Khater Method Research Articles

Investigation of Space-Time Dynamics of Akbota Equation using Sardar Sub-Equation and Khater Methods: Unveiling Bifurcation and Chaotic Structure

This paper focuses on obtaining exact solutions of nonlinear Akbota equation through the application of the modified Khater method and Sardar sub-equation method. Renowned as one of the latest and precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The equation is crucial in the study of optical solitons, which are stable pulses of light that maintain their shape over long distances. The Akbota equation helps in understanding the behavior and stability of these solitons. The governing equation undergoes transformation into an ordinary differential equation through a well-suited wave transformation. This analytical simplification paves the way for the derivation of trigonometric, hyperbolic, and rational solutions through the proposed methods. To illuminate the physical behavior of the model, the study presents graphical plots of the selected solutions of Khater and Sardar sub-equation method. This visual representation, achieved by selecting appropriate values for arbitrary parameters, enhances the understanding of the system’s dynamics. All calculations in this study are meticulously conducted using the Mathematica and Maple software, ensuring accuracy and reliability in the analysis of the obtained solution. Furthermore we investigate the sensitivity analysis of the dynamical system.

Modulation instability and optical wave profiles for the conformable Schrödinger–Poisson dynamical system

This manuscript studies the exact solitary wave profiles for the conformable Schrödinger–Poisson dynamical system. This system has a significant role in gravity’s quantum state operation approximates the interaction between quantum mechanics and gravitation. The diverse exact solitary wave profiles are constructed by using the Khater method. The different closed-form solutions are obtained such as dark, singular, hyperbolic, periodic, and plane waves. Moreover, the modulation instability is also discussed for this dynamical system. To, show the physical significance of desired results we plot the 3D and 2D plots along with the related contour graphs, which are also drawn. The constraint criteria for the exact solutions are also exposed.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Novel soliton insights into generalized fractional Tzitzéica-type evolution equations using the modified Khater method

Novel soliton insights into generalized fractional Tzitzéica-type evolution equations using the modified Khater method

Bifurcation analysis and soliton solutions to the doubly dispersive equation in elastic inhomogeneous Murnaghan's rod.

The doubly dispersive (DD) equation finds extensive utility across scientific and engineering domains. It stands as a significant nonlinear physical model elucidating nonlinear wave propagation within the elastic inhomogeneous Murnaghan's rod (EIMR). With this in mind, we have focused on the integration of the DD model and the modified Khater (MK) method. Through the wave transformation, this model is effectively converted into an ordinary differential equation. In this paper, the goal of our work is to explore new wave solutions to the DD model by using the MK scheme. These solutions provide extremely helpful insights into the operation of the system. The three-dimensional (3D) plot and two-dimensional (2D) combined plot via the impacts of the parameters are provided for various parameters in this manuscript. We also discussed the dynamical properties of the model, which are accomplished through the bifurcation analysis, and also found the Hamiltonian function. This research makes a substantial contribution to the area by increasing our understanding of wave solutions in the DD, introducing novel investigation tools, and carrying out an in-depth investigation of the bifurcation and stability aspects of the system. As a direct result of this research, novel openings have been uncovered for further investigation and application in the various disciplines of science and engineering.

Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations

In this study, we use the Khater Method (KM) as an efficient analytical tool to solve (3+1)-dimensional fractional extended shallow water wave equations (FESWWEs) with conformable derivatives. The KM transforms fractional partial differential equations to ordinary differential equations (ODEs) via strategic variable transformation. Then, series-form solutions to these ODEs are proposed, which turn them into nonlinear algebraic systems. The solution to this set of algebraic equations yields shock travelling wave solutions expressed in hyperbolic, trigonometric, exponential, and rational functions. The study’s findings are corroborated by 2D, 3D, and contour graphs that show the changing patterns of the detected shock travelling waves. These findings have important significance for the discipline, offering vital insights into the intricate dynamics of FESWWEs. The effectiveness of KM is demonstrated by its capacity to produce varied solutions and contribute to a thorough knowledge of such complex phenomena.

The analytical study of soliton dynamics in fractional coupled Higgs system using the generalized Khater method

The analytical study of soliton dynamics in fractional coupled Higgs system using the generalized Khater method

Dynamics of propagation patterns: An analytical investigation into fractional systems

Recent years have seen a growing interest in fractional differential equations, particularly the fractional Chaffee-Infante ([Formula: see text]) equation, pivotal for understanding dynamics governed by fractional orders in specific physical systems. Exploring solitary wave solutions, this study employs the extended Khater method and truncated Mittag-Leffler function properties to formulate tailored solutions for the ([Formula: see text]) model. Through a traveling wave ansatz, the equation transforms into a nonlinear ordinary differential equation, revealing intricate propagation patterns of solitary waves. Visual representations aid comprehension, while rigorous validation ensures solution precision, ultimately providing a comprehensive understanding of system responses to external stimuli. This study effectively integrates analytical and numerical methodologies to derive precise solitary wave solutions, with significant implications for advancing comprehension of complex phenomena in various disciplines governed by fractional-order dynamics.

Comment on the article published in Math Meth Appl Sci. 2021;44:2682–2691

Khater method is wrong. This wrong method has been amended by El‐Ganaini et al. in Pramana – J. Phys. (2019) 93:13. The so‐called modified Khater method has been applied in Sec. 2.1 of Math Meth Appl Sci. 2021;44:2682–2691. In this comment, we show that the modified Khater method does not belong to Mostafa M.A. Khater. It should be called El‐Ganaini et al. method.

Solutions of (3+1)-dimensional extended quantum nonlinear Zakharov–Kuznetsov equation using the generalized Kudryashov method and the modified Khater method

Solutions of (3+1)-dimensional extended quantum nonlinear Zakharov–Kuznetsov equation using the generalized Kudryashov method and the modified Khater method

A novel approach to the motion equations of null Cartan curves via the compatible Hasimoto map

In this manuscript, we try to understand the links between nonlinear equations of motion and equations attained by taking geometric differential description of Hasimoto map of null Cartan curves and some physical results. First, we obtain the equations of Hasimoto transformation occurred by applying Hasimoto map for null Cartan curves. Then, based on the Hasimoto map of null Cartan curves, we examine both Landau–Lifshitz system and visco Landau–Lifshitz system. Lastly, we portray the links of findings obtained by geometric physical issues. After all, we propose a new method, known as Khater method, to obtain solutions of fractional order differential equations. Also, these solutions are simulated by graphically using physical constructions.

Solitary wave solution of (2+1)-dimensional Chaffee–Infante equation using the modified Khater method

Most of the nonlinear phenomena are described by partial differential equation in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology and mathematical finance. The solutions of a wide range of nonlinear evolution equations exhibit the wave behavior corresponding to the underlying physical systems. In particular, solitary wave solutions and soliton solutions are of great interest for researchers owing to many applications in different areas of science. The behavior of gas diffusion in a homogeneous medium is described by the (2+1)-dimensional Chaffee–Infante equation. In the present study, the modified Khater method is used to solve (2+1)-dimensional Chaffee–Infante equation because it provides various forms of solitons. The bright, dark, and periodic traveling wave patterns are produced by choosing different values of parameters. The solutions are presented graphically through 2D, 3D and contour graphs. The obtained solutions are demonstrated that the modified Khater method is a more useful tool than the existing techniques for solving such nonlinear problems.

An investigation of a closed-form solution for non-linear variable-order fractional evolution equations via the fractional Caputo derivative

Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.

Optical modelling of the space-time fractional Eckhaus equation

In this paper, the space-time fractional Eckhaus equation is considered and solved using the a direct method (Khater method) to obtain exact solutions. This meth?od produces more solutions when compared to other known methods. The real solutions of this equation are classified as travelling wave, kink, periodic and sol?itary wave solutions. These solutions are searched with the help of the fractional conformable derivative sense. Some graphs and tables are drawn to interpret the solutions and method. With the interpretation of the results, it is explained that the method used is a reliable, effective, powerful and easily applicable technique for obtaining the solutions of fractional differential equations classes in many fields.

UNRAVELING THE COMPLEX DYNAMICS OF FLUID FLOW IN POROUS MEDIA: EFFECTS OF VISCOSITY, POROSITY, AND INERTIA ON THE MOTION OF FLUIDS

This study investigates novel solitary wave solutions of the Gilson–Pickering ([Formula: see text]) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The [Formula: see text] equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences.

Dimensionless Zakharov equation; high-frequency Langmuir waves and low-frequency ion-acoustic waves’ interaction

This study investigates the physical significance of the dimensionless Zakharov equation ZE, which theoretically describes the interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves. Coastal engineering, fluid dynamics, nonlinear optics, and plasma wave physics are all fields of study that explore this relationship. We use the modified Khater method, the auxiliary equation approach, and the Khater II method, all cutting-edge ways to compute, to come up with a wide range of new and different solutions. Other physical aspects of this interaction are briefly explained below. To figure out the originality of our solutions, we have to compare our results to those of other published studies.

Prorogation of waves in shallow water through unidirectional Dullin–Gottwald–Holm model; computational simulations

This paper investigates novel solitary wave solutions of the unidirectional Dullin–Gottwald–Holm model and employs the modified Khater (MKhat) method for studying the dynamical characterization of the prorogation of waves in shallow water. There are various solution types obtained such as kink, periodic, cone, anti-kink, etc. The accuracy of these solutions is checked by implementing He’s variational iteration technique. The analytical and numerical solutions are numerically simulated through 3D, 2D and contour plots for a visual explanation of the shallow water waves’ propagation and the match between both kinds of solutions. Additionally, the interaction between solutions is explained by some stream plots to show the local direction of the vector field at each point and a roughly uniform density throughout the property, which indicates no background scalar field. The novelty of the study’s solutions is explained by comparing it with the previously published articles.

New distinct optical dynamics of the beta-fractionally perturbed Chen–Lee–Liu model in fiber optics

In this paper, we have studied the new distinct optical solutions of the fractional form of the perturbed Chen–Lee–Liu model (CLLM). The beta-fractional derivative form of the CLLM is considered here. The modified Khater method is used to generate the solutions. The solutions yield distinct wave characteristics for different fractional and parametric values. Anti-peakon, two-dark soliton solution, periodic, w-shape, bright, dark, combined dark–bright, bell shape, anti-bell shape, two-soliton solution, grey type periodic solution, and combined anti-peakon-bright solutions are the obtained wave solutions. Graphical representations are used to examine the physical interpretation of the resulting solutions. These solutions are extremely useful in the field of fiber optics as well as plasma physics.

TRAVELING WAVE SOLUTIONS TO A MATHEMATICAL MODEL OF FRACTIONAL ORDER (2+1)-DIMENSIONAL BREAKING SOLITON EQUATION

The aim of this study is to consider solving an important mathematical model of fractional order ([Formula: see text])-dimensional breaking soliton (Calogero) equation by Khater method. The derivatives are in the local fractional derivative sense. The fractional transformation equation is utilized to convert the proposed nonlinear fractional order differential equation into nonlinear ordinary differential equation. The Khater method is used to construct the closed-form traveling wave solutions of the said fractional differential equation. In addition, many new exact solutions are constructed. This shows that the Khater method is more convenient, powerful, and easy to solve the nonlinear fractional differential equation arising in mathematical physics.

The new optical behaviour of the LPD model with Kerr law and parabolic law of nonlinearity

In this paper, the newly obtained optical analytical solutions of the Lakshmanan–Porsezian–Daniel model with two forms of nonlinearity is investigated. These are Kerr law and parabolic law of nonlinearity. The novel analytical method namely, the modified Khater method is used to achieve the newly derived solutions. The attained solutions are bright, dark, anti-kink, u-shaped periodic cusp wave, singular solution and evolving shape of anti-kink solutions. The analysed characteristics of the reported solutions with different forms of nonlinearity are the most compelling parts of the proposed method. This ensures the efficacy and constructiveness of the method. The novel aspect of this study is the newly achieved solutions that can be used in predicting the performance of the information carrier.