Three-Point Boundary Value Problems for Conformable Fractional Differential Equations

Abstract

We study a fractional differential equation using a recent novel concept of fractional derivative with initial and three-point boundary conditions. We first obtain Green's function for the linear problem and then we study the nonlinear differential equation.

New properties of conformable derivative

“…However, new derivatives should be proposed in order to deal better with the dynamics of the complex systems [2][3][4][5][6][7][8][9]. We have noticed that recently a new derivative has been suggested in [1,17,21,22] and it seems to satisfy all the requirements of the standard derivative. However, there is no discretization of this version in the literature.…”

Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with New Fractional Differentiation

“…There are few definitions of operators with fractional order, the Liouville-Caputo fractional derivative involving a kernel with singularity, and this definition is based on the power law and present singularity at the origin [9]. Recently, in order to solve the problem of singularity at the origin, Caputo and Fabrizio used the exponential decay law to construct a derivative with no singularity; however, the used kernel was local [10][11][12][13][14][15][16][17][18]. Thus, Atangana and Baleanu used the generalized Mittag-Leffler function to construct a derivative with no-singular and non-local kernel [19][20][21][22].…”

Lyapunov Inequality for a Boundary Value Problem Involving Conformable Derivative

“…Following this new conformable derivative, several papers have been presented, in particular some studies about boundary value problems for conformable differential equations have been the subject of some papers [3][4][5][6][7][8]18,24,26]. Furthermore, in [6], Batarfi et al studied a conformable differential equation of order α ∈ (1, 2], with three point boundary conditions and proved the existence and uniqueness of solution by using fixed point theorems.…”

“…Furthermore, in [6], Batarfi et al studied a conformable differential equation of order α ∈ (1, 2], with three point boundary conditions and proved the existence and uniqueness of solution by using fixed point theorems. In [7], Bayour et al solved an initial conformable differential value problem for α ∈ (0, 1) by the help of the tube solution method which is a generalization of the lower and upper solutions method.…”