Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

Abstract

Physical phenomena and natural disasters, such as tsunamis and floods, are caused due to dispersive water waves and shallow waves caused by earthquakes. In order to analyze and minimize damaging effects of such situations, mathematical models are presented by different researchers. The Wu–Zhang (WZ) system is one such model that describes long dispersive waves. In this regard, the current study focuses on a non-linear (2 + 1)-dimensional time-fractional Wu–Zhang (WZ) system due to its importance in capturing long dispersive gravity water waves in the ocean. A Caputo fractional derivative in the WZ system is considered in this study. For solution purposes, modification of the homotopy perturbation method (HPM) along with the Laplace transform is used to provide improved results in terms of accuracy. For validity and convergence, obtained results are compared with the fractional differential transform method (FDTM), modified variational iteration method (mVIM), and modified Adomian decomposition method (mADM). Analysis of results indicates the effectiveness of the proposed methodology. Furthermore, the effect of fractional parameters on the given model is analyzed numerically and graphically at both integral and fractional orders. Moreover, Caputo, Caputo–Fabrizio, and Atangana–Baleanu approaches of fractional derivatives are applied and compared graphically in the current study. Analysis affirms that the proposed algorithm is a reliable tool and can be used in higher dimensional fractional systems in science and engineering.

From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

“…Moreover, the classical Halley method is a method without memory and needs three function evaluations per iteration as shown by its algorithm in (2). An attempt is made in [14] to obtain two memory-based methods while extending the idea of the Halley method when derived from a hyperbolic equation.…”

“…In order to numerically solve a number of ordinary and partial differential equations, it is necessary to transform them into linear or nonlinear equations. Root‐finding algorithms can be used in this context [2].…”

Exploring the Dynamics of Dark and Singular Solitons in Optical Fibers Using Extended Rational Sinh–Cosh and Sine–Cosine Methods

“…Plugging these expressions into Equation ( 8) with (41), one obtains the following soliton solution of (7):…”

“…Nonlinear wave phenomena manifest across a spectrum of engineering and scientific domains, encompassing fluid dynamics, plasma physics, optics, biology, condensed matter physics, and beyond [1,9]. To understand the nonlinear wave phenomena and construct the solitary wave solutions in the nonlinear partial differential equations, endeavors have been attempted to utilize the various techniques, one may refer [36][37][38][39][40][41]. In particular, in ref.…”

Analysis of the proportional Caputo-Fabrizio derivative

“…Since the proportional derivative is an important tool especially for engineering applications, the properties of this derivative should be investigated. For more details see [1,12,18,20].…”