Equal Width Equation Research Articles

Parameter identification of shallow water waves using the generalized equal width equation and physics-informed neural networks: a conservative approximation scheme

Parameter identification of shallow water waves using the generalized equal width equation and physics-informed neural networks: a conservative approximation scheme

Numerical treatment of the KP–MEW equation and the Konno–Oono equation using the first-integral method

Finding new, more widely applicable accurate traveling wave solutions to nonlinear partial differential equations is the aim of this work. The Konno–Oono equation and the Kadomtsev–Petviashvili modified equal width equation have various possible applications in mathematical physics and engineering disciplines, and their exact solutions are presented in this article. Using the first integral approach as an analytical method and the traveling wave transformation, we explore the precise traveling wave solutions. This approach is effective and yields unique exact solutions that fall into two categories: solutions in the form of trigonometric functions and solutions in the form of hyperbolic functions. Furthermore, graphical illustrations in two and three dimensions are showcased to offer a thorough explanation of their dynamic nature. In addition, it is critical to emphasize the significance of mastering the Konno–Oono equation and the KP–MEW equation, both of which have applications in a variety of fields Our findings are also crucial for understanding the many oceanographic applications that involve ocean gravity waves, offshore rigs in the water, energy from moving ocean waves, and several other related phenomena. This approach requires less computing work and is straightforward and succinct. That is its main advantage.

Dynamics of geometric shape solutions for space-time fractional modified equal width equation with beta derivative

The modified equal width equation describes the propagation of shallow water waves in which nonlinear and dispersive effects are significant, including phenomena such as wave breaking, soliton interactions, ion-acoustic waves, energy transfer in plasma, and nonlinear stress waves. The aim of this article is to establish some novel and generic solutions to the space-time fractional modified equal width (MEW) equation using the two variable (θ′/θ, 1/θ)-expansion technique, a modification of the (θ′/θ)-expansion method. A wide range of geometric shapes and inclusive soliton solutions have been constructed, comprising rational, trigonometric, and hyperbolic solutions, along with their integration to the equation under consideration. The two- and three-dimensional graphs of the solitons, including periodic, V-shaped, bell-shaped, singular periodic, flat kink-shaped, plane-shaped, peakon, and parabolic, illustrate the physical aspects of the solitary wave solution and the effect of the fractional parameter. The results demonstrate the efficiency, appropriateness, and reliability of the adopted technique for investigating fractional-order nonlinear evolution equations in science, technology, and engineering.

Analysis of Truncated M-Fractional Mathematical and Physical (2+1)-Dimensional Nonlinear Kadomtsev–Petviashvili-Modified Equal-Width Model

This study focuses on the mathematical and physical analysis of a truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width model. The distinct types of the exact wave solitons of an important real-world equation called the truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width (KP-mEW) model are achieved. This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, and long-wavelength water waves. The diverse patterns of waves on the oceans are yielded by the Kadomtsev–Petviashvili-modified equal-width (KP-mEW) equation. We obtain kink-, bright-, and periodic-type soliton solutions by using the expa function and modified extended tanh function methods. The solutions are more valuable than the existing results due to the use of a truncated M-fractional derivative. These solutions may be useful in different areas of science and engineering. The methods applied are simple and useful.

Construction of some new traveling wave solutions to the space-time fractional modified equal width equation in modern physics

Nonlinear fractional evolution equations are important for determining various complex nonlinear problems that occur in various scientific fields, such as nonlinear optics, molecular biology, quantum mechanics, plasma physics, nonlinear dynamics, water surface waves, elastic media and others. The space-time fractional modified equal width (MEW) equation is investigated in this paper utilizing a variety of solitary wave solutions, with a particular emphasis on their implications for wave propagation characteristics in plasma and optical fibre systems. The fractional-order problem is transformed into an ordinary differential equation using a fractional wave transformation approach. In this article, the polynomial expansion approach and the sardar sub-equation method are successfully used to evaluate the exact solutions of space-time fractional MEW equation. Additionally, in order to graphically represent the physical significance of created solutions, the acquired solutions are shown on contour, 3D and 2D graphs. Based on the results, the employed methods show their efficacy in solving diverse fractional nonlinear evolution equations generated across applied and natural sciences. The findings obtained demonstrate that the two approaches are more effective and suited for resolving various nonlinear fractional differential equations.

Modified simple equation technique for first-extended fifth-order nonlinear equation, medium equal width equation and Caudrey–Dodd–Gibbon equation

Modified simple equation technique for first-extended fifth-order nonlinear equation, medium equal width equation and Caudrey–Dodd–Gibbon equation

Analytic wave solutions to the beta-time fractional modified equal width equation based on two efficient approaches

By using the two distinct methods known as the exp(-ϕ(η))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\phi (\\eta ))$$\\end{document} and the Expa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Exp_a$$\\end{document}-function methods, various forms of soliton solutions of the modified equal width wave equation (MEWE) with beta time derivative (BTD) are produced in this study. This model is used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. The obtained solutions are in the form of rational, trigonometry and hyperbolic trigonometry functions. Using Mathematica software, the resulting solitons are validated. Graphs are also used at the conclusion to explain the findings. These soliton solutions imply that these two methods are more dependable, simple and efficient than other methods. The findings can be used to explain how studious structures and other comparable non-linear physical structures are substantially understood. The obtained results are very helpful in the fields of optics, kinetics solid-state physics and hydro-magnetic waves in cold plasma.

A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L2 and L∞ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.

Weakly restoring forces and shallow water waves with dynamical analysis of periodic singular solitons structures to the nonlinear Kadomtsev–Petviashvili-modified equal width equation

In this study, the nonlinear two-dimensional Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation is under investigation, which is described as a nonlinear wave model in weakly restoring forces and shallow water waves in the way of ferromagnetic, solitary waves in two dimensions with short amplitude and long wavelength and are also helpful in the investigation of various behaviors in nonlinear sciences. We successfully found the interesting and novel solitary wave results in bright solitons, kink solitons, dark solitons, anti-kink solitons, periodic singular solitons for nonlinear two-dimensional KP-MEW equation on the base of extension of simple equation technique under symbolic computational software Mathematica. The created solitons play an important role in nonlinear engineering and physics such as nonlinear optics, nonlinear dynamics, soliton wave theory, optical fiber and communication system. We are sure that these constructed results are novel and interesting which does not exist in the previous literature works. The graphical representation of constructed results is shown by 2D, 3D and contour graphs. The investigated research work proved that utilized technique is more concise, straightaway and efficient for study of other nonlinear partial differential equations.

Lie vector fields, conservation laws, bifurcation analysis, and Jacobi elliptic solutions to the Zakharov–Kuznetsov modified equal-width equation

Lie vector fields, conservation laws, bifurcation analysis, and Jacobi elliptic solutions to the Zakharov–Kuznetsov modified equal-width equation

Retracted: Study of Fuzzy Fractional Nonlinear Equal Width Equation in the Sense of Novel Operator

Retracted: Study of Fuzzy Fractional Nonlinear Equal Width Equation in the Sense of Novel Operator

Solitary wave solution to the space–time fractional modified Equal Width equation in plasma and optical fiber systems

Nonlinear fractional evolution equations play a crucial role in characterizing assorted complex nonlinear phenomena observed in different scientific fields, including plasma physics, quantum mechanics, elastic media, nonlinear optics, surface water waves, nonlinear dynamics, molecular biology, and some other domains. In this study, we investigate diverse solitary wave solutions to the space–time fractional modified Equal Width equation, focusing on their significance for wave propagation behaviors in plasma and optical fiber systems. Two effective approaches, namely the improved Bernoulli sub-equation function and the new generalized (G′/G)-expansion methods are exploited. A fractional wave transformation technique is employed to convert the fractional-order equation into an ordinary differential equation. The solitary wave solutions obtained in terms of exponential, hyperbolic, rational, and trigonometric functions, have wide applications in various nonlinear phenomena. The 3D, spherical, contour, and vector plots are presented to elucidate the physical implications of the obtained solutions for specific parameter values. The soliton structures reveal various wave types, including bell-shaped, multi-soliton, kink, single soliton, anti-bell-shaped, periodic soliton, compacton, and other types for definite values of the free parameters. We compare the attained solutions with the solutions available in the literature to demonstrate the feasibility of the solutions. It is observed that both approaches are efficient, straightforward, and significant for investigating diverse nonlinear models that modulate complex phenomena.

Further physical study about solution structures for the fractional equal width equation (EW) in shallow water hydrodynamics

In this article, with the help of Matlab R2021a software, the exact solutions of the EW are successfully examined by Sub-Equation method and the unified F-expansion method. The main novelty of this paper lies in the six aspects: (1) Singular periodic wave U-shaped solition is discovered by the sub equation method which did not happen in previous papers [35–37]. (2) More and more intensive multi-periodic M-shape wave solutions are also discovered by the unified F-expansion method which did not happen in previous papers [32,34–37]. (3) The dynamic wave behaviours about the got results and the effects of fractionality α are illustrated by Figs. 1–7. The fractional parameter α masters the appearance of the solitons and makes the involved alphabetic shaped solitions due to the increase in the fractionality number α in Figs. 1–7. (4) The effect about free parameters c,k can be seen from Figs. 8–11 in this paper. If only the increase of wave velocity parameter c or wave number parameter k, the energy trend of the wave is always positive and intensive. This also never happens in previous literatures [34–37]. (5) The Jumarie’s modified Riemann–Liouville and beta fractional and conformable derivatives are tested for accuracy in Figs. 12 and 13. We seem firstly compare the relations and distinctions among different fractional order derivatives in EW model. (6) In our best cognitive, the planar dynamical system about EW model whose phase portraits are also first studied. Furthermore, phase portraits for the space–time fractional equal width equation are plotted, and the corresponding sensitivity and chaotic behaviors are analyzed. The geometrical representations of the EW provide the corporal information to explain the physically phenomenons.

On soliton solutions of the modified equal width equation

PurposeThe soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived.Design/methodology/approachThe main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal.FindingsThus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained.Originality/valueIllustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems.

Exploring the dynamics of soliton waves: A comparative analysis of analytical and numerical methods for the modified Equal-Width equation

The modified Equal-Width (MEW) equation is a fundamental model in the realm of partial differential equations that is used to simulate one-dimensional wave propagation in nonlinear media, incorporating dispersion processes. In this study, we have employed two contemporary analytical and numerical methods to obtain exact traveling wave solutions for this model. The derived solutions have been evaluated using a numerical technique, demonstrating excellent accuracy. Various graphical representations, including a contour plot, a two-dimensional graph, and a three-dimensional graph, have been utilized to illustrate and analyze the results. Our study offers significant contributions to the field by providing new insights into the dynamics of solitary wave solutions of the MEW model. Furthermore, our method proves to be a valuable and effective mathematical tool that can be utilized to solve various nonlinear wave problems, making it highly applicable in a range of physical and mathematical studies. By showcasing the properties of the investigated model and the primary advantages of the employed techniques, our study highlights the potential for further developments and advancements in this field.

Two finite difference methods for solving the Zakharov–Kuznetsov-Modified Equal-Width equation

We derive the implementation of two meshless methods, the Space–Time Cloud Method and the Generalized Finite Difference Method, for solving the Zakharov–Kuznetsov-Modified Equal-Width equation, a nonlinear wave equation used to model the propagation of waves in nonuniform media. Also, we prove convergence of the GFD explicit scheme. We compare both methods in terms of accuracy and efficiency (execution times).

Structure of analytical and symbolic computational approach of multiple solitary wave solutions for nonlinear Zakharov‐Kuznetsov modified equal width equation

AbstractThe nonlinear two dimensional Zakharov‐Kuznetsov modified equal width equation investigated under the observation of extended modified rational expansion method and determined the multiple solitary wave solutions. The interested and important things in this work is the multiple solitary wave solutions which have various kinds of physical structure including anti‐kink soliton, travelling wave solutions, bright soliton, kink soliton, dark soliton, kink bright and dark solitons, anti‐kink bright and dark solitons. In our knowledge investigated various kinds of solitary solutions found first time under one method in the existing literatures. The constructed multiple solutions for nonlinear ZK‐MEW equation will play keen role in the investigation of different physical structure in nonlinear sciences. The investigated work prove that applied method is very efficient, reliable, and powerful.

Group Analysis of Equal-Width Equation

We study a third-order nonlinear equal width equation, which has been used for simulation of a one-dimensional wave propagation in a non-linear medium with dispersion process, by symmetry analysis. First, Lie point symmetries are obtained and used to reduce reduce the equal width equation thereby constructing exact solutions. Traveling waves are constructed using of a linear combination of space and time translation symmetries. We have used the multiplier technique to compute conservation laws.

A New Algorithm Based on the Decic (tenth degree) B-spline Functions for Numerical Solution of the Equal Width Equation

In this study, a new algorithm is introduced for the numerical solution of equal width (EW) equation. This algorithm is created by using the collocation finite element method based on decic B-spline functions for the space discretization of the EW equation and the Crank-Nicolson method for the time discretization of his equation. The obtained results are compared with the previous ones to see the efficiency and accuracy of the proposed method.

A novel study of the nonlinear Kadomtsev–Petviashvili-modified equal width equation describing the behavior of solitons

A novel study of the nonlinear Kadomtsev–Petviashvili-modified equal width equation describing the behavior of solitons