M-Wright Function Research Articles

An analytical solution of the time-fractional telegraph equation describing neutron transport in a nuclear reactor

Fractional telegraph equation is introduced to promote a sub-diffusive transport model of the neutrons inside a nuclear reactor. Here, the time derivatives of the classical telegraph equation have been replaced by the Caputo fractional derivative. In this paper, the fractional solutions are obtained through a technique that relies heavily on Fourier and Laplace transformations and is given in terms of the M-Wright function. The neutron flux density is clarified for different values of the time-fractional order.

On fractional approximations of the Fokker–Planck equation for energetic particle transport

To determine the main effects of anomalous diffusion upon the propagation of energetic particles, the fractional modified telegraph equation is solved using the Laplace–Fourier technique and given in terms of the Fox H and M-Wright functions. Examples are used to illustrate the qualitative differences between the fractional telegraph, fractional advection–diffusion and fractional diffusion equations. The profiles of the particles densities are discussed in each case for different values of fractional orders.

Existence results for a generalization of the time-fractional diffusion equation with variable coefficients

In this paper we consider the Cauchy problem of a generalization of time-fractional diffusion equation with variable coefficients in mathbb {R}_{+}^{n+1}, where the time derivative is replaced by a regularized hyper-Bessel operator. The explicit solution of the inhomogeneous linear equation for any ninmathbb {Z}^{+} and its uniqueness in a weighted Sobolev space are established. The key tools are Mittag-Leffler functions, M-Wright functions and Mikhlin multiplier theorem. At last, we obtain the existence of solution of the semilinear equation for n=1 by using a fixed point theorem.

The fractional Dodson diffusion equation: a new approach

In this paper, after a brief review of the general theory concerning regularized derivatives and integrals of a function with respect to another function, we provide a peculiar fractional generalization of the $(1+1)$-dimensional Dodson's diffusion equation. For the latter we then compute the fundamental solution, which turns out to be expressed in terms of an M-Wright function of two variables. Then, we conclude the paper providing a few interesting results for some nonlinear fractional Dodson-like equations.

On M-Wright transforms and time-fractional diffusion equations

ABSTRACTIn this paper, we consider the Mittag-Leffler operator as an analytical solution of time-fractional diffusion equation in the Caputo sense. This solution is presented by an integral representation in terms of the M-Wright functions and the exponential operators. Further, we study the Mittag-Leffler operators associated with the Legendre and Bessel diffusion equations. Finally, we extend the obtained integral representation for the time-fractional diffusion equation of distributed order.

On The Volterra μ-Functions and the M-Wright Functions as Kernels and Eigenfunctions of Volterra Type Integral Operators

Abstract In this note we consider some classes of integral equations with the M-Wright functions as kernels. We show that the Volterra μ−functions are eigenfunctions of the introduced integral operators. We derive also some new integral identities related to the Weyl fractional integrals and derivatives of the M-Wright (Mainardi) functions.

Time-fractional wave-diffusion equation in an inhomogeneous half-space

We consider the fundamental solution of the time-fractional wave-diffusion equation in a three-dimensional half-space medium which contains an inhomogeneity in form of a plane parallel layer. The corresponding Green’s function which is derived by means of the Fourier and Laplace transforms can be accurately and efficiently evaluated without recourse to the Mittag-Leffler or the Fox H-function. Moreover, it is shown that in the one-dimensional case the fundamental solution in an inhomogeneous half-space is no longer a probability density function. In addition, we consider the advection equation for the fractional Laplacian and the Caputo time-fractional derivative of orders on a bounded domain. Simple algorithms for accurate evaluation of the M-Wright function and the Mittag-Leffler function are enclosed at the end of this article.

On the Fourier transform of the products of M-Wright functions

In this note, by applying the Bromwich's integral for the inverse Mellin transform we find a new integral representation for the M-Wright function $$ M_\alpha(x)=\sum _{k=0}^{\infty }\frac{(-x)^{k} }{k!\Gamma (-\alpha k+1-\alpha )},\quad \alpha=\frac{1}{2n+1}, n\in \mathbb{N},$$ and state the Fourier transform of this function. Also, using the new integral representations for the products of the M-Wright functions, we get the Fourier transform of it.

Numerical computation for backward time-fractional diffusion equation

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward problem of time-fractional diffusion equation (BTFDE). The kernels used in the approximation are the fundamental solutions of the time-fractional diffusion equation which can be expressed in terms of the M-Wright functions. To stably and accurately solve the resultant highly ill-conditioned system of equations, we successfully combine the standard Tikhonov regularization technique and the L-curve method to obtain an optimal choice of the regularization parameter and the location of source points. Several 1D and 2D numerical examples are constructed to demonstrate the superior accuracy and efficiency of the proposed method for solving both the classical backward heat conduction problem (BHCP) and the BTFDE.

The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

Abstract The leading role of a special function of the Wright-type, referred to as M-Wright or Mainardi function, within a parametric class of self-similar stochastic processes with stationary increments, is surveyed. This class of processes, known as generalized grey Brownian motion, provides models for both fast and slow anomalous diffusion. In view of a subordination-type formula involving M-Wright functions, these processes emerge to have all finite moments and be uniquely defined by their mean and auto-covariance structure like Gaussian processes. The corresponding master equation is shown to be a fractional differential equation in the Erdélyi-Kober sense and the diffusive process is named Erdélyi-Kober fractional diffusion. In Appendix, an historical overview on the M-Wright function is reported.

Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

Estimation and Simulation for the M-Wright Function

In this article, a structural form of an M-Wright distributed random variable is derived. The mixture representation then led to a random number generation algorithm. A formal parameter estimation procedure is also proposed. This procedure is needed to make the M-Wright function usable in practice. The asymptotic normality of the estimator is established as well. The estimator and the random number generation algorithm are then tested using synthetic data.

On the convergence of quadratic variation for compound fractional Poisson processes

Abstract The relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

Characterizations and simulations of a class of stochastic processes to model anomalous diffusion

In this paper, we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions. This class, called generalized grey Brownian motion (ggBm), is made up of self-similar with stationary increments processes (H-sssi) and depends on two real parameters α ∊ (0, 2) and β ∊ (0, 1]. It includes fractional Brownian motion when α ∊ (0, 2) and β = 1, and time-fractional diffusion stochastic processes when α = β ∊ (0, 1). The latter have a marginal probability density function governed by time-fractional diffusion equations of order β. The ggBm is defined through the explicit construction of the underlying probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite-dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of the M-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that the ggBm can be represented in terms of the product of a random variable, which is related to the M-Wright function, and an independent fractional Brownian motion. This representation highlights the H-sssi nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differential equation of a fractional type.