On fractional Schrödinger equation in α-dimensional fractional space

The Schrödinger equation is solved in $\alpha$-dimensional fractional space with a Coulomb potential proportional to $\frac{1}{r^{\beta ?2}}$, $2\le \beta \le 4$. The wave functions are studied in terms of spatial dimensionality $\alpha$ and $\beta$ and the results for $\beta =3$ are compared with those obtained in the literature.     

Comment on “Exact explicit travelling wave solutions for (n+1)-dimensional Klein–Gordon–Zakharov equations” [Chaos,solitons and fractals 34(2007) 867–871]

Jibin Li studied “exact explicit travelling wave solutions for (n+1)-dimensional Klein–Gordon–Zakharov equations.” By using the approach of dynamical system, the author claimed that they had obtained five classes of exact explicit travelling wave solutions. However, we checked the results and found two typographical technical errors in the main results. Furthermore, the author fails to note, for $a=0,b<0,h\in (0,\infty )$ and $a<0,b>0,h=0,$ there is a family of periodic wave solutions and a series of breaking wave solutions, respectively.     

Coloration des graphes de reines

Until 2003 no chromatic numbers (${\chi }_n$) for the queen graphs were available for $n>9$ except where n is not a multiple of 2 or 3. In this research announcement we present an exact algorithm which provides coloring solutions for $n=12,14,15,16,18,20,21,22,24,26,28$ and 32 such as ${\chi }_n=n$. Then we prove that there exists an infinite number of values for n such that $n=2p$ or $n=3p$, and ${\chi }_n=n$. To cite this article: M. Vasquez, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation ${(?\mathrm{\Delta })}^{\alpha }$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta }$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q({\mathrm{BMO}}_x^{?\gamma })$ with $q>2$ and $\gamma \in (0,2\alpha ?1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.     

The maximum rate of convergence for the approximation of the fractional Lévy area at a single point

In this note we study the approximation of the fractional Lévy area with Hurst parameter $H>1/2$, considering the mean square error at a single point as error criterion. We derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an equidistant discretization of the driving fractional Brownian motion. This rate is $n^{-2H+1/2}$, where $n$ denotes the number of evaluations of the fractional Brownian motion, and is obtained by a trapezoidal rule.     

Lattice BGK simulations of double diffusive natural convection in a rectangular enclosure in the presences of magnetic field and heat source

In this paper, we develop a temperature-concentration lattice Bhatnagar-Gross-Krook (TCLBGK) model, with a robust boundary scheme for simulating the two-dimensional, hydromagnetic, double-diffusive convective flow of a binary gas mixture in a rectangular enclosure, in which the upper and lower walls are insulated, while the left and right walls are at a constant temperature and concentration and a uniform magnetic field is applied in the $x$-direction. In the model, the velocity, temperature and concentration fields are solved by three independent LBGK equations which are combined into a coupled equation for the whole system. In our simulations, we take the Prandtl number $Pr=1$, the Lewis number $Le=2$, the thermal Rayleigh number $R{a}_T={10}^5,{10}^6$, the Hartmann number $Ha=0,10,25,50$, the dimensionless heat generation or absorption $?=0.0,?1.0$, the buoyancy ratio $N=0.8,1.3$, and the aspect ratio $A=2$ for the enclosure. The numerical results are found to be in good agreement with those of previous studies [A.J. Chamkha, H. Al-Naser, Hydromagnetic double-diffusive convection in a rectangular enclosure with opposing temperature and concentration gradients, Int. J. Heat Mass Transfer 45 (2002) 2465C2483].