We introduce a free boundary model to study the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length as a function of the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

This paper looks at adapting the method of Medvedev and Scaillet for pricing short-term American options to evaluate short-term convertible bonds. However unlike their method, we provide explicit formulae for the coefficients of our series solution. This means that we do not need to solve complicated recursive systems, and can efficiently provide fast solutions. We also compare the method with numerical solutions, and find that it performs extremely well, giving accurate bond prices as well as accurate optimal conversion prices.

We study the problem of stopping a Brownian bridge X in order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.

Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to “shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry.

In this paper we analyze in detail the equilibrium configurations of classical polytropic stars with a multi-parametric differential rotation law of the literature using the standard numerical method introduced by Eriguchi and Mueller. Specifically we numerically investigate the parameters’ space associated with the velocity field characterizing both equilibrium and non-equilibrium configurations for which the stability condition is violated or the mass-shedding criterion is verified.

British put options are financial derivatives with an early exercise feature whereby on payoff, the holder receives the best prediction of the European put payoff under the hypothesis that the true drift of the stock price is equal to a contract drift. In this paper, we derive simple analytic approximations for the optimal exercise boundary and the option valuation, valid for short expiry times – which is a common feature of most options traded in the market. Empirical results show that the approximations provide accurate results for expiries of at least up to two months.

The superheating that usually occurs when a solid is melted by volumetric heating can produce irregular solid–liquid interfaces. Such interfaces can be visualised in ice, where they are sometimes known as Tyndall stars. This paper describes some of the experimental observations of Tyndall stars and a mathematical model for the early stages of their evolution. The modelling is complicated by the strong crystalline anisotropy, which results in an anisotropic kinetic undercooling at the interface; it leads to an interesting class of free boundary problems that treat the melt region as infinitesimally thin.

We consider Hele-Shaw flows driven by injection of a highly shear-thinning power-law fluid (of exponent n) in the absence of surface tension. We formulate the problem in terms of the streamfunction ψ, which satisfies the p-Laplacian equation ∇·(|∇ψ|p−2∇ψ) = 0 (with p = (n+1)/n) and use the method of matched asymptotic expansions in the large n (extreme-shear-thinning) limit to find an approximate solution. The results show that significant flow occurs only in (I) segments of a (single) circle centred on the injection point, whose perimeters comprise the portion of free boundary closest to the injection point and (II) an exponentially small region around the injection point and (III) a transition region to the rest of the fluid: while the flow in the latter is exponentially slow it can be characterised in detail.

In this paper the free boundary problem for groundwater phreatic surface is represented in the form of a variational principle. It is proved that the flow domain Ω that solves the problem is a minimizer of some functional Λ(Ω). Weak solutions are introduced as minimizers of the lower semi-continuous regularization of Λ(⋅). Within this approach the existence of weak solutions is proved for a wide class of input data.

In this paper we prove first the existence and uniqueness results for the weak solution, to the stationary equations for Bingham fluid in a three dimensional bounded domain with Fourier and Tresca boundary condition; then we study the asymptotic analysis when one dimension of the fluid domain tend to zero. The strong convergence of the velocity is proved, a specific Reynolds limit equation and the limit of Tresca free boundary conditions are obtained.

We are concerned with the proof of a generalized solution to an ill-posed variational inequality. This is determined as a solution to an appropriate minimization problem involving a nonconvex functional, treated by an optimal control technique.

The behaviour of two-dimensional finite blobs of conducting viscous fluid in a Hele-Shaw cell subject to an electric field is considered. The time-dependent free boundary problem is studied both analytically using the Schwarz function of the free boundary and numerically using a boundary integral method. Various problems are considered, including (i) the behaviour of an initially circular blob of conducting fluid subject to an electric point charge located arbitrarily within the blob, (ii) the delay in cusp formation on the free boundary in sink-driven flow due to a strategically placed electric charge and (iii) the stability of exact steady solutions having both hydrodynamic and electric forcing.

Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer isinitiated by gene mutations that result in local proliferation of abnormal cells and theirmigration to other parts of the human body, a process called metastasis. The metastasizedcancer cells then interfere with the normal functions of the body, eventually leading todeath. There are two hundred types of cancer, classified by their point of origin. Most ofthem share some common features, but they also have their specific character. In thisarticle we review mathematical models of such common features and then proceed to describemodels of specific cancer diseases.

Waxy Crude Oils (WCO’s) are characterized by the presence of heavy paraffins insufficiently large concentrations. They exhibit quite complex thermodynamical andrheological behaviour and present the peculiar property of giving rise to the formation ofsegregated wax deposits, when temperature falls down the so called WAT, or Wax AppearanceTemperature. In extreme cases, segregated waxes may lead to pipeline occlusion due todeposition on cold walls. In this paper we review the mathematical models formulated todescribe: (i) wax cystallization or thawing in cooling/heating cycles; (ii) the mechanismsof mass transport in saturated non-isothermal solutions; (iii) the experimental deviceused to measure wax solubility and wax diffusivity; (iv) wax deposition in pipelinescarrying a warm, wax-saturated WCO through cold regions; (v) wax deposition accompanied bygelification during the cooling of a WCO under a thermal gradient.

The present paper introduces a tumormodel with two time scales, the time t during which the tumorgrows and the cycle time of individual cells. The model alsoincludes the effects of gene mutations on the population densityof the tumor cells. The model is formulated as a free boundaryproblem for a coupled system of elliptic, parabolic and hyperbolicequations within the tumor region, with nonlinear and nonlocalterms. Existence and uniqueness theorems are proved, andproperties of the free boundary are established.

Mathematical models of tumour spheroids, proposed since the early seventies, have beengenerally formulated in terms of a single diffusive nutrient which is critical for cell replicationand cell viability. Only recently, attempts have been made to incorporate in the models the cellenergy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH).By assuming glucose and lactate as the only fuel substrates, we propose a simple model for thecell ATP production which takes into account the main reactions that occur in the glycolytic andthe oxidative pathway. Under the assumption that cell death occurs when ATP production fallsbelow a critical level, we have studied the free boundary problem for the concentration of glucose,lactate and oxygen inside the spheroid viable rim. We show that the existence of a necrotic core isguaranteed for a sufficiently large size of the spheroid.

The aim of this work is to deduce the existence of solutionof a coupled problem arising in elastohydrodynamiclubrication. The lubricant pressure and concentration aremodelled by Reynolds equation, jointly with the free-boundaryElrod-Adams model in order to take into account cavitationphenomena. The bearing deformation is solution of Koitermodel for thin shells. The existence of solution to thevariational problem presents some difficulties: the coupledcharacter of the equations, the nonlinear multivaluedoperator associated to cavitation and the fact of writing theelastic and hydrodynamic equations on two different domains.In a first step, we regularize the Heaviside operator.Additional difficulty related to the differentdomains is circumvented by means of prolongation andrestriction operators, arriving to a regularized coupledproblem. This one is decoupled into elastic and hydrodynamicparts, and we prove the existence of a fixed point for theglobal operator. Estimations obtained for theregularized problem allow us to prove the existence ofsolution to the original one. Finally, a numerical method is proposed in orderto simulate a real journal-bearing device and illustrate the qualitative andquantitative properties of the solution.