Application of the Caputo-Fabrizio Fractional Derivative Without Singular Kernel to Korteweg-De Vries-Burgers Equation∗

Abstract

In order to bring a broader outlook on some unusual irregularities observed in wave motions and liquids’ movements, we explore the possibility of extending the analysis of Korteweg–de Vries–Burgers equation with two perturbation’s levels to the concepts of fractional differentiation with no singularity. We make use of the newly developed Caputo-Fabrizio fractional derivative with no singular kernel to establish the model. For existence and uniqueness of the continuous solution to the model, conditions on the perturbation parameters ν, µ and the derivative order α are provided. Numerical approximations are performed for some values of the perturbation parameters. This shows similar behaviors of the solution for close values of the fractional order α.

Stability analysis of epidemic models of Ebola hemorrhagic fever with non-linear transmission

“…New fractional derivatives with no singular kernel were recently proposed by many authors including Caputo et al in [13], Doungmo Goufo [19], and a version with non-local and non-singular kernel was introduced by Atangana and Baleanu [5]. However, Caputo fractional derivative [12], for instance, remains the one mostly used for modelling real world problems in the field [9,11,17,18,20].…”

A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations

“…Baleanu et al [22] obtained the approximate solution for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations. Goufo [23] used the fractional derivative of the newly developed Caputo-Fabrizio without singular kernel to establish the Korteweg-de Vries-Burgers equation with two perturbation levels. Atangana and Nieto [24] studied the numerical approximation of this new fractional derivative and established an improved RLC circuit model.…”

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

“…These kinds of fractional power kernel put more weight in the present than in the past, which is particularly suitable to describe physical processes in the material with short memory, but is incapable of properly describing some behavior observed in materials with huge heterogeneities and structures with different scales. In [22], a new kind of fractional KdV model was established by using the developed Caputo-Fabrizio fractional derivative, and the conditions of existence and uniqueness of the continuous solution were provided. In [23], a time-fractional Shamel-KdV equation with Riesz fractional derivatives was derived to describe nonlinear behavior of dust-ion acoustic waves.…”