Singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel

In a recent paper [3], Cao and Xu established the Galerkin method for weakly singular Fredholm integral equations that preserves the singularity of the solution. Their Galerkin method provides a numerical solution that is a linear combination of a certain class of basis functions which includes elements that reflect the singularity of the solution. The purpose of this paper is to extend the result of Cao and Xu and to establish the singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. The iterated singularity preserving Galerkin method is also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

The direction of an automorphism of a totally disconnected locally compact group

We describe the asymptotic behavior of automorphisms of totally disconnected locally compact groups in terms of a set of `directions' which comes equipped with a natural pseudo-metric. The structure at infinity obtained by completing the induced metric quotient space of the set of directions recovers familiar objects such as: the set of ends of the tree for the group of inner automorphisms of the group of isometries of a regular locally finite tree; and the spherical Bruhat-Tits building for the group of inner automorphisms of the set of rational points of a semisimple group over a local field. Research supported by A.R.C. Grant DP0208137.     

Decay rates to the planar rarefaction waves for a model system of the radiating gas in n dimensions

In this paper, we study the asymptotic decay rates to the planar rarefaction waves to the Cauchy problem for a hyperbolic-elliptic coupled system called as a model system of the radiating gas in Rn (n=3,4,5) if the initial perturbations corresponding to the planar rarefaction waves are sufficiently small in (H²∩L¹∩W2,6) (Rn). The analysis is based on the Lp-energy method and several special interpolation inequalities.     

Long term dynamics of a reaction-diffusion system

We study a reaction-diffusion system of two parabolic differential equations describing the behavior of a nuclear reactor. We provide existence results for nontrivial periodic solutions, nonexistence results for stationary solutions and we prove that, depending on the value of the parameters, either the system admits a compact global attractor, or the solutions are unbounded.     

The external multiplication for the Conley index

We construct an additional operation of the external multiplication on the cohomological Conley index defined by Mrozek for discrete semidynamical systems. The construction is based on the notion of the Conley index over a phase space introduced by Szybowski. We show how to apply the external multiplication to solve the problem of continuation of two isolated invariant sets and illustrate it by an example.     

Asymptotic equipartition and long time behavior of solutions of a thin-film equation

We investigate the large-time behavior of classical solutions to the thin-film type equation ut=−x(uuxxx). It was shown in previous work of Carrillo and Toscani that for non-negative initial data u₀ that belongs to H¹(R) and also has a finite mass and second moment, the strong solutions relax in the L¹(R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H¹(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H¹(R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy.     

Convergence of the Grünwald–Letnikov scheme for time-fractional diffusion

Using bivariate generating functions, we prove convergence of the Grünwald–Letnikov difference scheme for the fractional diffusion equation (in one space dimension) with and without central linear drift in the Fourier–Laplace domain as the space and time steps tend to zero in a well-scaled way. This implies convergence in distribution (weak convergence) of the discrete solution towards the probability of sojourn of a diffusing particle. The difference schemes allow also interpretation as discrete random walks. For fractional diffusion with central linear drift we show that in the Fourier–Laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation.     

Periodic orbits on Riemannian manifolds under the action of an at most quadratic potential

In this paper we look for periodic orbits for a Lagrangian system in a complete Riemannian manifold under the action of an eventually unbounded potential. An upper bound on the fixed period is obtained by means of variational tools involving penalization arguments and Morse theory.