Analysis of Nonlinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative

On Nonlinear Hybrid Fractional Differential Equations with Atangana-Baleanu-Caputo Derivative

In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving Atangana--Baleanu--Caputo (ABC) fractional derivative. We construct the equivalent fractional integral equation and establish the existence results through it. Further, we build up the theory of inequalities for ABC--hybrid fractional differential equations and use it to examine the uniqueness, existence of a maximal and minimal solution and the comparison results.

Fractional integral inequalities via Atangana-Baleanu operators for convex and concave functions

Recently, many fractional integral operators were introduced by different mathematicians. One of these fractional operators, Atangana-Baleanu fractional integral operator, was defined by Atangana and Baleanu in [2]. In this study, firstly, a new identity by using Atangana-Baleanu fractional integral operators are proved. Then, new fractional integral inequalities have been obtained for convex and concave functions with the help of this identity and some certain integral inequalities

Initial and Boundary Value Problems for Fractional differential equations involving Atangana-Baleanu Derivative

Initial value problem involving Atangana-Baleanu derivative is considered. An Explicit solution of the given problem is obtained by reducing the differential equation to Volterra integral equation of second kind and by using Laplace transform. To find the solution of the Volterra equation, the successive approximation method is used and a lemma simplifying the resolvent kernel has been presented. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form of series expansion using orthogonal basis obtained by …

New numerical scheme of Atangana-Baleanu fractional integral: an application to groundwater flow within leaky aquifer

Many physical problems can be described using the integral equations known as Volterra equations. There exists quite a number of analytical methods that can handle these equations for linear cases. For non-linear cases, a numerical scheme is needed to obtain an approximation. Recently a new concept of fractional differentiation was introduced using the Mittag-Leffler function as kernel, and the associate fractional integral was also presented. Up to this point there is no numerical approximation of this new integral in the literature. Therefore, to accommodate researchers working in the field …

基于Atangana-Baleanu演化方程的解析方法：拉普拉斯符号、弱解与正则性

Resolvent Approach to Atangana--Baleanu Evolution Equations: Laplace Symbols, Mild Solutions, and Regularity

本文提出基于Atangana-Baleanu导数的解析方法，通过引入分数解析和Mittag-Leffler核，建立了AB型演化方程的弱解表示，并推导了稳定性与正则性估计。

Comment on: "Stokes' first problem for heated flat plate with Atangana--Baleanu fractional derivative" [Chaos Solitons Fractals 117 (2018) 68]

In the sense of distributions, the derivative of the Heaviside unit step function $H(t)$ is a generalized Dirac-$δ$ distribution. If the velocity $V(t)$ of a flat plate is impulsive, as $V(t)=H(t)$ (i.e., it is suddenly set into motion with unit velocity at $t=0^+$), then its acceleration is $V'(t)=δ(t)$. The Dirac-$δ$ distribution has no point values. However, when the Dirac-$δ$ is the forcing term of an ODE (in $t$), it contributes to the solution. The recently published paper [Chaos Solitons Fractals 117 (2018) 68] incorrectly treats the Dirac-$δ$ function as being identically 0. This Comme…