Subordination Pathways to Fractional Diffusion

The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and under power law regime is splitted into three distinct random walks: (rw_1), a random walk along the line of natural time, happening in operational time; (rw_2), a random walk along the line of space, happening in operational time;(rw_3), the inversion of (rw_1), namely a random walk along the line of operational time, happening in natural time. Via the general integral equation of CTRW and appropriate rescaling, the transition to the diffusion limit is carried out for each of these three random walks. Combining the limits of (rw_1) and (rw_2) we get the method of parametric subordination for generating particle paths, whereas combination of (rw_2) and (rw_3) yields the subordination integral for the sojourn probability density in space-time fractional diffusion.Comment: 20 pages, 4 figure

Some applications of Wright functions in fractional differential equations

In this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to obtain some new particular solutions for nonlinear fractional partial differential equations

V-Langevin Equations, Continuous Time Random Walks and Fractional Diffusion

The following question is addressed: under what conditions can a strange diffusive process, defined by a semi-dynamical V-Langevin equation or its associated Hybrid kinetic equation (HKE), be described by an equivalent purely stochastic process, defined by a Continuous Time Random Walk (CTRW) or by a Fractional Differential Equation (FDE)? More specifically, does there exist a class of V-Langevin equations with long-range (algebraic) velocity temporal correlation, that leads to a time-fractional superdiffusive process? The answer is always affirmative in one dimension. It is always negative in two dimensions: any algebraically decaying temporal velocity correlation (with a Gaussian spatial correlation) produces a normal diffusive process. General conditions relating the diffusive nature of the process to the temporal exponent of the Lagrangian velocity correlation (in Corrsin approximation) are derived.Comment: Latex 69 pages including 23 EPS figure

Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy

Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality

Variable-order fractional calculus: A change of perspective

Several approaches to the formulation of a fractional theory of calculus of “variable order” have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative

Analysis of surface atrial signals using spectral methods for time series with missing data

In this work, the analysis of atrial signals recorded during atrial fibrillation was pursued using two spectral estimators designed for series with missing data: the Lomb periodogram (LP) and the Iterative Singular Spectrum Analysis (ISSA). The main aim is to verify if subtraction ofthe ventricular activity might be avoided by performing spectral analysis on those ECG intervals where such activity is absent, (i.e. the T-Q intervals), at least to estimate the dominant atrial Fibrillatory Frequency (FF). Recordings coming from the 2004 Computers in Cardiology Termination Challenge Database were analyzed. Fibrillatory frequencies were then compared with those obtained from the analysis ofthe correspondent atrial signals extracted using a modified Average Beat Substraction (ABS) technique. We observed that the mean absolute difference was 0.42 \ub1 0.66 Hz for LP, (mean\ub1SD), and 0.39 \ub1 0.64 Hz for ISSA. We concluded that estimation of FF is feasible without applying QRS-T subtraction

Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation

A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page

Assessment of spatial heterogeneity of ventricular repolarization after multi-channel blocker drugs in healthy subjects

Background and objectives: In contrast to potassium channel blockers, drugs affecting multiple channels seem to reduce torsadogenic risks. However, their effect on spatial heterogeneity of ventricular repolarization (SHVR) is still matter of investigation. Aim of this work is to assess the effect of four drugs blocking the human ether-\ue0-go-go-related gene (hERG) potassium channel, alone or in combination with other ionic channel blocks, on SHVR, as estimated by the V-index on short triplicate 10 s ECG. Methods: The V-index is an estimate of the standard deviation of the repolarization times of the myocytes across the entire myocardium, obtained from multi-lead surface electrocardiograms. Twenty-two healthy subjects received a pure hERG potassium channel blocker (dofetilide) and 3 other drugs with additional varying degrees of sodium and calcium (L-type) channel block (quinidine, ranolazine, and verapamil), as well as placebo. A one-way repeated-measures Friedman test was performed to compare the V-index over time. Results: Computer simulations and Bland-Altman analysis supported the reliability of the estimates of V-index on triplicate 10 s ECG. Ranolazine, verapamil and placebo did not affect the V-index. On the contrary, after quinidine and dofetilide administration, an increase of V-index from predose to its peak value was observed (\u394\u394V-index values were 19 ms and 27 ms, respectively, p < 0.05). Conclusions: High torsadogenic drugs (dofetilide and quinidine) affected significantly the SHVR, as quantified by the V-index. The metric has therefore a potential in assessing drug arrhythmogenicity