Some Applications of the Wright Function in Continuum Physics: A Survey

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed.

Double integral-balance method to the fractional subdiffusion equation: Approximate solutions, optimization problems to be resolved and numerical simulations

An approximate integral-balance solution of the fractional subdiffusion equation by a double-integration technique has been conceived. The time-fractional linear subdiffusion equation with Dirichlet boundary condition (and zero initial condition) has been chosen as a test example. Approximations of time-fractional Riemann-Liouville and Caputo derivatives when the distribution is assumed as a parabolic profile with unspecified exponent have been developed. Problems pertinent to determination of the optimal exponent of the parabolic profile and approximations of the time-fractional derivative of by different approaches have been formulated. Solved and unresolved problems in determination of the optimal exponents have been demonstrated. Examples with predetermined quadratic and cubic assumed profiles are analyzed, too. Comparative numerical studies with exact solutions expressed by the Mainardi function in terms of a similarity variable have been performed.

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

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.

Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.

Erdélyi-Kober fractional diffusion

The aim of this Short Note is to highlight that the generalized grey Brownian motion (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as Erdélyi-Kober fractional diffusion. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: 0 < α ≤ 2 and 0 < β ≤ 1. It includes the fractional Brownian motion when 0 < α ≤ 2 and β = 1, the time-fractional diffusion stochastic processes when 0 < α = β < 1, and the standard Brownian motion when α = β = 1. In the ggBm framework, the 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.