Global Existence and Uniqueness of Solution of Atangana–Baleanu Caputo Fractional Differential Equation with Nonlinear Term and Approximate Solutions

Abstract

In this paper, a class of fractional order differential equation expressed with Atangana–Baleanu Caputo derivative with nonlinear term is discussed. The existence and uniqueness of the solution of the general fractional differential equation are expressed. To present numerical results, we construct approximate scheme to be used for producing numerical solutions of the considered fractional differential equation. As an illustrative numerical example, we consider two Riccati fractional differential equations with different derivatives: Atangana–Baleanu Caputo and Caputo derivatives. Finally, the study of those examples verifies the theoretical results of global existence and uniqueness of solution. Moreover, numerical results underline the difference between solutions of both examples.

A decreasing operator method for a fractional initial value problem on infinite interval

“…It is known that there have a large of fractional derivatives, in which Caputo fractional derivative is important, and the theory of Caputo fractional differential equations also greatly attracted the attention for their wide applications; see, e.g. [1,2,7,10,15,17,19,21,23] and the related references therein.…”

Existence Solutions of ABC-Fractional Differential Equations with Periodic and Integral Boundary Conditions

“…Different techniques for fractional derivatives have been proposed in research investigations, including Riemann-Liouville, Caputo, Caputo-Fabrizio, Caputo-Hadamard, Grunwald-Letnikov, and Atangana-Baleanu derivatives. Many mathematicians have worked on difficulties concerning fractional differential equations' existence and uniqueness [8][9][10][11]. Various well-known approaches related to fixed point theory, such as Banach and Krasnoselskii's fixed point theorems [11], are frequently applied.…”

“…Many mathematicians have worked on difficulties concerning fractional differential equations' existence and uniqueness [8][9][10][11]. Various well-known approaches related to fixed point theory, such as Banach and Krasnoselskii's fixed point theorems [11], are frequently applied.…”

Taylor’s Formula for Generalized Weighted Fractional Derivatives with Nonsingular Kernels

“…That gave rise to the appearance of fractional derivatives with nonsingular kernels [31,32] and, as a consequence, to the need to obtain Taylor's formulas for such kinds of operators [33]. In particular, in [34], Fernandez and Baleanu established analogues of Taylor's theorem for fractional differential operators defined using a Mittag-Leffler kernel and a mean value theorem for the Atangana-Baleanu-Caputo (ABC) fractional derivative, introduced in [35] and now under strong current investigations [36][37][38]. Here, we consider the generalized weighted fractional derivative in Caputo sense, as introduced in 2020 by Hattaf [39,40].…”