IMA Journal of Applied Mathematics - Advance Access

Solutions and symmetry reductions of the n-dimensional non-linear convection-diffusion equations

Ji, L., Qu, C., Ye, Y. — Fri, 20 Nov 2009 05:13:52 PST

This paper discusses a wide class of n-dimensional non-linear convection–diffusion equations with source term. It is shown that the radially symmetric equations admit certain types of conditional Lie–Bäcklund symmetries. As a result, exact solutions and symmetry reductions to 2D dynamical systems of the resulting equations are obtained. Those solutions extend the known ones such as self-similar solutions and instantaneous source-type solutions of the porous medium equation with absorption term. The behaviour of extinction and blow-up to many of the solutions are described.

Energy and helicity dissipation rates of the NS-alpha and NS-alpha-deconvolution models

Layton, W., Rebholz, L., Sussman, M. — Thu, 12 Nov 2009 01:34:00 PST

We consider the Navier-Stokes (NS)-alpha and the family of high-accuracy NS-alpha-deconvolution models of turbulence on = [0, L]³ subject to periodic boundary conditions. For body-force-driven turbulence, we prove directly from the model equations of motion the following bounds on the time-averaged modified energy dissipation rate, <_{, N}(w_{, N})>, and unmodified helicity dissipation rate, <(w_{, N})>, for the Nth model (N = 0, 1, 2, ...): Here, N is the degree of the approximate deconvolution operator, U_N and L_N are global velocity and length scales and C₁ and C₂ are constants that do not depend on U_N.

The evolution of travelling wavefronts in a hyperbolic Fisher model. III. The initial-value problem when the initial data has exponential decay rates

Leach, J. A., Needham, D. J. — Mon, 09 Nov 2009 10:27:51 PST

In this paper, we consider an initial-value problem for a non-linear hyperbolic Fisher equation. The non-linear hyperbolic Fisher equation is given by where > 0 is a parameter and F(u) = u(1 – u) is the classical Fisher kinetics. The initial data considered is positive, having unbounded support with exponential decay of O(e^–x) at large x (dimensionless distance), where > 0 is a parameter. It is established, via the method of matched asymptotic expansions, that the large time structure of the solution to the initial-value problem involves the evolution of a propagating wavefront which is either of reaction–diffusion or of reaction–relaxation type. In particular, the wave speed for the large t (dimensionless time) permanent form travelling wave (PTW), which may be subsonic (reaction–diffusion), sonic (reaction–relaxation) or supersonic (reaction–relaxation), the asymptotic correction to the wave speed and the rate of convergence of the solution onto the PTW are obtained for all values of the parameters and .

A surface energy approach to the mass reduction problem for elastic bodies

D'Ambrosio, P., De Tommasi, D., Granieri, L., Maddalena, F. — Tue, 03 Nov 2009 02:13:03 PST

We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this work, this problem is related to the minimization of a surface energy, depending on the stress tensor in the original equilibrium configuration. Special cases of mechanical interest are also analysed.

Periodic motion of a mass-spring system

Shearer, M., Gremaud, P., Kleiner, K. — Tue, 27 Oct 2009 09:33:09 PDT

The equations of planar motion of a mass attached to two anchored massless springs form a symmetric Hamiltonian system. The system has a single dimensionless parameter L, corresponding to the spacing between the anchors. For L > 1, there is a stable equilibrium at which the springs are in tension and lie on a line, but for L < 1, this equilibrium has both springs in compression and is unstable. However, there are then two stable equilibria at which both springs carry no force. Oscillations are studied in both regimes, but more systematically in the tension case, where techniques of bifurcation theory, numerical approximation and numerical simulation are used to explore the rich variety of periodic solutions.

Analysis of delay-dependent stability of linear {theta}-methods for linear delay-integro-differential equations

Xu, Y., Zhao, J.-J. — Tue, 27 Oct 2009 09:33:09 PDT

In this paper, the stability regions of linear -methods are considered with respect to the linear integro-differential equation with an arbitrary but fixed delay. A necessary condition is proved for -methods to be weakly (0)-stable and a sufficient condition is conjectured by some numerical investigation.

Chaotic synchronization in lattices of two-variable maps coupled with one variable

Lin, W.-W., Peng, C.-C., Wang, Y.-Q. — Tue, 27 Oct 2009 09:33:08 PDT

In this paper, we study chaotic synchronization in 1D lattices of two-variable maps coupled with one variable. We give a rigourous proof for the occurrence of chaotic synchronization of spatially homogeneous solutions in such coupled map lattices (CMLs) of lattice size n = 4 with suitable coupling coefficients. For the case of lattice size n > 4, we demonstrate numerical results of synchronized chaotic behaviour of the CMLs. Moreover, we show numerically that the difference between two variables manifests chaotic behaviour. This behaviour combined with the special coupling method in the CMLs guarantees high security in applications using our new model.

Construction of orthogonal multi-wavelets using generalized-affine fractal interpolation functions

Bouboulis, P. — Sun, 11 Oct 2009 20:22:41 PDT

We present a new construction of fractal interpolation surfaces defined on arbitrary rectangular lattices. We use this construction to form finite sets of fractal interpolation functions (FIFs) that generate multiresolution analyses of L₂(R²) of multiplicity r. These multiresolution analyses are based on the dilation properties of the construction. The associated multi-wavelets are orthogonal and discontinuous functions. We give concrete examples to illustrate the method and generalize it to form multiresolution analyses of L₂(R^d), d > 2. To this end, we prove some results concerning the Hölder exponent of FIFs defined on [0, 1]^d.

The transmission problem to thermoelastic plate of hyperbolic type

Vila Bravo, J. C., Munoz Rivera, J. E. — Mon, 24 Aug 2009 03:42:43 PDT

In this paper, we consider the thermoelastic plate equations with localized thermal dissipation of memory type, proposed by Gurtin & Pipkin (1968, Arch. Ration. Mech. Anal., 31, 113–126). We will show that the solution of the corresponding model decays exponentially as time goes to infinity, provided the relaxation function decays exponentially. The main difference between the current model and other thermoelastic systems is that the whole system is of hyperbolic type and the ‘dissipation’ is weaker (indefinite) than that given by the Fourier law for the heat flux.

Regularization of parabolic equations backward in time by a non-local boundary value problem method

Hao, D. N., Van Duc, N., Lesnic, D. — Thu, 20 Aug 2009 08:00:39 PDT

Let H be a Hilbert space with norm ||·||, A: D(A) H -> H a positive definite, self-adjoint operator with compact inverse on H, and T and are given positive numbers. The ill-posed parabolic equation backward in time is regularized by the well-posed non-local boundary value problem with a > 1 being given and > 0, the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order optimal regularization methods. Numerical results based on the boundary element method are presented and discussed to confirm the theory.