For hyperbolic diffeomorphisms, we describe the variational properties of the dimension spectrum of equilibrium measures on locally maximal hyperbolic sets, when the measure or the dynamical system are perturbed. We also obtain explicit expressions for the first derivative of the dimension spectra and the associated Legendre transforms. This allows us to establish a local version of multifracta...

An adaptive conservative or dissipative numerical method for nonlinear partial differential equations is established. The method not only inherits the welcome conservation or dissipation property of the equation but also uses suitable non-uniform grids at each time step. Our numerical experiments indicate that the method is useful especially for localized solutions such as solitary wave solutio...

We study the existence of minimizers for a constrained variational problems in H(R). These minimizers are stable waves solutions for the Generalized Camassa-Holm equation, and their derivative may have a singularity (in which case the travelling wave is called a peakon). The existence result is based on a method developed by the same author in a previous work. By giving examples, we show how ou...

We consider the problem of variational interpolation of subsets of Euclidean spaces by curves such that the L norm of the second derivative is minimized. It is well known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types of subsets such as subspaces, polyhedra, and submanifolds, and we indicate how solutions can be computed in th...

We study a dynamic contact problem for a thermoelastic von Kármán plate vibrating against a rigid obstacle. Dynamics is described by a hyperbolic variational inequality for deflections. The plate is subjected to a perpendicular force and to a heat source. The parabolic equation for a thermal strain resultant contains the time derivative of the deflection. We formulate a weak solution of the sys...

In this paper, we discuss the steady state Fractional Advection Dispersion Equation (FADE) on bounded domains in IRd . Fractional differential and integral operators are defined and analyzed. Appropriate fractional derivative spaces are defined, and shown to be equivalent to the fractional dimensional Sobolev spaces. A theoretical framework for the variational solution of the steady state FADE ...

We consider the numerical integration of the Degasperis–Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at a same time. We also prove the unique solvability of the schemes, and show som...

We present a generating functional which describes the equilibrium thermodynamic response of a relativistic system to external sources. A variational principle gives rise to constraints on the response parameters of relativistic hydrodynamics without making use of an entropy current. Our method reproduces and extends results available in the literature. It also provides a technique for efficien...

We give a new proof of the small excess regularity theorems for integer multiplicity recti able currents of arbitrary dimension and codimension minimizing an elliptic parametric variational integral. This proof does not use indirect blow-up arguments, it covers interior and boundary regularity, it applies to almost minimizing currents, and it gives an explicit and often optimal modulus of conti...

This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, m...