Novel Recursive Approximation for Fractional Nonlinear Equations within Caputo-Fabrizio Operator

Mehmet Yavuz,C Cattani,Z Hammouch,H Bulut,H.M Baskonus,S Agoujil,T Mekkaoui,A Atangana

doi:10.1051/itmconf/20182201008

Mehmet Yavuz, C Cattani + Show 6 more

Open Access

https://doi.org/10.1051/itmconf/20182201008

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Journal: ITM Web of Conferences	Publication Date: Jan 1, 2018
Citations: 5	License type: CC BY 4.0

Affiliation: Necmettin Erbakan University

Abstract

This study displays a novel method for solving time-fractional nonlinear partial differential equations. The suggested method namely Laplace homotopy method (LHM) is considered with Caputo-Fabrizio fractional derivative operator. In order to show the efficiency and accuracy of the mentioned method, we have applied it to time-fractional nonlinear Klein-Gordon equation. Comparisons between obtained solutions and the exact solutions have been made and the analysis shows that recommended solution method presents a rapid convergence to the exact solutions of the problems.

Highlights

In the last decades, fractional calculus were applied to various fields of mathematical and physical analysis such as modelling some common viruses, estimation financial processes, diffusion, relaxation processes and so on [1,2,3,4,5,6,7,8,9,10,11]
In order to find out the efficiency and superiority of the fractional differential equations, a lot of studies were emerged by some scientists [12,13,14,15,16,17]
Laplace homotopy method which is defined with a new fractional operator is considered

Summary

Introduction

Fractional calculus were applied to various fields of mathematical and physical analysis such as modelling some common viruses, estimation financial processes, diffusion, relaxation processes and so on [1,2,3,4,5,6,7,8,9,10,11]. In order to find out the efficiency and superiority of the fractional differential equations, a lot of studies were emerged by some scientists [12,13,14,15,16,17]. Some important fractional derivative operators have been developed such as Caputo-Fabrizio [18] and Atangana-Baleanu [19]. These operators are very important to model the complex nonlinear fractional dynamical systems and to solve them. We use the Caputo-Fabrizio fractional operator to solve nonlinear KleinGordon equation by considering the Laplace homotopy method

Caputo-Fabrizio Operator and its Main Properties

Role of the Suggested Fractional Operator in the Fundamental Method

Solution of Nonlinear Fractional Klein-Gordon Equation

Conclusions