IMA Journal of Applied Mathematics - current issue

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.