Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform

Abstract

A new integral transform method for regularized long-wave (RLW) models having fractional-order is presented in this study. Although analytical approaches are challenging to apply to such models, semianalytical or numerical techniques have received much attention in the literature. We propose a new technique combining integral transformation, the Elzaki transform (ET), and apply it to regularized long-wave equations in this study. The RLW equations describe ion-acoustic waves in plasma and shallow water waves in seas. The results obtained are extremely important and necessary for describing various physical phenomena. This work considers an up-to-date approach and fractional operators in this context to obtain satisfactory approximate solutions to the proposed problems. We first define the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for solving RLW equations. We can readily obtain numerical results that provide us with improved approximations after only a few iterations. The derived solutions were found to be in close contact with the exact solutions. Furthermore, the suggested procedure has attained the best level of accuracy. In fact, when compared to other analytical techniques for solving nonlinear fractional partial differential equations, the present method might be considered one of the finest.

Novel Analysis of Nonlinear Seventh Order Kaup-Kupershmidt Equation via the Caputo Operator

“…A variety of mathematical methods that have been created and studied [14,15,16,17,18,19] have been used to obtain the precise solution of NLFDEs. For example, the q-homotopy analysis transform approach for Navier-Stokes equations having fractional-order [20], Natural transform decomposition method for fractional modiőed Boussinesq and approximate long wave equations [24] and fractional-order kaup-kupershmidt equation [23], Yang transform decomposition method for time-fractional Fisher's equation [22] and for time-fractional Noyes-Field model [21], Elzaki homotopy perturbation technique for solving regularized long-wave equations of order fraction [25], Variational iteration transform method for fractional third order Burgers and KdV nonlinear systems [26], Modiőed Khater method for solving nonlinear fractional Ostrovsky equation [27], modiőed ( G ′ G )-expansion scheme for travelling wave solutions of fractional Boussinesq equation [28], generalized Kudryashov method for nonlinear FPDEs of Burgers type [29], Laplace residual power series approach for solving Black-Scholes Option pricing equations having fractional-order [30], őrst integral method for solving fractional Cahn-Allen equation and fractional DSW system [31] and many more [32,33,34,35,36,37,38].…”

A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator

“…The main focus of their studies was the systematic comprehension of FC, including uniqueness and existence. The theory of fractional-order calculus has been linked to real-world projects and used in a variety of fields, including electrodynamics [12], chaos theory [13], optics [14], signal processing [15], and other areas [16][17][18][19][20][21][22].…”

Fractional Hamilton’s Canonical Equations and Poisson Theorem of Mechanical Systems with Fractional Factor

“…In the study of dynamics, Lagrange mechanics and Hamiltonian mechanics can be established by using fractional derivatives, which write non-conservative forces directly into the Lagrange and Hamiltonian functions in terms of fractional derivatives [13][14][15][16]. In addition to the application of the above problems, fractional calculus has also been applied to Maxwell fluid [17,18], unsmooth boundary [19][20][21][22], porous medium [23,24], numerical calculations [25,26], etc. This is an active field, and various fractional models have been proposed by many researchers studying in it, such as fractional Biswas-Milovic model [27], fractional (1+1)-dimensional SRLW equation [28], fractional Riccati differential equation [29], time-fractional K(m, n) equation [30], fractional optical fiber Schrodinger models [31], etc.…”