Analysis of a class of boundary value problems depending on left and right Caputo fractional derivatives

In this work we study boundary value problems associated to a nonlinear fractional ordinary differential equation involving left and right Caputo derivatives. We discuss the regularity of the solutions of such problems and, in particular, give precise necessary conditions so that the solutions are C1([0, 1]). Taking into account our analytical results, we address the numerical solution of those problems by the augmented-RBF method. Several examples illustrate the good performance of the numerical method.P.A. is partially supported by FCT, Portugal, through the program “Investigador FCT” with reference IF/00177/2013 and the scientific projects PEstOE/MAT/UI0208/2013 and PTDC/MAT-CAL/4334/2014. R.F. was supported by the “Fundação para a Ciência e a Tecnologia (FCT)” through the program “Investigador FCT” with reference IF/01345/2014.info:eu-repo/semantics/publishedVersio

Coupled systems of fractional equations related to sound propagation: analysis and discussion

In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in the constitutive equations. By doing so, we embrace a vast phenomenology, including subdiffusive, superdiffusive and also memoryless processes like classical diffusions. From a mathematical point of view, we study systems of coupled fractional equations, leading to fractional diffusion equations or to equations with sequential fractional derivatives. In this framework we also propose a method to solve partial differential equations with sequential fractional derivatives by analysing the corresponding coupled system of equations

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 a fractional differential equation with infinitely many solutions

We present a set of restrictions on the fractional differential equation $x^{(\alpha)}(t)=g(x(t))$, $t\geq0$, where $\alpha\in(0,1)$ and $g(0)=0$, that leads to the existence of an infinity of solutions starting from $x(0)=0$. The operator $x^{(\alpha)}$ is the Caputo differential operator

Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations

We propose and study a class of numerical schemes to approximate time fractional differential equations. The methods are based on the approximation of the Caputo fractional derivative by continuous piecewise polynomials, which is strongly related to the backward differentiation formulae for the integer-order case. We investigate their theoretical properties, such as the local truncation error and global error analyses with respect to a sufficiently smooth solution, and the numerical stability in terms of the stability region and $A(\frac{\pi}{2})$-stability by refining the technique proposed in \cite{LubichC:1986b}. Numerical experiments are given to verify the theoretical investigations.Comment: 34 pages, 3 figure

Complex variable approach to analysis of a fractional differential equation in the real line

The first aim of this work is to establish a Peano type existence theorem for an initial value problem involving complex fractional derivative and the second is, as a consequence of this theorem, to give a partial answer to the local existence of the continuous solution for the following problem with Riemann-Liouville fractional derivative: \begin{equation*} \begin{cases} &D^{q}u(x) = f\big(x,u(x)\big), \\ &u(0)=b, \ \ \ (b\neq 0). \\ \end{cases} \end{equation*} Moreover, in the special cases of considered problem, we investigate some geometric properties of the solutions.Comment: 14 page