involving arbitrary functions. The following equations are investigated in examples: Ito, Liouville, Burgers, Kadomtsev–Petviashvili, Karney–Sen–Chu–Verheest, Boussinesq, Tzetzeica, Benney.

We examine how symmetry and computer algebra can assist in solving the Cauchy problem for Pfaffian systems. We use recent results on integrating Frobenius integrable distributions via solvable symmetry structures to develop two techniques that when used in conjunction with symmetry determination software DIMSYM, allow us to solve the Cauchy problem for the special situation when there exists a one-dimensional Cauchy characteristic space. We also illustrate how our work can assist in extracting local solutions of a certain class of first and second order non-linear partial differential equations.

We consider an American call option and let C(S, T0) be the price of an option corresponding to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various asymptotic limits. These include situations where the interest and dividend rates are large or small, compared to the volatility of the asset. We also analyze the optimal exercise boundary for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies, or a nonlinear integral equation that is satisfied by the optimal exercise boundary.

The behaviour of liquid crystal materials used in display devices is discussed. The underlying continuum theory developed by Frank, Ericksen and Leslie for describing this behaviour is reviewed. Particular attention is paid to the approximations and extensions relevant to existing device technology areas where mathematical analysis would aid device development. To illustrate some of the special behaviour of liquid crystals and in order to demonstrate the techniques employed, the specific case of a nematic liquid crystal held between two parallel electrical conductors is considered. It has long been known that there is a critical voltage below which the internal elastic strength of the liquid crystal exceeds the electric forces and hence the system remains undeformed from its base state. This bifurcation behaviour is called the Freedericksz transition. Conventional analytic analysis of this problem normally considers a magnetic, rather than electric, field or a near-transition voltage since in these cases the electromagnetic field structure decouples from the rest of the problem. Here we consider more practical situations where the electromagnetic field interacts with the liquid crystal deformation. Assuming strong anchoring at surfaces and a one dimensional deformation, three nondimensional parameters are identified. These relate to the applied voltage, the anisotropy of the electrical permittivity of the liquid crystal, and to the anisotropy of the elastic stiffness of the liquid crystal. The analysis uses asymptotic methods to determine the solution in a numerous of different regimes defined by physically relevant limiting cases of the parameters. In particular, results are presented showing the delicate balance between an anisotropic material trying to push the electric field away from regions of large deformation and the deformation trying to be maximum in regions of high electric field.

The free boundary model of diffusion-induced grain boundary motion derived in Cahn et al. [3], Fife et al. [6] and Cahn & Penrose [4] is extended, in the case of thin metallic films, to account for bidirectional motion, together with the appearance of S-shapes and double seam configurations. These are often observed in the laboratory. Computer simulations based on the extended model are given to illustrate these and other features of bidirectional motion. More generally, the extension accounts for the motion of grain boundaries whose traces on the film's surface are curved. The new free boundary model is one of forced motion by curvature, the forcing term possibly changing sign due to the bidirectionality. The thin film model is derived systematically under explicit assumptions, and an adjustment for grooving is included.

We consider a one-dimensional stochastic model of sediment deposition in which the complete time history of sedimentation is the sum of a linear trend and a fractional Brownian motion wH(t) with self-similarity parameter H ∈ (0, 1). The thickness of the sedimentary layer as a function of time, d(t), looks like the Cantor staircase. The Hausdorff dimension of the points of growth of d(t) is found. We obtain the statistical distribution of gaps in the sedimentary record, periods of time during which the sediments have been eroded. These gaps define sedimentary unconformities. In the case H = 1/2 we obtain the statistical distribution of layer thicknesses between unconformities and investigate the multifractality of d(t). We show that the multifractal structures of d(t) and the local time function of Brownian motion are identical; hence d(t) is not a standard multifractal object. It follows that natural statistics based on local estimates of the sedimentation rate produce contradictory estimates of the range of local dimension for d(t). The physical object d(t) is interesting in that it involves the above anomalies, and also in its mechanism of fractality generation, which is different from the traditional multiplicative process.

We consider the Ginzburg–Landau equation in dimension two. We introduce a key notion of the vortex (interaction) energy. It is defined by minimizing the renormalized Ginzburg–Landau (free) energy functional over functions with a given set of zeros of given local indices. We find the asymptotic behaviour of the vortex energy as the inter-vortex distances grow. The leading term of the asymptotic expansion is the vortex self-energy while the next term is the classical Kirchhoff–Onsager Hamiltonian. To derive this expansion we use several novel techniques.

We prove the existence of energy equilibria of the di-block copolymer problem in the unit disk. They consist of concentrically layered micro-domains rich in one of the two monomer building units. We construct them by solving the proper singular limit of the free energy functional. The same limit also explains how under a dynamic law of the free energy, circular interfaces of non-equilibria may move to the origin and vanish, or collapse to each other, thereby reducing the number of layers.

A general formalism is described whereby some regular singular points are effectively removed and substantial simplifications ensue for a class of Fuchsian ordinary differential equations, and related confluent equations. These simplifications follow provided the exponents at the singular points satisfy certain relations; explicit, illustrative examples are constructed to demonstrate the ideas.

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. A method of multiple scales, with restrictions on the form of the microscopic free boundaries, is used to derive a macroscopic model for the mushy region. The final model depends both on the microscopic structure and on how the free-boundary temperature varies with curvature (Gibbs–Thomson effect), kinetic undercooling, or, for an alloy, composition.

Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions). Moreover, we discuss the approximations of such statistical quantities by the spectral Galerkin method. This is an important question, as the numerical computation of statistical quantities plays a key role in the verification of the models.

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.

We analyze the front structures evolving under the difference-differential equation ∂tCj = −Cj+C2j−1 from initial conditions 0 [les ] Cj(0) [les ] 1 such that Cj(0) → 1 as j → ∞ suffciently fast. We show that the velocity v(t) of the front converges to a constant value v* according to v(t) = v*−3/(2λ*t)+(3√π/2) Dλ*/(λ*2Dt)3/2+[Oscr ](1/t2). Here v*, λ* and D are determined by the properties of the equation linearized around Cj = 1. The same asymptotic expression is valid for fronts in the nonlinear diffusion equation, where the values of the parameters λ*, v* and D are specific to the equation. The identity of methods and results for both equations is due to a common propagation mechanism of these so-called pulled fronts. This gives reasons to believe that this universal algebraic convergence actually occurs in an even larger class of equations.

We consider the one-dimensional and two-dimensional filtration-absorption equation ut = uΔu−(c−1)(∇u)2. The one-dimensional case was considered previously by Barenblatt et al. [4], where a special class of self-similar solutions was introduced. By the analogy with the 1D case we construct a family of axisymmetric solutions in 2D. We demonstrate numerically that the self-similar solutions obtained attract the solutions of non-self-similar Cauchy problems having the initial condition of compact support. The main analytical result we provide is the linear stability of the above self-similar solutions both in the 1D case and in the 2D case.

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 consider the two-dimensional Rayleigh–Taylor problem for the dynamics of the free interface Γ between two layers of immiscible viscous liquids. For a slow flow model (which corresponds to the case of a small relative jump of density) and under sufficiently wide assumptions on the geometry of Γ, we analyze the time dynamics of Γ. In particular, we prove that its increase in time t is bounded by an exponential function with exponent independent of Γ.

We consider a mathematical model for the quasi-static deformation of a thinning sheet. The model couples a first-order equation for the thickness of the sheet to a prescribed curvature equation for the displacement of the sheet. We prove a local in time existence and uniqueness theorem for this system when the sheet can be written as a graph. A contact problem is formulated for a sheet constrained to be above a mould. Finally we present some computational results.

We study the annihilation dynamics arising in the KPP-Fisher equation, proposed by Fisher in 1936 to model the propagation of a mutant gene and subsequently studied rigorously in the seminal work of Kolmogorov, Petrovskii and Piskunov. The approach is via a comparison theorem, where the comparison functions satisfy equations which are linearizable to the heat equation. In some sense, we have obtained a ‘linearization’ of the KPP-Fisher equation.