IMA Journal of Applied Mathematics - Advance Access

Quasi-separation of the biharmonic partial differential equation

Everitt, W. N., Johansson, B. T., Littlejohn, L. L., Markett, C. — 2009-04-16

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.

Travelling waves in the Oregonator model for the BZ reaction

Merkin, J.H. — 2009-04-16

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 .

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

Zhuang, P., Liu, F., Anh, V., Turner, I. — 2009-04-06

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.

Asymptotic solution of slender viscous jet break-up

Decent, S. P. — 2009-03-25

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.

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

Yuan, S., Song, Y. — 2009-03-19

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.

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

Duncan, A. J., Mackay, T. G., Lakhtakia, A. — 2009-03-11

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.

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

Liu, Z., Wu, J., Tan, R. — 2009-03-05

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.

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

Eden, A., Gurel, T. B., Kuz, E. — 2009-03-04

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.).

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

Li, K., Li, X. — 2009-02-24

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.

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. — 2008-12-18

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.

An approximation for a subclass of the Riemann-Hilbert problems

Kucerovsky, D., Payandeh Najafabadi, A. T. — 2008-11-06

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. — 2008-09-09

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.