In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by Nigmatullin (1986), and Zaslavsky (1994) in R for modeling some physical phenomena. The fractional derivative models time delays in a diffusion process. We will give a survey of the recent results on the fractional Cauchy problem and its generalizati...

The paper presents a new fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as special cases. Conditions are given for such a generalized fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for th...

This paper builds upon our recent paper on generalized fractional variational calculus FVC . Here, we briefly review some of the fractional derivatives FDs that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives GFDs which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases o...

The Cauchy problem is considered for a homogeneous Hamilton–Jacobi equation with fractional-order coinvariant derivatives, which arises in problems of dynamic optimization systems described by differential equations Caputo fractional derivatives. A generalized solution the minimax sense defined. It proved that such exists, unique, depends continuously on parameters problem, and consistent class...

The aim of the present paper is to introduce and use the generalized exponential shift operators, operators on the base a (a > 0, = 1), to deal with the families of pseudo-Kampé de Fériet polynomials, which can be viewed as the natural complement for the theory of fractional derivatives and partial fractional differential equations of evolutive type. We show that these families allow the possib...

The object of this paper is to solve a fractional integro-differential equation involving a generalized Lauricella confluent hypergeometric function in several complex variables and the free term contains a continuous function f (τ). The method is based on certain properties of fractional calculus and the classical Laplace transform. A Cauchy-type problem involving the Caputo fractional derivat...