Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities

Abstract

After the recent introduction of the Caputo-Fabrizio derivative by authors of the same names, the question was raised about an eventual comparison with the old version, namely, the Caputo derivative. Unlike Caputo derivative, the newly introduced Caputo-Fabrizio derivative has no singular kernel and the concern was about the real impact of this nonsingularity on real life nonlinear phenomena like those found in shallow water waves. In this paper, a nonlinear Sawada-Kotera equation, suitable in describing the behavior of shallow water waves, is comprehensively analyzed with both types of derivative. In the investigations, various fixed-point theories are exploited together with the concept of Piccard K-stability. We are then able to obtain the existence and uniqueness results for the models with both versions of derivatives. We conclude the analysis by performing some numerical approximations with both derivatives and graphical simulations being presented for some values of the derivative order γ. Similar behaviors are pointed out and they concur with the expected multisoliton solutions well known for the Sawada-Kotera equation. This great observation means either of both derivatives is suitable to describe the motion of shallow water waves.

Analysis and numerical simulation of cross reaction–diffusion systems with the Caputo–Fabrizio and Riesz operators

“…Fractional calculus has been described as the generalization of classical derivatives and integrals to fractional order types. The concept of fractional operators has been used to model a number of physical and real-life phenomena in applied physics, control system [15,31,32,34,39], economics and finance [6], mathematical ecology and epidemiology [14,29,35,37], underground water, hydrology [3-5, 13, 42], among several other processes.…”

High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo–Fabrizio fractional derivative

“…Following the definition in , Losada and Nieto analyzed and discussed some properties of the new Caputo–Fabrizio fractional derivative. So far, some scholars have started analytical and numerical studies based on the new Caputo–Fabrizio fractional derivative; see . However, the studies on the numerical methods for FPDEs with the Caputo–Fabrizio fractional derivatives have been rarely reported.…”

Notes on fuzzy fractional Sumudu transform

“…The amalgamation of fuzzy theory [3,11,23,24] with fractional calculus [2,8,17] has multiplied the practicality and expediency of calculus theory. Owing to the advantageous applications, both theories together have gained considerable attention in modeling different physical and engineering problems.…”