The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations

Abstract

In this paper, the existence and uniqueness problem of the initial and boundary value problems of the linear fractional Caputo-Fabrizio differential equation of order σ ∈ (1, 2] have been investigated. By using the Laplace transform of the fractional derivative, the fractional differential equations turn into the classical differential equation of integer order. Also, the existence and uniqueness of nonlinear boundary value problem of the fractional Caputo-Fabrizio differential equation has been proved. An application to mass spring damper system for this new fractional derivative has also been presented in details.

A Bilocal Problem Associated to a Fractional Differential Inclusion of Caputo-Fabrizio Type

“…(I), D α CF x(t) = f (t), t ∈ I and x(0) = x 0 , x(1) = x 1 . In order to prove our results we also need the next result proved in [11] (namely, Theorem 3.4).…”

“…Properties of this definition have been studied in [5][6][7][8]. Several recent papers are devoted to qualitative results for fractional differential equations and inclusions defined by Caputo-Fabrizio fractional derivative [9][10][11][12]. The aim of the present paper is to study the set-valued framework for problems defined by Caputo-Fabrizio operator.…”

On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions

“…Geometric and physical interpretation of fractional differentiation and integration can be found in the paper [27]. Existence results for fractional differential equations have studied and developed by many authors; see the books [26,4,2] and references [11,12,24,15,9,26,4,2,17,18,19,30,31,39,41,42,43,44,45] and references therein.…”

On Lyapunov-type inequalities for boundary value problems of fractional Caputo-Fabrizio derivative

“…We follow the idea of the paper [12] and we assume that the nontrivial solution satisfies u ′′ (a) = 0 . We mention that the related works involving the Caputo-Fabrizio derivative of order α ∈ (1, 2] with the Dirichlet boundary condition is presented in [8] and the Caputo-Fabrizio derivative of order α ∈ (2, 3] with the mixed boundary condition is studied in [14].…”