Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation

In this paper, we study the 2D Boussinesq equations with fractional Laplacian dissipation. In particular, we prove the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The main ingredient of the proof is the utilization of the Hölder estimates for advection fractional-diffusion equations as well as Littlewood–Paley technique.     

Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli et al. (J Eur Math Soc (JEMS) 12(5):1151–1179, 2010). A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be zero along the boundary.     

Adjoint Methods for the Infinity Laplacian Partial Differential Equation

To study fine properties of certain smooth approximations ${u^\varepsilon}$ to a viscosity solution u of the infinity Laplacian partial differential equation (PDE), we introduce Green??s function ${\sigma^\varepsilon}$ for the linearization. We can then integrate by parts with respect to ${\sigma^\varepsilon}$ and derive various useful integral estimates. We are, in particular, able to use these estimates (i) to prove the everywhere differentiability of u and (ii) to rigorously justify interpreting the infinity Laplacian equation as a parabolic PDE.     

The asymptotics for the eigenelements of the Laplacian in a cylinder with frequently alternating boundary conditions

The singular perturbed eigenvalue problem for the Laplace operator in a cylindrical body with frequently alternating boundary conditions is considered. The asymptotics for the eigenelements are constructed.     

The Time-Scaled Trapezoidal Integration Rule for Discrete Fractional Order Controllers

In this paper, the time-scaled trapezoidal integration rule for discretizing fractional order controllers is discussed. This interesting proposal is used to interpret discrete fractional order control (FOC) systems as control with scaled sampling time. Based on this time-scaled version of trapezoidal integration rule, discrete FOC can be regarded as some kind of control strategy, in which strong control action is applied to the latest sampled inputs by using scaled sampling time. Namely, there are two time scalers for FOC systems: a normal time scale for ordinary feedback and a scaled one for fractional order controllers. A new realization method is also proposed for discrete fractional order controllers, which is based on the time-scaled trapezoidal integration rule. Finally, a one mass position 1/s^k control system, realized by the proposed method, is introduced to verify discrete FOC systems and their robustness against saturation non-linearity.     

The Time-Scaled Trapezoidal Integration Rule for Discrete Fractional Order Controllers

In this paper, the time-scaled trapezoidal integration rule for discretizing fractional order controllers is discussed. This interesting proposal is used to interpret discrete fractional order control (FOC) systems as control with scaled sampling time. Based on this time-scaled version of trapezoidal integration rule, discrete FOC can be regarded as some kind of control strategy, in which strong control action is applied to the latest sampled inputs by using scaled sampling time. Namely, there are two time scalers for FOC systems: a normal time scale for ordinary feedback and a scaled one for fractional order controllers. A new realization method is also proposed for discrete fractional order controllers, which is based on the time-scaled trapezoidal integration rule. Finally, a one mass position 1/s ^k control system, realized by the proposed method, is introduced to verify discrete FOC systems and their robustness against saturation non-linearity.     

A Fractional Linear System View of the Fractional Brownian Motion

A definition of the fractional Brownian motion based on the fractional differintegrator characteristics is proposed and studied. It is shown that the model enjoys the usually required properties. A discrete-time version based in the backward difference and in the bilinear transformation is considered. Some results are presented.     

The First Eigenvalue of the Laplacian and the Conductance of a Compact Surface

We present some results whose central theme is the phenomenon of the first eigenvalue of the Laplacian and conductance of the dynamical system. Our main tool is a method for studying how the hyperbolic metric on a Riemann surface behaves under deformation of the surface. With this model, we show that there are variation of the first eigenvalue of the laplacian and the conductance of the dynamical system, with the Fenchel–Nielsen coordinates, that characterize the surface.     

A Fractional Linear System View of the Fractional Brownian Motion

A definition of the fractional Brownian motion based on the fractional differintegrator characteristics is proposed and studied. It is shown that the model enjoys the usually required properties. A discrete-time version based in the backward difference and in the bilinear transformation is considered. Some results are presented.     

Fractional plasticity for over-consolidated soft soil

The stress–strain response of over-consolidated soft soil, e.g., clay, is dependent on its pre-consolidation history and material state. In this study, a state-dependent constitutive model for over-consolidated soft soils is proposed by extending the fractional plasticity originally developed for granular soil. The state-dependent stress-dilatancy and peak failure behaviour of over-consolidated soft soil are analytically captured through stress-fractional gradient of the current yielding surface. In addition, a reference yielding surface describing the pre-consolidation history of soft soil is proposed. A combined hardening rule expressed as a function of both the incremental plastic volumetric and shear strains is suggested. To validate the proposed model, a series of drained and undrained test results of different soft soils with a wide range of over-consolidation ratios are simulated and compared. It is found that without using additional plastic potentials and state parameters, the developed fractional model can capture the state-dependent hardening and softening responses of soft soils subjected to different over-consolidation states.

     

The Fractional Fourier Transform and Harmonic Oscillation

The ath-order fractional Fourier transform is a generalization ofthe ordinary Fourier transform such that the zeroth-order fractionalFourier transform operation is equal to the identity operation and thefirst-order fractional Fourier transform is equal to the ordinaryFourier transform. This paper discusses the relationship of thefractional Fourier transform to harmonic oscillation; both correspondto rotation in phase space. Various important properties of thetransform are discussed along with examples of commontransforms. Some of the applications of the transform are brieflyreviewed.     

Minimization of the k-th eigenvalue of the Dirichlet Laplacian

For every ${k \in \mathbb{N}}$ , we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.     

Laplace-Transform Based Inversion Method for Fractional Dispersion

Partial differential equations with memory are challenging models for mass transport in porous media where fluid and tracer may be stored by the solid matrix, and then released. Moreover, integral transforms (generalizing time moments) of solutions to such models are linked to the corresponding transport parameters. Inverting that link provides a method to determine model parameters on the basis of solutions. It is checked using numerically generated profiles before passing to experimental data.     

Fractional central pattern generators for bipedal locomotion

Locomotion has been a major research issue in the last few years. Many models for the locomotion rhythms of quadrupeds, hexapods, bipeds and other animals have been proposed. This study has also been extended to the control of rhythmic movements of adaptive legged robots. In this paper, we consider a fractional version of a central pattern generator (CPG) model for locomotion in bipeds. A fractional derivative D ^α f(x), with α non-integer, is a generalization of the concept of an integer derivative, where α=1. The integer CPG model has been proposed by Golubitsky, Stewart, Buono and Collins, and studied later by Pinto and Golubitsky. It is a network of four coupled identical oscillators which has dihedral symmetry. We study parameter regions where periodic solutions, identified with legs’ rhythms in bipeds, occur, for 0<α≤1. We find that the amplitude and the period of the periodic solutions, identified with biped rhythms, increase as α varies from near 0 to values close to unity.     

Fractional step algorithm for estuarine mass transport

A successful and economical fractional step algorithm for the convection-dispersion-reaction equation is described. Exact solutions are adopted for the reaction and convection steps, the latter by the introduction of a moving co-ordinate system. The dispersion step uses an optimized finite difference algorithm which specifically accommodates the grid non-uniformity. The excellent performance of the algorithm is confirmed by numerical experiments together with computations of the Fourier response and integrated square error characteristics.     

Fractional Diffusion Limit for Collisional Kinetic Equations

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.     

Fractional diffusion equations for open quantum system

To describe non-local interactions of quantum systems with environment we consider a fractional generalization of the quantum Markovian equation. Quantum analogs of fractional Laplacian operator for coordinate and momentum spaces are suggested. In phase-space form of quantum mechanics we obtain fractional equations for Wigner distribution function, where fractional Laplacian operators of the Grünvald–Letnikov type are used.     

On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System

Let Ω be a bounded Lipschitz domain in ℝ ⁿ with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x _n) with x′=(x1, ..., x _n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations. This revised version was published online in August 2006 with corrections to the Cover Date.