Existence of solutions for fractional interval--valued differential equations by the method of upper and lower solutions

In this work we firstly study some important properties of fractional calculus for interval-valued functions and introduce the concepts of upper and lower solutions for intervalvalued Caputo fractional differential equations. Then, we prove an existence result for intervalvalued Caputo fractional differential equations by use of the method of upper and lower solutions. Finally several examples will be presented to illustrate our abstract results

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Fractional Vector Calculus and Fractional Maxwell's Equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered.Comment: 42 pages, LaTe

Stochastic Calculus with respect to Gaussian Processes

Stochastic integration with respect to Gaussian processes, such as fractional Brownian motion (fBm) or multifractional Brownian motion (mBm), has raised strong interest in recent years, motivated in particular by applications in finance, Internet traffic modeling and biomedicine. The aim of this work to define and develop, using White Noise Theory, an anticipative stochastic calculus with respect to a large class of Gaussian processes, denoted G, that contains, among many other processes, Volterra processes (and thus fBm) and also mBm. This stochastic calculus includes a definition of a stochastic integral, It\^o formulas (both for tempered distributions and for functions with sub-exponential growth), a Tanaka Formula as well as a definition, and a short study, of (both weighted and non weighted) local times of elements of G . In that view, a white noise derivative of any Gaussian process G of G is defined and used to integrate, with respect to G, a large class of stochastic processes, using Wick products. A comparison of our integral wrt elements of G to the ones provided by Malliavin calculus in [AMN01] and by It\^o stochastic calculus is also made. Moreover, one shows that the stochastic calculus with respect to Gaussian processes provided in this work generalizes the stochastic calculus originally proposed for fBm in [EVdH03, BS{\O}W04, Ben03a] and for mBm in [LLV14, Leb13, LLVH14]. Likewise, it generalizes results given in [NT06] and some results given in [AMN01]. In addition, it offers alternative conditions to the ones required in [AMN01] when one deals with stochastic calculus with respect to Gaussian processes.Comment: (26/07/2014). Previously this work appeared as arXiv:1703.08393 which was incorrectly submitted as a new paper (and has now been withdrawn

Fractional Derivative as Fractional Power of Derivative

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.Comment: 20 pages, LaTe

Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^{o} SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_n^{-1}$, where $\gamma_n=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $\gamma_n=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $\gamma_n=n$ if $H>\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $\gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.Comment: Published at http://dx.doi.org/10.1214/15-AAP1114 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org