Volume 16, Issue 2

In this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation $\int_0^2 {p(\beta )D_t^\beta u(x,t)d\beta } = \frac{{\partial ^2 }} {{\partial x^2 }}u(x,t) $ is considered. Here, the time-fractional derivative D tβ is understood in the Caputo sense and p(β) is a non-negative weight function with support somewhere in the interval [0, 2]. By employing the technique of the Fourier and Laplace transforms, a representation of the fundamental solution of the Cauchy problem in the transform domain is obtained. The main focus is on the interpretation of the fundamental solution as a probability density function of the space variable x evolving in time t. In particular, the fundamental solution of the time-fractional distributed order wave equation (p(β) ≡ 0, 0 ≤ β < 1) is shown to be non-negative and normalized. In the proof, properties of the completely monotone functions, the Bernstein functions, and the Stieltjes functions are used.

We deal with the stability problem of the fractional order Black-Scholes model driven by fractional Brownian motion (fBm). First, necessary and sufficient conditions are established for almost sure asymptotic stability and pth moment asymptotic stability by means of the largest Lyapunov exponent and the pth moment Lyapunov exponent, respectively. Moreover, we are able to present large deviations results for this fractional process. In particular, for the first time it is found that the Hurst parameter affects both stability conclusions and large deviations. Interestingly, large deviations always happen for the considered system when 1/2 < H < 1. This fact is due to the long-range dependence (LRD) property of the fBm. Numerical simulation results are presented to illustrate the above findings.

The speed of many one-line transformation methods for the production of, for example, Lévy alpha-stable random numbers, which generalize Gaussian ones, and Mittag-Leffler random numbers, which generalize exponential ones, is very high and satisfactory for most purposes. However, fast rejection techniques like the ziggurat by Marsaglia and Tsang promise a significant speed-up for the class of decreasing probability densities, if it is possible to complement them with a method that samples the tails of the infinite support. This requires the fast generation of random numbers greater or smaller than a certain value. We present a method to achieve this, and also to generate random numbers within any arbitrary interval. We demonstrate the method showing the properties of the transformation maps of the above mentioned distributions as examples of stable and geometric stable random numbers used for the stochastic solution of the space-time fractional diffusion equation.

The time-fractional heat conduction equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in an infinite medium with a spherical hole in the central symmetric case under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of the values of temperature and the values of its normal derivative at the boundary and the physical condition with the prescribed linear combination of the values of temperature and the values of the heat flux at the boundary. The integral transforms techniques are used. Several particular cases of the obtained solutions are analyzed. The numerical results are illustrated graphically.

We show that the Riesz fractional integration operator I α(·) of variable order on a bounded open set in Ω ⊂ ℝn in the limiting Sobolev case is bounded from L p(·)(Ω) into BMO(Ω), if p(x) satisfies the standard logcondition and α(x) is Hölder continuous of an arbitrarily small order.

This paper surveys one of the last contributions by the late Professor Anatoly Kilbas (1948–2010) and research made under his advisorship. We briefly describe the historical development of the theory of the discussed multi-parametric Mittag-Leffler functions as a class of the Wright generalized hypergeometric functions. The method of the Mellin-Barnes integral representations allows us to extend the considered functions to the case of arbitrary values of parameters. Thus, the extended Mittag-Leffler-type functions appear. The properties of these special functions and their relations to the fractional calculus are considered. Our results are based mainly on the properties of the Fox H-functions, as one of the widest class of special functions.

In Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions. In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.

A necessary and sufficient condition is given for holomorphic functions to be represented by series of the kind $\sum\limits_{n = 0}^\infty {a_n J_0 (nz),z,a_n \in \mathbb{C},} $ where J 0(z) is the Bessel function of first kind with zero index. To derive the result, we use an Erdélyi-Kober operator of fractional order.

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.

This work focuses on an investigation of the (n+1)-dimensional time-dependent fractional Schrödinger type equation. In the early part of the paper, the wave function is obtained using Laplace and Fourier transform methods and a symbolic operational form of the solutions in terms of Mittag-Leffler functions is provided. We present an expression for the wave function and for the quantum mechanical probability density. We introduce a numerical method to solve the case where the space component has dimension two. Stability conditions for the numerical scheme are obtained.

Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in various problems of mathematics, statistics and natural sciences. In this survey we start with considering the case of a Gauss hypergeometric function with the argument being a rectangular matrix. Subsequently, some fractional integral operators are introduced which complement theresults available on fractional operators in the matrix variate cases. Several properties and limiting forms are derived. Then the pathway idea is incorporated to move among several different functional forms. When these are used as models for problems in the natural sciences, then these can cover the ideal situations, neighborhoods, in between stages and paths leading to optimal situations.

During the last fifty years the area of Fractional Calculus verified a considerable progress. This paper analyzes and measures the evolution that occurred since 1966.

In this short communication, an attempt is made to continue beyond paragraph 29 of Euler’s famous paper in Vol. 5 of Comment. Acad. Sci. Petropol. (1738), using his style of storytelling to extrapolate the audacity of his approach from fractional differentiation to fractional integration. To add the authenticity and the amusement to the imitation, the emulated paragraphs 30–32 are first presented in Latin version followed by the English translation. This reconstruction aims to demonstrate that Euler could consider not only fractional differentiation, but also fractional-order integration and its inverse relationship with differentiation of the same fractional order.