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

Meryeme Hassouna,El Hassan El Kinani,Abdelaziz Ouhadan,Gaston Mandata N Gu R Kata

doi:10.1155/2021/5675789

Meryeme Hassouna, El Hassan El Kinani + Show 2 more

Open Access

https://doi.org/10.1155/2021/5675789

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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.

Highlights

Fractional calculus is considered to be a generalization of the integer-order calculus. is field has a long motivated history that started with Leibniz’s letter to Hospital, where the meaning of half derivative was first considered
Where some known methods related to the fixed point theory are often used such as Banach and Kranoselkii fixed point theorems. ey have established many fundamental theorems of local and global existence and uniqueness, and the conditions of each theorem could change by changing the derivative of the fractional differential problem. en, many authors investigated the existence and uniqueness of general fractional differential equation with different forms
We are interested by a fractional differential equation involving Atangana–Baleanu Caputo and a nonlinear term N(y)

Summary

Introduction

Fractional calculus is considered to be a generalization of the integer-order calculus. is field has a long motivated history that started with Leibniz’s letter to Hospital, where the meaning of half derivative was first considered. Lin in [9] have proposed an organized analysis of local and global existence and uniqueness of a general problem expressed with Caputo fractional derivative CDα0,t: International Journal of Differential Equations. Chidouh et al in [20] have considered the existence and uniqueness of local positive solution in C[0, 1] of the fractional differential equation using the Caputo fractional derivative:. Where the authors proved the local existence using Banach fixed point theorem for solutions y ∈ H1[0, 1] or y ∈ C[a, b]; ABCDα0,t is the Atangana–Baleanu Caputo fractional derivative. Theorems of existence and uniqueness and stability of solutions are related directly to the properties of the differential operator of the studied problem. U is closed convex. en, Schauder’s fixed point theorem [25] claims that A has a fixed point, which is a solution of our initial value problem (4)

Global Existence and Uniqueness of Solution

Illustrative Example

Numerical Results

Conclusion