We are concerned with the problem of determining the diffusivity D of a diffusion process governed by the equation ut = (Dux)x', under the assumption that D depends on u. The main point consists in the observation that there exist solutions of travelling-wave type and that the dependence D = D(u) can be explicitly found in terms of the profile of such solutions. The property of finite propagation speed is required for this method to work. We propose two concrete implementations of the inverse problem, and give a rigorous mathematical proof of our statements. We also describe the application of the travelling-wave method to another interesting class of nonlinear parabolic equations.

The distortion of a two-dimensional bubble (or drop) in a corner flow of an inviscid incompressible fluid is considered. Numerical solutions are obtained by series truncation. The results confirm and extend previous calculations.

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 → +∞.

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 determines the asymptotic solution of certain initial-boundary value problems for singularly-perturbed reaction-diffusion equations, including the Allen–Cahn and Cahn–Hilliard equations, on bounded one-dimensional spatial domains for r[ges ]0. Attention is focused on the metastable evolution of a transition layer over an asymptotically exponentially-long time interval.

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.

We develop a mathematical model in hydrodynamic lubrication that takes into account three phenomena: cavitation, surface roughness and compressibility of the fluid. Like the classical Reynolds equation, the model is mass preserving. We compute the homogenized coefficients in the case of unidirectional roughness. A one-dimensional problem is also solved explicitly.

In this paper, we perform a sensitivity analysis for shape optimization problems arising in models we suggest for a lithotripter. More precisely, we use two models based on high intensity ultrasound focusing by an acoustic lens, where through changing the shape of the lens we try to achieve a favourable focusing. The models are based on acoustic wave equations with piecewise constant coefficients, a linear and a nonlinear one, where for the nonlinear one we use the Westervelt equation. The sensitivity analysis is performed using an adjoint approach.

This paper concerns the reflection of high-frequency, monochromatic linear waves of wavenumber k(≫ 1) from smooth boundaries which are O(k−1/2) perturbations away from either a specified near-planar boundary or else from a given smooth, two-dimensional curve of general O(1) curvature. For each class of perturbed boundary, we will consider separately plane and cylindrical wave incidence, with general amplitude profiles of each type of incident field. This interfacial perturbation scaling is canonical in the sense that a ray approach requires a modification to the standard WKBJ ‘ray ansatz’ which, in turn, leads to a leading-order amplitude (or ‘transport’) equation which includes an extra term absent in a standard application of the geometrical theory of diffraction. This extra term is unique to this scaling, and the afore-mentioned modification that is required is an application of a generalised type of ray expansion first posed by F. G. Friedlander and J. B. Keller (1955 Commun. Pure Appl. Math.6, 387–394).

Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We consider general interpolation functions, which give rise to a variety of different dynamics, for example, the nonlocal interaction dynamics coming from a solution-dependent velocity field. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen.

We apply the method of matched asymptotic expansions to link the outgoing wave solution at infinity of the differential equation y″(x)+(lgr;+єxn)y(x) = 0, x∈(0, ∞),λ∈Cє>0, n∈N across the turning point x = (—λ/є)1/n nearest to the positive real axis to a linear homogeneous boundary condition at the origin. The equation with n = 1 models the leakage of energy from the core of a bent fibre optic waveguide, the rate of leakage corresponding to Im λ, which was shown to be exponentially small like O(exp[— 1/є]) by Paris & Wood (1989). The extension n = 2 by Brazel et al. (1990) obtained Im λ = O(exp[— 1/є1/2]). Both these papers involve delicate analysis of the asymptotics of special functions near to Stokes' lines. When n > 2 no special functions are available, and completely different methods must be employed to obtain the result Im λ = O(exp[— 1/є1/n]).

In this paper, we study a class of linear functional differential equations of which the pantograph equation is a prominent member. Specifically, we study the existence of solutions holomorphic at a fixed point of the functional argument. The local theory for equations with attracting fixed points is known [17, 13], but little is known about the case where the fixed point is repelling. We formulate an eigenvalue problem for the repelling fixed point case and show that the corresponding spectrum is discrete. Hence, that holomorphic solutions occur only as special cases. The second order pantograph equation is used to illustrate this result. A key step in this process is to reformulate the problem in terms of a compact operator. Aside from exploiting well known results for the spectra of such operators, we use results such as the Fredholm Alternative to derive existence results for the non-homogeneous problem.

Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.

The characterisation of those shapes that can be made by the gravity sag-bending manufacturing process used to produce car windscreens and lenses is modelled as an inverse problem in linear plate theory. The corresponding second-order partial differential equation for the Young's modulus is shown to change type (possibly several times) for certain target shapes. We consider the implications of this behaviour for the existence and uniqueness of solutions of the inverse problem for some frame geometries. In particular, we show that no general boundary conditions for the inverse problem can be prescribed if it is desired to achieve certain kinds of target shapes.

Similar to how Hopf–Lax–Oleinik-type formula yield variational solutions for Hamilton–Jacobi equations on Euclidean space, optimal mass transportations can sometimes provide variational formulations for solutions of certain mean-field games. We investigate here the particular case of transports that maximize and minimize the following ‘ballistic’ cost functional on phase space TM, which propagates Brenier’s transport along a Lagrangian L,

$$b_T(v, x):=\inf\left\{\langle v, \gamma (0)\rangle +\int_0^TL(t, \gamma (t), {\dot \gamma}(t))\, dt; \gamma \in C^1([0, T], M); \gamma(T)=x\right\}\!,$$

$M = \mathbb{R}^d$

\begin{align*} \underline{B}_T^s(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t, X,\beta(t,X))\,dt\right]\!; X\in \mathcal{A}, V\sim\mu,X_T\sim \nu\right\}\!, \end{align*}

$\mathcal{A}$

This paper establishes the existence of smooth solutions for the Doi–Edwards rheological model of viscoelastic polymer fluids in shear flows. The problem turns out to be formally equivalent to a K-BKZ equation but with constitutive functions spanning beyond the usual mathematical framework. We prove, for small enough initial data, that the solution remains in the domain of hyperbolicity of the equation for all t≥0.

We present a new asymptotic formula for the maximum static voltage in a simplified model for on-chip power distribution networks of array bonded integrated circuits. In this model the voltage is the solution of the Poisson's equation in an infinite planar domain whose boundary is an array of circular pads of radius ϵ, and we deal with the singular limit ϵ → 0 case. In comparison with approximations that appear in the electronics engineering literature, our formula is more complete, since we have obtained terms up to order ϵ15. A procedure will be presented to compute all the successive terms, which can be interpreted by using multipole solutions of equations involving spatial derivatives of δ-functions. To deduce the formula, we use the method of matched asymptotic expansions. Our results are completely analytical and we make an extensive use of special functions and the Gauss constant G.

Biofilms, and collections of embedded microbial communities, present structural heterogeneities with functional consequences for important processes, such as transport. The origin of such structures has been unclear. Here, we propose that they can arise as a consequence of diffusive transport limitation. To illustrate, a model allowing internal heterogeneity is developed. Linear analysis is applied to a simplified version of the model suggesting that heterogeneity forms on (or below) the active layer length, a length scale that may not be suitable for homogenization, with non-trivial implications for system scale properties such as reduction in system-wide diffusive transport efficiency. Numerics suggest that the simplified model provides useful insight into behaviour of the full model. We then show examples based on microcolony formation in host domains and argue that internal heterogeneity can be related to community function.

We consider the steady two-dimensional thin-film version of a problem concerning a weightless non-isothermal free fluid film subject to thermocapillarity, proposed and analysed by Pukhnachev and co-workers. Specifically, we extend and correct the paper by Karabut and Pukhnachev (J. Appl. Mech. Tech. Phys. 49, 568–579, 2008), in which the problem is solved numerically, and in which it is claimed that there exists a unique solution for any value of a prescribed heat-flux parameter in the model. We present a closed-form (parametric) solution of the problem, and from this show that, on the contrary, solutions exist only when the heat-flux parameter is less than a critical value found numerically by Karabut and Pukhnachev, and that when this condition is satisfied there are in fact two solutions, one of which recovers that obtained numerically by Karabut and Pukhnachev, the other being new.

The paper is devoted to convective instability of reaction fronts. New approaches are developed to study some eigenvalue problems arising in chemical hydrodynamics. For gaseous combustion in the case of equality of transport coefficients, a linear stability analysis of an upward propagating front is carried out. A minimax representation of the stability boundary is obtained.