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Preview: IMA Journal of Applied Mathematics - current issue

IMA Journal of Applied Mathematics Current Issue

Published: Mon, 03 Jul 2017 00:00:00 GMT

Last Build Date: Wed, 19 Jul 2017 10:47:45 GMT


Limit cycles bifurcating from discontinuous centres


In this article, we study limit cycles in discontinuous piecewise linear vector fields in $\mathbb R^2$ and $\mathbb R^3$. More precisely, we address the problem of understanding the dynamics around a degenerated two-fold singularity in $\mathbb R^2$ and $\mathbb R^3$, where it is also called T-singularity, which behave as discontinuous centres after suitable perturbations of the separation boundaries. We prove that, in both systems, it is possible to obtain $k$ hyperbolic limit cycles bifurcating from these discontinuous centres, for any positive integer $k$. The same holds if $k$ is infinity.

Finite difference methods of the spatial fractional Black–Schloes equation for a European call option


We study finite difference methods for solving a spatial fractional Black–Scholes equation on a finite domain which extends the classical Black–Scholes model by replacing the second-order spatial derivative with a fractional derivative. We design first order and second order implicit numerical schemes for the spatial fractional Black–Scholes equation, and then analyse their stability and convergence. Some numerical examples are given to confirm our theoretic results and discuss the influence of the order of the fractional derivative on the option price.

Spatio-temporal patterns in a diffusive model with non-local delay effect


Steady states and periodic oscillations are two important solution forms in reaction–diffusion equations. In this article, we deal with a diffusive Lotka–Volterra system with non-local delay effect and Dirichlet boundary condition. Firstly, the existence and multiplicity of spatially non-homogeneous synchronous/mirror-reflecting steady-state solutions are investigated by means of Lyapunov–Schmidt reduction. Then using centre manifold reduction, normal form analysis and equivariant bifurcation theory, we show that each of the standard Hopf bifurcations gives rise to only one branch of spatially non-homogeneous time-periodic synchronous waves, whereas each of the equivariant Hopf bifurcations gives rise to eight branches of periodic orbits, including two phase-locked oscillations, three mirror-reflecting waves and three standing waves. In particular, we derive the formula for determining the Hopf bifurcation direction and stability of these Hopf bifurcating periodic orbits. These theoretical results give an accurate picture of the bifurcation structure, in the neighbourhood of the bifurcation point, for any set of kernel functions.

Solitary gravity waves and free surface flows past a point vortex


Nonlinear free surface flows past a disturbance in a channel of finite depth are considered. The fluid is assumed to be incompressible and inviscid and the flow to be two-dimensional, irrotational and supercritical. The disturbance is chosen to be a point vortex. Highly accurate numerical solutions are computed. The basic idea of the numerical approach is first to develop codes to compute solitary waves and then to introduce appropriate modifications to model the point vortex. Previous results are recovered and new solutions are presented.

Radiating solitary waves in coupled Boussinesq equations


In this article, we are concerned with the analytical description of radiating solitary wave solutions of coupled regularized Boussinesq equations. This type of solution consists of a leading solitary wave with a small-amplitude co-propagating oscillatory tail and emerges from a pure solitary wave solution of a symmetric reduction of the full system. We construct an asymptotic solution, where the leading order approximation in both components is obtained as a particular solution of the regularized Boussinesq equations in the symmetric case. At the next order, the system uncouples into two linear non-homogeneous ordinary differential equations with variable coefficients, one correcting the localized part of the solution, which we find analytically, and the other describing the co-propagating oscillatory tail. This latter equation is a fourth-order ordinary differential equation and is solved approximately by two different methods, each exploiting the assumption that the leading solitary wave has a small amplitude, and thus enabling an explicit estimate for the amplitude of the oscillating tail. These estimates are compared with corresponding numerical simulations.

A note on the large-time development of the solution to an initial-value problem for a variable coefficient Korteweg-de Vries equation


In this article, we consider an initial-value problem for a variable coefficient Korteweg-de Vries equation. The normalized variable coefficient Korteweg-de Vries equation considered is given by ut+uux+eαtuxxx=0,−∞0, where $x$ and $t$ represent dimensionless distance and time respectively, and $\alpha (>0)$ is a constant. In particular, we consider the case when the initial data has a discontinuous expansive step, where $u(x,0)=u_{+}$ for $x \ge 0$ and $u(x,0)=u_{-}$ for $x<0$. The method of matched asymptotic coordinate expansions is used to obtain the large-$t$ asymptotic structure of the solution to this problem. We find that the large-$t$ attractor for the solution $u(x,t)$ of the initial-value problem is based on the integral of the standard Airy function, where u(zeα3t,t)→[(u−+2u+)3+(u+−u−)∫0(α3)13zAi(s)ds] as $t \to \infty$ with $z=x e^{-\frac{\alpha}{3}t}=O(1)$. Further, this large-$t$ attractor forms in a stretching frame of reference of thickness $x=O\left(e^{\frac{\alpha t}{3}}\right)$ as $ t \to \infty$.

A revisited Johnson–Mehl–Avrami–Kolmogorov model and the evolution of grain-size distributions in steel


The classical Johnson–Mehl–Avrami–Kolmogorov approach for nucleation and growth models of diffusive phase transitions is revisited and applied to model the growth of ferrite in multiphase steels. For the prediction of mechanical properties of such steels, a deeper knowledge of the grain structure is essential. To this end, a Fokker–Planck evolution law for the volume distribution of ferrite grains is developed and shown to exhibit a log-normally distributed solution. Numerical parameter studies are given and confirm expected properties qualitatively. As a preparation for future work on parameter identification, a strategy is presented for the comparison of volume distributions with area distributions experimentally gained from polished micrograph sections.

A nonlinear elasticity approach to modelling the collapse of a shelled microbubble


There is considerable interest in using shelled microbubbles as a transportation mechanism for localized drug delivery, specifically in the treatment of various cancers. In this paper a theoretical model is proposed which predicts the dynamics of an oscillating shelled microbubble. A neo-Hookean, compressible strain energy density function is used to model the potential energy per unit volume of the shell. The shell is stressed by applying a series of small radially directed stress steps to the inner surface of the shell whilst the outer surface is traction free. Once a certain radial deformation is reached, the stress load at the inner radius is switched off causing the shell to collapse and oscillate about its equilibrium (stress free) position. The inflated shell configuration is used as an initial condition to model the time evolving collapse phase of the shell. The collapse phase is modelled by applying the momentum balance law and mass conservation. The dynamical model which results is then used to predict the collapse time of the shelled microbubble as it oscillates about its equilibrium position. A linear approximation is used in order to gain analytical insight into both the quasistatic inflationary and the oscillating phases of the shelled microbubble. Results from the linearized model are then analysed which show the influence of the shell’s thickness, Poisson ratio and shear modulus on the rate of oscillation of the shelled microbubble. The nonlinear model for the quasistatic state is solved numerically and compared to the linearized quasistatic solution. At present, there is no solution to the nonlinear collapsed state. This is a future area of research for the current authors.

On the blow-up of some complex solutions of the 3D Navier–Stokes equations: theoretical predictions and computer simulations


We consider some complex-valued solutions of the Navier–Stokes equations in ${\mathbb R}^{3}$ for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the ‘fluid’ remains quiet.

A two-scale Stefan problem arising in a model for tree sap exudation


The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze–thaw cycles, gives rise to an interesting multiscale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.