Equal Width Equation Research Articles

Bifurcation of some new traveling wave solutions for the time–space M- fractional MEW equation via three altered methods

In this work,1/G′, modified G′/G2 and new extended direct algebraic methods are proposed to construct the novel exact traveling wave solutions in the form of trigonometric, hyperbolic and exponential functions of the time-fractional modified equal-width (MEW) equation in the sense of M- truncated fractional derivative. These methods contribute a variety of exact solutions in terms of the hyperbolic, trigonometric and rational functions to the scientific literature. The obtained solutions are verified for aforesaid equation through symbolic soft computations. To promote the essential propagated features, some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the precise values to the parameters under the constrain conditions. Further, the dynamical behavior is investigated. Based on the bifurcation constrains on the system’s parameters, we constructed also some new wave solution which are assorted into solitary, kink, periodic, and super periodic wave solutions. The influence of the included parameters on the solution is clarified. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.

Soliton Solutions and Sensitive Analysis of Modified Equal-Width Equation Using Fractional Operators

In this manuscript, the novel auxiliary equation methodology (NAEM) is employed to scrutinize various forms of solitary wave solutions for the modified equal-width wave (MEW) equation. M-truncated along with Atangana–Baleanu (AB)-fractional derivatives are employed to study the soliton solutions of the problem. The fractional MEW equations are important for describing hydro-magnetic waves in cold plasma. A comparative analysis is utilized to study the influence of the fractional parameter on the generated solutions. Secured solutions include bright, dark, singular, periodic and many other types of soliton solutions. In compared to other methods, the solutions demonstrate that the proposed technique is particularly effective, straightforward, and trustworthy that contains families of solutions. In addition, the symbolic soft computation is used to verify the obtained solutions. Finally, the system is subjected to a sensitive analysis. Integer-order results calculated by the symmetry method present in the literature can be addressed as limiting cases of the present study.

A variety of solitons on the oceans exposed by the Kadomtsev Petviashvili-modified equal width equation adopting different techniques

The diverse patterns of waves on the oceans yielded by the Kadomtsev Petviashvili-modified equal width (KP-mEW) equation are highlighted in this paper. Two recent established approaches such as the improved auxiliary equation technique and the enhanced rational (G′/G)-expansion scheme are utilized to construct wave solutions of the proposed governing model. Numerous rational, trigonometric, exponential, and hyperbolic wave solutions bearing many free parameters are successfully acquired in appropriate form. The obtained solutions are plotted in various profiles as three-dimension, two-dimension, and contour to illustrate their physical appearances. The plotting outlines appear in the shapes of singular kink, anti-kink, kink, compacton, anti-compacton, bell, anti-bell, periodic, singular periodic etc. The computational software Maple is used for plotting and checking the validity of the found solutions. This paper claims to be novel for generating new results regarding the earlier results.

Soliton solutions and fractional effects to the time-fractional modified equal width equation

The one-dimensional long water wave propagation in a nonlinear medium, including the dispersion process, is well simulated by the fractional-order modified equal-width (MEW) equation. This article establishes several recognized, standard, inclusive, and scores of typical exact wave solutions to the MEW equation using the double G'/G,1/G-expansion method. For specific parameter values, kink, periodic, periodic-singular, singular-kink, and other forms of solitons can be recovered from general solutions. The effect of the fractional parameter on wave forms has also been analyzed by depicting several graphs for different values of the fractional-order α. In order to illustrate the potential characteristics, three- and two-dimensional combined plots using Maple have been drawn. It has been established that the introduced approach is a potential tool for extracting new exact solutions to various nonlinear evolution equations (NLEEs) arising in engineering, science, and applied mathematics.

New traveling wave solutions for space-time fractional modified equal width equation with beta derivative

This work is based on finding the new and interesting traveling wave solutions of space-time fractional modified equal width equation, which is a pulse-like solitary wave nonlinear wave equation with cubic nonlinearity. The most efficient and reliable methodology, the unified method has been used coupled with the beta derivative to get the required results. This approach allows one to discover solutions to nonlinear evolution equations in polynomial and rational forms. All of the developed solutions are consistent with the requirements. Also, the obtained solutions are displayed on 3D and 2D graphs to represent the physical meaning of constructed solutions graphically. The findings show that the applied method is useful and can be used to solve other fractional non-linear evolution equations raised in natural and applied sciences.

Phase characterization and new optical solitons of a pulse passing through nonlinear dispersive media

In this paper, modified equal width Burgers’ equation has been investigated with the aid of unified method and bifurcation. This model has many applications in long wave transmission with dispersion and dissipation in nonlinear medium. The applied technique is efficient to retrieve exact solutions and their dynamic behaviors. The obtained solutions are polynomial and rational function solutions. The behavior of dynamical planer system has been analyzed by assigning different values to the parameters, also each possible case has been shown as phase portraits in this research paper. The estimated solutions demonstrate that the proposed approaches are simple, practical, and promising for investigating further equal width equation’s soliton wave solutions and phase portraits.

The space–time generalized finite difference scheme for solving the nonlinear equal-width equation in the long-time simulation

In this research, the space–time (ST) coupled generalized finite difference method (GFDM) was extended to cooperate with the Newton–Raphson and time-marching methods for stably solving the nonlinear equal-width equation in long-time simulation. The equal-width equation is a time-dependent nonlinear partial differential equation and has a third-order derivation that is mixed space and time. In such a case, the action of fluid flow is over a long period, and the numerical scheme needs to rely on the state-steady condition or set up a specific time to stop the simulation. In the ST-GFDM, the temporal and spatial discretizations can be performed using the GFDM in an ST-domain. That makes the proposed scheme can easily deal with the mixed derivative. By applying the GFDM, which is advanced from the Taylor series expansion and the moving-least square method, the process of the numerical discretizations is only related to nearby nodes on the central node. Thus, the Jacobian matrix is sparse and can be solved by the Newton–Raphson method efficiently. Furthermore, the time-marching method is used to proceed with an ST-domain along the time axis. In this paper, two numerical examples are solved to verify the capability of the proposed ST-GFDM.

Travelling wave solutions for a Zakharov–Kuznetsov modified equal width equations

We consider a family of generalized Zakharov–Kuznetsov modified equal width equations in (2＋1)-dimensions which involves an arbitrary function and three parameters. By applying the Lie group theory, we perform a complete classification of the Lie point symmetries of these equations depending on the arbitrary elements involved. We carry out symmetry reductions and obtain exact solutions by using the Lie point symmetries admitted by this family. Moreover, we show that this family of equations admits local low-order multipliers. Finally, we derive all local low-order conservation laws of this family through the multiplier approach.

On a Class of Partial Differential Equations and Their Solution via Local Fractional Integrals and Derivatives

This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details.

New dynamical soliton propagation of fractional type couple modified equal-width and Boussinesq equations

The space–time fractional coupled modified equal-width equation and the coupled Boussinesq equation are a category of fractional partial differential equations, which might be crucial mathematical feathers in nonlinear optics, solid-state physics, vibrations in the nonlinear string, ion sound waves in plasma, hydro-magnetic waves in cold plasma and many more. To assemble such new exact solutions of the mentioned equations, the Sine-Gordon expansion (SGE) technique has been proposed with inside the sense of conformable derivative and the fractional order partial differential equation that is capable to change into an ordinary differential equation by using the traveling wave transform. In this article, the SGE technique has been employed to search the higher-dimensional fractional nonlinear evolution equations and hooked up consistent soliton solutions to the faster thought fractional nonlinear evolution equations through installing use of the prolonged higher-dimensional SGE technique. The compatibility of the extended SGE technique confirms through the scoring of soliton solutions. Moreover, we explored a couple of varieties of solutions over the maple calculations, including soliton, kink types, bell types, single soliton type, dark soliton, singular kink type, and anti-bell type solutions for distinct values of constants, which have been illustrated by the usage of 3D, list-point, contour analysis, and vector plotting. It is far incredible to understand that the feature of the solutions relies upon the selection of the parameters from the figures. This takes a look at an impactful position in studying higher-dimensional fractional nonlinear evolution equations through the prolonged SGE technique.

An analytical approach to the solution of fractional-coupled modified equal width and fractional-coupled Burgers equations

We opted to construct a traveling wave solution to the nonlinear space-time fractional coupled modified equal width (CMEW) equation and the space-time fractional-coupled Burgers equation, which are often used as an electro-hydro-dynamical model to advance the local electric field and particle acoustic waves in plasma, the shallow water wave issues and portray the variety additional time of an actual structure on the partial liquid mechanics framework, particle acoustic waves through a gas-filled line, and certain consistent state gooey liquid. In this study, we employ the two variable (G′/G,1/G)-expansion method to create further general solitary wave solutions to those equations based on Riemann– Liouville fractional derivative. The fractional differential wave transform simplifies by generating ordinary differential equations (ODE) from fractional-order differential equations. We identified multiple types of solutions through the maple that are illustrated using 3D shape, 3D list point plot, and contour narratives. Additionally, we proposed that the methodology be changed to be more pragmatic, economical, and dependable and that we investigate more generalized precise solutions for traveling waves, such as solitary wave solutions.

Newly developed analytical method and its applications of some mathematical models

This paper presents a newly developed method, namely, the rational sine-Gordon expansion method to find novel exact solutions to nonlinear differential equations. This method is based on the sine-Gordon expansion method. To generalize the approach, we utilize the ansatz, which is a rational function as different to a polynomial function. In this way, we have more general wave solutions for nonlinear dynamic systems. We apply this method to the ([Formula: see text])-dimensional conformable Zakharov–Kuznetsov modified equal width (ZK-MEW) equation and the modified regularized long wave (MRLW) equation. Some new solutions are reported. Consequently, we submit the new soliton solutions to the literature.

Exact analytical wave solutions for space-time variable-order fractional modified equal width equation

The variable-order evolution equation is an impressive mathematical model that explain complex dynamical problems efficiently and accurately. This latest research investigates the modified equal width nonlinear space–time variable-order fractional differential equation and fractional derivative operator in the sense of Caputo. Using transformation, an ordinary differential equation is obtained from the variable-order differential equation. For accuracy, the space–time variable-order fractional modified equal width equation is solved by the modified (G'/G) -expansion method. This model describes the simulation for one-dimensional wave transmission in nonlinear media with dispersion processes. As a result, new traveling wave solutions are developed for various values of parameters. The obtained solutions have several applications in a recent area of research in mathematical physics. The obtained graphical solutions are in the form of singular solitons, kink solitons and periodic solitons waves which demonstrate that the physical signifance and dynamical behaviors of fractional variable-order differential equations and the proposed method are more effective, powerful, and easy in solving problems arising in mathematical physics.

New Traveling Wave Solutions for Space-Time Fractional Modified Equal Width Equation with Beta Derivative

New Traveling Wave Solutions for Space-Time Fractional Modified Equal Width Equation with Beta Derivative

New Traveling Wave Solutions for Space-Time Fractional Modified Equal Width Equation with Beta Derivative

New Traveling Wave Solutions for Space-Time Fractional Modified Equal Width Equation with Beta Derivative

Evaluation of fractional-order equal width equations with the exponential-decay kernel

<abstract><p>In this article we consider the homotopy perturbation transform method to investigate the fractional-order equal-width equations. The homotopy perturbation transform method is a mixture of the homotopy perturbation method and the Yang transform. The fractional-order derivative are defined in the sense of Caputo-Fabrizio operator. Several fractions of solutions are calculated which define some valuable evolution of the given problems. The homotopy perturbation transform method results are compared with actual results and good agreement is found. The suggested method can be used to investigate the fractional perspective analysis of problems in a variety of applied sciences.</p></abstract>

Analytical Solutions for the Equal Width Equations Containing Generalized Fractional Derivative Using the Efficient Combined Method

In this paper, the efficient combined method based on the homotopy perturbation Sadik transform method (HPSTM) is applied to solve the physical and functional equations containing the Caputo–Prabhakar fractional derivative. The mathematical model of this equation of order μ ∈ 0,1 with λ ∈ ℤ + , θ , σ ∈ ℝ + is presented as follows: D t μ C u x , t + θ u λ x , t u x x , t − σ u x x t x , t = 0 , where for λ = 1 , θ = 1 , σ = 1 s and λ = 2 , θ = 3 , σ = 1 , equations are changed into the equal width and modified equal width equations, respectively. The analytical method which we have used for solving this equation is based on a combination of the homotopy perturbation method and Sadik transform. The convergence and error analysis are discussed in this article. Plots of the analytical results with three examples are presented to show the applicability of this numerical method. Comparison between the obtained absolute errors by the suggested method and other methods is demonstrated.

Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

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

In this paper, we designed an algorithm by applying the Laplace transform to calculate some approximate solutions for fuzzy fractional-order nonlinear equal width equations in the sense of Atangana-Baleanu-Caputo derivatives. By analyzing the fuzzy theory, the suggested technique helps the solution of the fuzzy nonlinear equal width equations be investigated as a series of expressions in which the components can be effectively recognised and produce a pair of numerical results with the uncertainty parameters. Several numerical examples are analyzed to validate convergence outcomes for the given problem to show the proposed method’s utility and capability. The simulation outcomes reveal that the fuzzy iterative transform method is an effective method for accurately and precisely studying the behavior of suggested problems. We test the developed algorithm by three different problems. The analytical analysis provided that the results of the models converge to their actual solutions at the integer-order. Furthermore, we find that the fractional derivative produces a wide range of fuzzy results.

Construction of abundant novel analytical solutions of the space–time fractional nonlinear generalized equal width model via Riemann–Liouville derivative with application of mathematical methods

Abstract The space–time fractional generalized equal width (GEW) equation is an imperative model which is utilized to represent the nonlinear dispersive waves, namely, waves flowing in the shallow water strait, one-dimensional wave origination escalating in the nonlinear dispersive medium approximation, gelid plasma, hydro magnetic waves, electro magnetic interaction, etc. In this manuscript, we probe advanced and broad-spectrum wave solutions of the formerly betokened model with the Riemann–Liouville fractional derivative via the prosperously implementation of two mathematical methods: modified elongated auxiliary equation mapping and amended simple equation methods. The nonlinear fractional differential equation (NLFDE) is renovated into ordinary differential equation by the composite function derivative and the chain rule putting together along with the wave transformations. We acquire several types of exact soliton solutions by setting specific values of the personified parameters. The proposed schemes are expedient, influential, and computationally viable to scrutinize notches of NLFDEs.