The equivalent system for a multiple-rational-order (MRO) fractional differential system is studied, where the fractional derivative is in the sense of Caputo or Riemann-Liouville. With the relationship between the Caputo derivative and the generalized fractional derivative, we can change the MRO fractional differential system with a Caputo derivative into a higher-dimensional system with the s...

In this paper, we apply the local fractional Laplace transform method (or Yang-Laplace transform) on Volterra integro-differential equations of the second kind within the local fractional integral operators to obtain the analytical approximate solutions. The iteration procedure is based on local fractional derivative operators. This approach provides us with a convenient way to find a solution ...

The main purpose of this paper is to address two open questions raised by Reichel (2009) in [2] on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded C1 domainΩ ⊂ RN , we consider the Riesz-potential u(x) = ∫ Ω 1 | x− y |N−α dy for 2 ≤ α = N . We show that u = constant on ∂Ω if and only if Ω is a ball. In the case of α = N , the similar characteri...

We perform asymptotic analysis for the Euler–Riesz system posed in either $${\mathbb {T}}^d$$ or {R}}^d$$ high-force regime and establish a quantified relaxation limit result from to fractional porous medium equation. provide unified approach regardless of presence pressure case repulsive Riesz interactions, based on modulated energy estimates, Wasserstein distance order 2, bounded Lipschitz di...

the aim of this work is to describe the qualitative behavior of the solution set of a givensystem of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. in order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. this is done by the extension of ...

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...

this work is devoted to the study of global solution for initialvalue problem of interval fractional integrodifferential equationsinvolving caputo-fabrizio fractional derivative without singularkernel admitting only the existence of a lower solution or an uppersolution. our method is based on fixed point in partially orderedsets. in this study, we guaranty the existence of special kind ofinterv...

We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operat...