By using logarithm of derivation, logarithm of a differential form is defined. It allows to define fractional order differential forms on a flat space. Regularized volume form on a flat infinite-dimensional space is defined by using fractional order differential forms. Applying noncommutative connection, we show regularized volume form can be defined on a mapping space, if its string class vani...

In this paper, we consider the numerical solutions of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two kinds of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection-dispersion equation (RFADE). RFDE is obtained from the standard diffusion equation by replacing the secondo...

Space fractional derivatives have been used instead of integer operators with success in a variety of practical applications to study and describe transport processes in media characterized by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides an implementation of space fractional FitzHugh-Nagumo model of excitable med...

Consider a Markov chain {X n } n≥0 with an ergodic probability measure π. Let Ψ a function on the state space of the chain, with α-tails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α N n Ψ(X n) to a α-stable law. A " martingale approximation " approach and " coupling " approach give two different sets of condition...

This paper presents a fractional mathematical model of a one-dimensional phase-change problem (Stefan problem) with a variable latent-heat (a power function of position). This model includes space–time fractional derivatives in the Caputo sense and time-dependent surface-heat flux. An approximate solution of this model is obtained by using the optimal homotopy asymptotic method to find the solu...

Consider a Markov chain {X n } n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with α-tails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α N n Ψ(X n) to an α-stable law. A " martingale approximation " approach and " coupling " approach give two different sets of condi...

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by fractional-order derivatives. This article considers a nonlocal inverse problem and shows that the exponents of the fractional time and space derivatives are determined uniquely by the data u(t, 0) = g(t), 0 < t < T . The uniqueness result is a theoretical background for determining experimental...

We consider Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise spaces, all represented through the corresponding orthogonal basis of the Hilbert space of random variables with finite second moments, given by the Hermite and the Charlier polynomials. There exist unitary mappings between the Gaussian and Poissonian white noise spaces. We investigate the relationship o...

In this paper we investigate composition operators induced on the Dirichlet space by linear fractional maps. We characterize the essential normality in this setting, obtain conditions for the linear fractional symbols φ and ψ of the unit disc for which CφC∗ ψ or C ∗ ψCφ is compact, and investigate the shape of the numerical range for linear fractional composition operators induced on the Dirich...