IMA Journal of Applied Mathematics - recent issues

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

Zhuang, P., Liu, F., Anh, V., Turner, I. — Wed, 30 Sep 2009 04:44:29 PDT

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and ₀D_t^1– is the Riemann–Liouville time fractional partial derivative of order 1 – . We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Analytic solution of an exterior Dirichlet problem in a non-convex domain

Baganis, G., Hadjinicolaou, M. — Wed, 30 Sep 2009 04:44:29 PDT

An exterior Dirichlet problem in a non-convex domain is solved analytically by combining two powerful methods: the well-known Kelvin transformation and the newly established method of generalized integral transforms introduced by Fokas (2001, Proc. R. Soc. A., 457, 371–393). In fact, our approach leads to an integral representation for the solution of Laplace's equation in the unbounded domain formed by the exterior of the Kelvin image of an equilateral triangle. First, we apply the Kelvin transformation of the given boundary with arbitrary data. Second, we use the Dirichlet-to-Neumann map (Dassios & Fokas, 2005, Proc. R. Soc. A., 461, 2721–2748). to obtain the unknown Neumann boundary values on the image boundary. Third, by inverting the Kelvin transformation, we derive the Neumann data on the original boundary. We demonstrate the 2D version of Kelvin transformation and we apply it to the equilateral triangle, which, through the Dirichlet-to-Neumann map, leads us in a natural way to a known integral representation of the solution.

Quasi-separation of the biharmonic partial differential equation

Everitt, W. N., Johansson, B. T., Littlejohn, L. L., Markett, C. — Wed, 30 Sep 2009 04:44:29 PDT

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Focusing and defocusing cases of the purely elliptic generalized Davey-Stewartson system

Eden, A., Gurel, T. B., Kuz, E. — Wed, 30 Sep 2009 04:44:29 PDT

We define the focusing and the defocusing cases for the purely elliptic generalized Davey–Stewartson system. These cases are mutually exclusive and exhaustive and therefore close the gap that was left in the previous studies. In the defocusing case, all solutions exist globally. In the focusing case, any initial data can be scaled to one with negative energy. The solution with the scaled initial data then blows up in finite time. We also show the existence of standing waves and the global existence and scattering of solutions with subminimal mass. Our results equally apply to the elliptic almost-cubic non-linear Schrödinger equation as described in Eden & Kuz (2009, Commun. Pure Appl. Anal.).

The Cauchy problem for coupled IMBq equations

Wang, S., Li, M. — Wed, 30 Sep 2009 04:44:29 PDT

In this paper, we study the Cauchy problem associated with two coupled IMBq equations. Under the assumptions for non-linear terms and initial data, we prove the existence and uniqueness of the global solution and give sufficient conditions of blow-up of the solution in finite time by convex methods. This supplements and improves some results by Dè Godefroy (1998, IMA J. Appl. Math., 60, 123–138).

Asymptotic solution of slender viscous jet break-up

Decent, S. P. — Wed, 30 Sep 2009 04:44:29 PDT

The break-up of a slender viscous jet is examined using the Needham–Leach asymptotic method. This method enables the calculation of the large time asymptotic structure of the model evolution equations using matched asymptotic expansions. An equation which describes the dynamics of non-linear travelling waves at large times is derived using this method. In particular, the wave speed, wavelength, growth rate and frequency of these travelling waves are determined. This provides information on how the jet breaks up, the region of break-up and the possibility for multiple break-up points. Also, this method gives information on how non-linear jets may be controlled.

Voltage and current spectra for a single-phase voltage source inverter

Cox, S. M. — Wed, 30 Sep 2009 04:44:29 PDT

An inverter converts a direct current power supply to an alternating current power supply. To do so, its output is switched at high frequency between the inputs in order to synthesize the desired alternating current output in the low-frequency part of the Fourier spectrum. Here, we calculate analytical expressions for the input and output current and voltage spectra for two inverter designs: so-called single-phase-leg and two-phase-leg inverters. The output voltage and current spectra are well known but are found here by a new, analytically compact means. More significantly, the input current spectra for the two designs are calculated here for the first time. A prior, approximate method for determining the input current spectrum (which assumes an exactly sinusoidal output current) is compared with our exact results.

Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains

Jaoua, M., Leblond, J., Mahjoub, M., Partington, J. R. — Thu, 23 Jul 2009 12:42:03 PDT

We consider the Cauchy problem of recovering both Neumann and Dirichlet data on the inner part of the boundary of an annular domain from measurements of a harmonic function on some part of the outer boundary. Using tools from complex analysis and best approximation in Hardy classes, we present a family of fast data completion algorithms which are shown to provide constructive and robust identification schemes. These are applied to the computation of an impedance or Robin coefficient and are validated by a thorough numerical study.

On the homogenization of orthotropic elastic composites by the strong-property-fluctuation theory

Duncan, A. J., Mackay, T. G., Lakhtakia, A. — Thu, 23 Jul 2009 12:42:03 PDT

The strong-property-fluctuation theory (SPFT) provides a general framework for estimating the constitutive parameters of a homogenized composite material (HCM). We developed the elastodynamic SPFT for orthotropic HCMs in order to undertake numerical studies. A specific choice of two-point covariance function—which characterizes the distributional statistics of the generally ellipsoidal particles that constitute the component materials—was implemented. Representative numerical examples revealed that the lowest-order SPFT estimate of the HCM stiffness tensor is qualitatively similar to the estimate provided by the Mori–Tanaka mean-field formalism, but the differences between the two estimates vary as the orthotropic nature of the HCM is accentuated. The second-order SPFT provides a correction to the lowest-order estimate of the HCM stiffness tensor and density. The correction, indicating effective dissipation due to scattering loss, increases as the HCM becomes less orthotropic but decreases as the correlation length becomes smaller.

An approximation for a subclass of the Riemann-Hilbert problems

Kucerovsky, D., Najafabadi, A. T. P. — Thu, 23 Jul 2009 12:42:03 PDT

Consider the problem of solving a Riemann–Hilbert problem with ‘zero index’. Abraham (2000, IMA J. Appl. Math., 65, 257–281) suggested to replace a possibly complicated kernel of a homogeneous Riemann–Hilbert problem with a Padé approximant that uniformly approximates the original kernel. Abraham's procedure fails whenever the kernel cannot be approximated uniformly by a Padé approximant (see Example 1). This article (i) provides an approximation technique to approximate solutions of a non-homogeneous Riemann–Hilbert problem with zero index in L^p(R) (1 < p < ) sense, which improves the result by Abraham in two directions (weaker conditions on approximating functions and solutions for a non-homogeneous Riemann–Hilbert problem with zero index). Also, we discussed an interesting case p = (uniformly approximation). (ii) Using the Egoroff's theorem provides a pointwise approximate solutions for a class of non-homogeneous Riemann–Hilbert problem with zero index. (iii) Using the Shannon sampling theorem provides explicit solutions for certain non-homogeneous Riemann–Hilbert problems with zero index. Some approximations which exploiting this fact will be discussed. (iv) Applications to integral equations are given.

Integral equation methods for the Robin problem in stationary oscillations of elastic plates

Thomson, G. R., Constanda, C. — Thu, 23 Jul 2009 12:42:03 PDT

The interior and exterior Robin boundary-value problems for the model of flexural vibrations of plates with transverse shear deformation are solved by means of layer potentials. Existence theorems are proved when certain conditions are satisfied by the elastic constants, the frequency parameter and the matrix connecting the tractions and displacements on the boundary.

Permanence and extinction of an impulsive delay competitive Lotka-Volterra model with periodic coefficients

Liu, Z., Wu, J., Tan, R. — Thu, 23 Jul 2009 12:42:03 PDT

In this paper, a periodic competitive system with delays and pulses is proposed. By using the comparison theorem for impulsive differential equations and the property of globally asymptotic stability of a periodic single-species growth population model with impulsive perturbations, sufficient conditions for permanence and extinction of the above system are derived, respectively. Our main results show that under appropriate conditions, the permanence and extinction of system are irrespective of the size of delays, however, impulsive perturbations play an important role and have effects on the permanence and extinction of system.

Bifurcation and stability analysis for a delayed Leslie-Gower predator-prey system

Yuan, S., Song, Y. — Thu, 23 Jul 2009 12:42:03 PDT

A delayed Leslie–Gower predator–prey system is considered in this paper. It is assumed that the predator and the prey species have the same feedback delay to their growth. Using the delay as a bifurcation parameter, our results show that the positive equilibrium can only be asymptotically stable or unstable depending on the delays and that Hopf bifurcations can occur as the delay crosses some critical values. The model can exhibit an interesting property, i.e. under certain conditions, the positive equilibrium may switch a finite number of times between being stable and unstable, but always becomes unstable eventually. By deriving the equation describing the flow on the centre manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of Wu (1998, Trans. Am. Math. Soc., 350, 4799–4838 for functional differential equations, we may show the global existence of periodic solutions. Computer simulations illustrate the results.

Travelling wave solutions in diffusive and competition-cooperation systems with delays

Li, K., Li, X. — Thu, 23 Jul 2009 12:42:03 PDT

In this paper, travelling wave solutions are considered for two species diffusive and competition–cooperation systems with delays. The method is Schauder's fixed-point theorem and a new cross-iteration scheme for delayed reaction–diffusion systems with partial monotonicity.

Travelling waves in the Oregonator model for the BZ reaction

Merkin, J. H. — Thu, 23 Jul 2009 12:42:03 PDT

Solutions to the travelling wave equations that arise in the two-variable version of the Oregonator model for the Belousov–Zhabotinsky reaction are obtained for small values of the kinetic parameter using the method of matched asymptotic expansions (MAEs). Single-pulse solutions are considered for both oxidation and reduction waves with the MAE approach clearly bringing out the structure of both types of wave. Various regions are derived where the concentrations of the active species HBrO₂ and M_ox, as well as the concentration of Br^– varying quasi-statically with HBrO₂ and M_ox, undergo significant changes. The lateral extent of these regions is also estimated in terms of the parameter .

A Papkovich-Neuber-based numerical approach to cracks with contact in 3D

Hintermuller, M., Kovtunenko, V. A., Kunisch, K. — Tue, 26 May 2009 22:57:25 PDT

The mathematical model of a crack with non-penetration conditions is considered in the framework of 3D elasticity. The spatial crack problem is investigated with respect to its numerical realization in the context of constrained optimization. Specifically, for homogeneous isotropic solids with planar cracks, a Papkovich–Neuber-based representation is adopted. It allows to employ a primal–dual active set strategy with an unconditional global and monotone convergence property. The iterates turn out to be primally feasible. Illustrative numerical examples are presented.

Energy decay in a transmission problem in thermoelasticity of type III

Messaoudi, S. A., Said-Houari, B. — Tue, 26 May 2009 22:57:25 PDT

In this paper, we consider a 1D linear thermoelastic transmission problem, where the heat conduction is described by the theories of Green and Naghdi. By using the energy method, we prove that the thermal effect is strong enough to produce an exponential stability of the solution, no matter how small the action domain is.

Stability of equilibria of some switched non-linear systems with applications to control synthesis for electrohydraulic servomechanisms

Halanay, A., Ursu, I. — Tue, 26 May 2009 22:57:25 PDT

Stability of the zero solution is analyzed for a family of switched systems indexed by a parameter, each system having = 0 in the spectrum of the Jacobian matrix calculated in zero. It is proved that existence of a common quadratic Lyapunov function for some lower dimensional linear systems is sufficient to ensure local uniform stability of the zero solution of the switched non-linear system and a regular asymptotic behaviour. An application to control synthesis for stabilizing equilibria in a switched non-linear control system modelling an electrohydraulic servomechanism is given.

Global solution for a quasi-linear plate system with boundary memory damping

Zhang, Q. — Tue, 26 May 2009 22:57:25 PDT

In this work, we consider a quasi-linear plate model with boundary memory damping. We prove that this system has a unique global solution when the initial data are small enough and the non-linear coefficient function, the memory damping as well as the geometry of the domain satisfy suitable assumptions. We also prove the exponential decay of the energy of the system.

Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy

Wang, Y. — Tue, 26 May 2009 22:57:25 PDT

In the paper, we consider the non-existence of global solutions of Cauchy problem for coupled Klein–Gordon equations of the form on R x Rⁿ. First, for the case n = 2, 3, we prove the existence of ground state of the corresponding Lagrange–Euler equations of the above equations. Then, we establish a blow-up result with low initial energy, which leads to instability of standing waves of the system above. Moreover, as a byproduct we also discuss the global existence. Next, based on concavity method, we prove the blow-up result for the system with non-positive initial energy in the general case: 1 <= n < 6. Finally, when the initial energy is given arbitrarily positive, we show that if the initial datum satisfies some conditions, the corresponding solution blows up in a finite time. In other words, in this paper we establish the complete blow-up result for the Klein–Gordon equation above in the sense of the initial energy, – < E(0) < + .

An extended notion of enthalpy. Electromagnetic solids

Trimarco, C. — Tue, 26 May 2009 22:57:25 PDT

Some procedures are here expounded in order to introduce the physical and the material or configurational stress in deformable and moving solids. In electromagnetic materials, the electromagnetic fields are preliminarily introduced in the Lagrangian form. Then a variational approach is proposed through two different procedures. Two energy–momentum tensors are excerpted from the suggested procedures. One of them corresponds to a Cauchy-like stress and the other one to the material stress. The latter rules the balance law for the material momentum. This balance law across a first-order discontinuity surface addresses an extended notion of enthalpy and also an extension of the Maxwell condition in thermodynamical phase transitions of electromagnetic solids.

The Kelvin transformation in potential theory and Stokes flow

Dassios, G. — Tue, 26 May 2009 22:57:25 PDT

Kelvin's transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method.

Micro/nanoparticle melting with spherical symmetry and surface tension

McCue, S. W., Wu, B., Hill, J. M. — Tue, 26 May 2009 22:57:25 PDT

The process of melting a small spherical particle is treated by setting up a two-phase Stefan problem. Surface tension is included through the Gibbs–Thomson condition, the effect of which is to decrease the melting temperature as the particle radius decreases. Analytical results are derived via a small-time expansion and also through large Stefan number asymptotics. Numerical solutions are computed with a front-fixing scheme, and these results suggest that the model exhibits finite-time blow-up, in the sense that both the interface speed and the temperature gradient in the solid phase (at the interface) will become unbounded at some time before complete melting. The near-blow-up behaviour appears to be similar to that encountered in the ill-posed problem of melting a superheated solid (without surface tension), and may help explain the onset of abrupt melting observed in some experiments with nanoscaled particles.

Stability of sticky particle dynamics and related scalar conservation laws

Moutsinga, O. — Tue, 26 May 2009 22:57:25 PDT

We show the stability of the sticky particle forward flow (x, s, t) ↦ (x, s, P_t, u_t) w.r.t. perturbations of the initial mass distribution P₀ and velocity function u₀. Then, we deduce the stability of related scalar conservation laws and pressureless gas system.

Asymptotic behaviour of ground state solutions for the Henon equation

Cao, D., Peng, S., Yan, S. — Tue, 26 May 2009 22:57:25 PDT

Let B₁(0) R^N be the unit ball centred at the origin, N ≥ 3. In this paper, we analyse the profile of the ground state solution of the Hénon equation – u = |x|u^p^–1 in B₁(0), u = 0 onB₁(0). We prove that for fixed p (2, 2^*), (2^* = 2N/(N – 2)), the maximum point x of the ground state solution u satisfies (1 – |x|) -> l (0, +) as -> . We also obtain the asymptotic behaviour of u, which shows that the ground state solution is non-radial. Moreover, we prove the existence of multi-peaked solutions and give their asymptotic behaviour.