Linearized Asymptotic Stability for Fractional Differential Equations

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector \{\lambda \in \C : |\arg \lambda| > \frac{\alpha \pi}{2}\} where $\alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable

On the Nonlinear Impulsive Ψ\Psi--Hilfer Fractional Differential Equations

In this paper, we consider the nonlinear $\Psi$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $\Psi$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.Comment: 2

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of viscoelasticity and in the system identfica- tion theory. Also, the existence and uniqueness of a solution to a general linear fractional differential equation in the space of tempered distributions is given

Fractional differential equations solved by using Mellin transform

In this paper, the solution of the multi-order differential equations, by using Mellin Transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands.Comment: 19 pages, 2 figure

Finite Domain Anomalous Spreading Consistent with First and Second Law

After reviewing the problematic behavior of some previously suggested finite interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space-time fractional differential equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space-time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.Comment: 14 figure

General Connectivity Distribution Functions for Growing Networks with Preferential Attachment of Fractional Power

We study the general connectivity distribution functions for growing networks with preferential attachment of fractional power, $\Pi_{i} \propto k^{\alpha}$, using the Simon's method. We first show that the heart of the previously known methods of the rate equations for the connectivity distribution functions is nothing but the Simon's method for word problem. Secondly, we show that the case of fractional $\alpha$ the $Z$-transformation of the rate equation provides a fractional differential equation of new type, which coincides with that for PA with linear power, when $\alpha = 1$. We show that to solve such a fractional differential equation we need define a transidental function $\Upsilon (a,s,c;z)$ that we call {\it upsilon function}. Most of all previously known results are obtained consistently in the frame work of a unified theory.Comment: 10 page