Fractional Hamilton formalism within Caputo's derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.Comment: 8 page

Reaction-diffusion systems and nonlinear waves

The authors investigate the solution of a nonlinear reaction-diffusion equation connected with nonlinear waves. The equation discussed is more general than the one discussed recently by Manne, Hurd, and Kenkre (2000). The results are presented in a compact and elegant form in terms of Mittag-Leffler functions and generalized Mittag-Leffler functions, which are suitable for numerical computation. The importance of the derived results lies in the fact that numerous results on fractional reaction, fractional diffusion, anomalous diffusion problems, and fractional telegraph equations scattered in the literature can be derived, as special cases, of the results investigated in this article.Comment: LaTeX, 16 pages, corrected typo

Solution of generalized fractional reaction-diffusion equations

This paper deals with the investigation of a closed form solution of a generalized fractional reaction-diffusion equation. The solution of the proposed problem is developed in a compact form in terms of the H-function by the application of direct and inverse Laplace and Fourier transforms. Fractional order moments and the asymptotic expansion of the solution are also obtained.Comment: LaTeX, 18 pages, corrected typo

Super-diffusive Transport Processes in Porous Media

The basic assumption of models for the transport of contaminants through soil is that the movements of solute particles are characterized by the Brownian motion. However, the complexity of pore space in natural porous media makes the hypothesis of Brownian motion far too restrictive in some situations. Therefore, alternative models have been proposed. One of the models, many times encountered in hydrology, is based in fractional differential equations, which is a one-dimensional fractional advection diffusion equation where the usual second-order derivative gives place to a fractional derivative of order α, with 1 < α ≤ 2. When a fractional derivative replaces the second-order derivative in a diffusion or dispersion model, it leads to anomalous diffusion, also called super-diffusion. We derive analytical solutions for the fractional advection diffusion equation with different initial and boundary conditions. Additionally, we analyze how the fractional parameter α affects the behavior of the solutions

Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals

Yor's generalized meander is a temporally inhomogeneous modification of the $2(\nu+1)$-dimensional Bessel process with $\nu > -1$, in which the inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$. We introduce the non-colliding particle systems of the generalized meanders and prove that they are the Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions $J_{\nu}$ used in the fractional calculus, where orders of differintegration are determined by $\nu-\kappa$. As special cases of the two parameters $(\nu, \kappa)$, the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.Comment: LaTeX, 35 pages, v3: The argument given in Section 3.2 was simplified. Minor corrections were mad

Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes

We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical objects and a generalized tensor calculus all being similar to respective integer dimension constructions. Such models of fractional gravity mimic the Einstein gravity theory and various Lagrange-Finsler and Hamilton-Cartan generalizations in nonholonomic variables. The approach suggests a number of new implications for gravity and matter field theories with singular, stochastic, kinetic, fractal, memory etc processes. We prove that the fractional gravitational field equations can be integrated in very general forms following the anholonomic deformation method for constructing exact solutions. Finally, we study some examples of fractional black hole solutions, fractional ellipsoid gravitational configurations and imbedding of such objects in fractional solitonic backgrounds.Comment: latex2e, 11pt, 40 pages with table of conten

Fractional reaction-diffusion equations

In a series of papers, Saxena, Mathai, and Haubold (2002, 2004a, 2004b) derived solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which provide the extension of the work of Haubold and Mathai (1995, 2000). The subject of the present paper is to investigate the solution of a fractional reaction-diffusion equation. The results derived are of general nature and include the results reported earlier by many authors, notably by Jespersen, Metzler, and Fogedby (1999) for anomalous diffusion and del-Castillo-Negrete, Carreras, and Lynch (2003) for reaction-diffusion systems with L\'evy flights. The solution has been developed in terms of the H-function in a compact form with the help of Laplace and Fourier transforms. Most of the results obtained are in a form suitable for numerical computation.Comment: LaTeX, 17 pages, corrected typo

Solutions of a particle with fractional δ\delta-potential in a fractional dimensional space

A Fourier transformation in a fractional dimensional space of order \la (0<\la\leq 1) is defined to solve the Schr\"odinger equation with Riesz fractional derivatives of order \a. This new method is applied for a particle in a fractional $\delta$-potential well defined by V(x) =- \gamma\delta^{\la}(x), where $\gamma>0$ and \delta^{\la}(x) is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if 0< \la<\a. The results for \la= 1 and \a=2 are in exact agreement with those presented in the standard quantum mechanics

PCSK6 and Survival in Idiopathic Pulmonary Fibrosis

Rationale: Idiopathic pulmonary fibrosis (IPF) is a devastating disease characterized by limited treatment options and high mortality. A better understanding of the molecular drivers of IPF progression is needed. Objectives: To identify and validate molecular determinants of IPF survival. Methods: A staged genome-wide association study was performed using paired genomic and survival data. Stage I cases were drawn from centers across the United States and Europe and stage II cases from Vanderbilt University. Cox proportional hazards regression was used to identify gene variants associated with differential transplantation-free survival (TFS). Stage I variants with nominal significance (P < 5 x 10(-5)) were advanced for stage II testing and meta-analyzed to identify those reaching genome-wide significance (P < 5 x 10(-8)). Downstream analyses were performed for genes and proteins associated with variants reaching genome-wide significance. Measurements and Main Results: After quality controls, 1,481 stage I cases and 397 stage II cases were included in the analysis. After filtering, 9,075,629 variants were tested in stage I, with 158 meeting advancement criteria. Four variants associated with TFS with consistent effect direction were identified in stage II, including one in an intron of PCSK6 (proprotein convertase subtilisin/kexin type 6) reaching genome-wide significance (hazard ratio, 4.11 [95% confidence interval, 2.54-6.67]; P = 9.45 x 10(-9)). PCSK6 protein was highly expressed in IPF lung parenchyma. PCSK6 lung staining intensity, peripheral blood gene expression, and plasma concentration were associated with reduced TFS. Conclusions: We identified four novel variants associated with IPF survival, including one in PCSK6 that reached genome-wide significance. Downstream analyses suggested that PCSK6 protein plays a potentially important role in IPF progression