In this paper, some theorems of the classical power series are generalized for the fractional power series. Some of these theorems are constructed by using Caputo fractional derivatives. Under some constraints, we proved that the Caputo fractional derivative can be expressed in terms of the ordinary derivative. A new construction of the generalized Taylor’s power series is obtained. Some applic...

In this paper, the generalized Taylor’s expansion is presented for fuzzy-valued functions. To achieve this aim, fuzzyfractional mean value theorem for integral, and some properties of Caputo generalized Hukuhara derivative are necessarythat we prove them in details. In application, the fractional Euler’s method is derived for solving fuzzy fractionaldifferential equations in the sense of Caputo...

The present study aims at indicating the existence and uniqueness result of system in extended colombeau algebra. The Caputo fractional derivative is used for solving the system of ODEs. In addition, Riesz fractional derivative of  Colombeau generalized algebra is considered. The purpose of introducing Riesz fractional derivative is regularizing it in Colombeau sense. We also give a solution to...

and Applied Analysis 3 fractional order to distributed order fractional. In Section 4, we introduce the distributed order fractional evolution systems C doD α t x t A C doD β t x t Bu t , x 0 x0, 0 < β < α ≤ 1, 1.5 where u t is control vector, and generalize the results obtained in Section 3 for this case. Finally, the conclusions are given in the last section. 2. Elementary Definitions and The...

In this note, we establish the sectorial property of the Caputo fractional derivative operator of order α ∈ (1, 2) with a zero Dirichlet boundary condition.

*Correspondence: [email protected] Department of Mathematics, School of Science, Xi’an University of Posts and Telecommunications, Chang’an Road, Xi’an, China Abstract In this paper, we study the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on a Caputo-Fabrizio fractional derivative. The new kernel of Capu...

in this paper, a new numerical method for solving fractional optimal control problems (focps) is presented. the fractional derivative in the dynamic system is described in the caputo sense. the method is based upon biorthogonal cubic hermite spline multiwavelets approxima-tions. the properties of biorthogonal multiwavelets are first given. the operational matrix of fractional riemann-lioville i...

Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary In the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann...

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canoni-cal Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange form...