In this paper, the long-time behavior of Cesaro mean fundamental solution for fractional Heat equation corresponding to random time changes in Brownian motion is studied. We consider both stable subordinators leading equations with Caputo-Djrbashian derivative and more general cases differential-convolution operators, particular, distributed order derivatives.

In this paper, we further discuss the properties of three kinds of fractional derivatives: the Grünwald–Letnikov derivative, the Riemann–Liouville derivative and the Caputo derivative. Especially, we compare the Riemann–Liouville derivative with the Caputo derivative. And sequential property of the Caputo derivative is also derived, which is helpful in translating the higher fractional-order di...

Keywords: Relative controllability Time delays Distributed delays Fractional derivative Mittag–Leffler function a b s t r a c t This paper is concerned with the controllability of nonlinear fractional dynamical systems with time varying multiple delays and distributed delays in control defined in finite dimensional spaces. Sufficient conditions for controllability results are obtained using the...

Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous diffus...

In this paper, the fractional Sturm-Liouville problems, in which the second order derivative is replaced by a fractional derivative, are derived by the Homotopy perturbation method. The fractional derivatives are described in the Caputo sense. The present results can be implemented on the numerical solutions of the fractional diffusion-wave equations. Numerical results show that HPM is effectiv...

The usual approach to the integration of fractional order initial value problems is based on Caputo derivative, whose conditions are used formulate classical integral equation. Thanks an elementary counter example, we demonstrate that this technique leads wrong free-response transients. solution fundamental problem use frequency-distributed model integrator and its distributed conditions. Using...

The aim of this work is to describe the qualitative behavior of the solution set of a given system 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...