Modified Kudryashov Method Research Articles

Analytical Behaviour of Travelling Wave Solutions to the Van der Waals Model

In this article, our work is related to the most popular model that occurred naturally and also industrially, known as the Van der Waals normal form for the fluidized matter. After using a wave transformation that converts the nonlinear partial differential equation into nonlinear ordinary differential equations, we employed two novel analytical methods namely the hyperbolic function method the modified Kudryashov method. As a consequence, many new exact solutions are established coupled with 2D and 3D graphical patterns for clarity and accuracy.

Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity

In this paper, the perturbed nonlinear Schrodinger equation with dual-power law of nonlinearity is studied by adopting four mathematical methods namely the Modified Kudryashov method, the extended Tanh–Coth method, the modified simple equation method and soliton ansatz method. As a results, a set of various types of solitons that contains dark, singular, dark–singular, bright optical solitons and other form of optical soliton solutions are obtained. Firstly, we solve the perturbed nonlinear Schrodinger equation by considering the dual power law parameter using the first three integration methods and secondly we set this parameter equal to one in order, to solve the problem with the fourth method. The used methods in this paper present various applications in fields of nonlinear wave equations. Comparing our new results with well-known in literature are also given. Moreover, the graphical representations of the modulus of some optical soliton solutions and their 2-D profile are also depicted.

Traveling wave solutions to nonlinear directional couplers by modified Kudryashov method

This work finds several new traveling wave solutions for nonlinear directional couplers with optical metamaterials by means of the modified Kudryashov method. This model can be used to distribute light from a main fiber into one or more branch fibers. Two forms of optical couplers are considered, namely the twin- and multiple- core couplers. These couplers, which have applications as intensity-dependent switches and as limiters, are studied with four nonlinear items namely the Kerr, power, parabolic, and dual-power laws. The restrictions on the parameters for the existence of solutions are also examined. The 3D- and 2D figures are introduced to discuss the physical meaning for some of the gained solutions. The performance of the method shows the adequacy , power, and ability for applying to many other nonlinear evolution models.

New soliton solutions for resonant nonlinear Schrödinger’s equation having full nonlinearity

In this paper, we discuss deep visual solutions of resonant nonlinear Schrödinger’s equation having full nonlinearity via taking the modified Kudryashov method. There are four types of nonlinearity in this paper. They are quadratic-cubic law, anti-cubic law, cubic-quintic-septic law and triple-power law. By performing this algorithm, logarithmical and rational solitons are deduced.

The Traveling Wave Solutions of Space-Time Fractional Partial Differential Equations by Modified Kudryashov Method

In this paper, the modified Kudryashov method is employed to find the traveling wave solutions of two well-known space-time fractional partial differential equations, namely the Zakharov Kuznetshov Benjamin Bona Mahony equation and Kolmogorov Petrovskii Piskunov equation, and as a helping tool, the sense of modified Riemann-Liouville derivative is also used. The propagation properties of obtained solutions are investigated where the graphical representations and justifications of the results are done by mathematical software Maple.

Dispersive optical dromions and domain walls with a few golden integration formulae

This paper reveals optical dromions and domain walls to (2+1)-dimensional nonlinear Schrödinger's equation in presence of higher order dispersion terms, together with Kerr law nonlinearity. Four notable integration algorithms, via, the sine-cosine method, the simplest equation method, the modified Kudryashov method and the unified Riccati equation expansion have been applied with grand success. The revealed solutions come with their respective existence criteria that are also presented.

Oblique resonance wave phenomena for nonlinear coupled evolution equations with fractional temporal evolution

This work investigates the obliquely propagating resonance wave phenomena described by coupled model equations with fractional temporal evolution arising in many branches of physics. The (2 + 1) -dimensional coupled nonlinear Schrodinger equation (NLSE), long wave-short wave resonance interaction (LSRI) equation and Maccari system (MS) having fractional time evolution are considered to study such physical phenomena. The modified Kudryashov method (mKM) with the properties of Khalil's conformable fractional derivatives is implemented to divulge scattered oblique wave solutions of considered equations. The effects of obliqueness and fractional parameter on obtained results are demonstrated graphically along with the physical descriptions. It is found that the resonance wave phenomena are changed with the increase of obliqueness as well as fractional parameter.

Analytic solutions of the chiral nonlinear schrödinger equations investigated by an efficient approach

This paper studies the chiral nonlinear Schrödinger equations, describing a central role in the developments of quantum me-chanics, particularly in the field of quantum Hall effect, where chiral excitations are known to appear. More precisely, in this paper, we acquired new exact solutions of the chiral nonlinear (1+1) and (1+2)-dimensional Schrödinger equations by using the modified Kudraysov method. As outcomes, some of the new exact traveling wave solutions for the equations above is formally produced. All solutions are plotted in the view of three-dimensional (3D) and two-dimensional (2D) line shape through the MATLAB software for investigating the real significance of the studied equations. The periodic type of solitons is generated by employing modified Kudryashov method which is different from other studied methods.Â

Solving fractional nonlinear partial differential equations by the modified Kudryashov method

There are more and more methods for transforming nonlinear partial differential equations into ordinary differential equations by using the traveling wave transform. In this paper, the modified Kudryashov method is used to use the new traveling wave transform, and the exact solution of the space-time fractional equal-width equation is obtained by means of symbolic computation. Search for solution of the equal-width equation becomes more direct and simple. It is also suitable for solving a large number of similar fractional-order nonlinear partial differential equations, the method provides a new idea for solving fractional-order nonlinear partial differential equations.

New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation

This study focuses on acquiring new complex hyperbolic and trigonometric function solutions to the generalized conformable fractional Gardner equation by using two distinct integration schemes, namely: the modified Kudryashov method and the hyperbolic function method with the help of symbolic computation package. We generate new closed-form solitary wave solutions to the governing model. Implemented schemes are pragmatically effective for solving various types of nonlinear partial differential equations with integer or fractional order arising in various fields of applied sciences.

Exact traveling wave solutions of the space\u2013time fractional complex Ginzburg\u2013Landau equation and the space-time fractional Phi-4 equation using reliable methods

Ultrashort pulse propagation in optical transmission lines and phenomena in particle physics can be investigated via the cubic–quintic Ginzburg–Landau equation and the Phi-4 equation, respectively. The main objective of this paper is to construct exact traveling wave solutions of the (2 + 1)-dimensional cubic–quintic Ginzburg–Landau equation and the Phi-4 equation of space-time fractional orders in the sense of the conformable fractional derivative. The method employed to solve the Ginzburg–Landau equation and the Phi-4 equation are the modified Kudryashov method and the (G'/G,1/G)-expansion method, respectively. Several types of exact analytical solutions are obtained including reciprocal of exponential function solutions, hyperbolic function solutions, trigonometric function solutions and rational function solutions. Graphical representations and physical explanations of some of the obtained solutions are demonstrated using a range of fractional orders. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. These methods are simple and efficient for solving the proposed equations.

Extraction of Solitary Wave Features to the Heisenberg Ferromagnetic Spin Chain and the Complex Klein–Gordon Equations

In this paper, the modified Kudryashov method is applied for obtaining the exact travelling wave solutions for the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation and the complex Klein–Gordon equation. The extracted solutions show distinct physical configurations. More precisely, the modified Kudryashov method is utilized for constructing the exact periodic and soliton solutions of these equations. As an outcome, in a broader sense, dark, bright, dark-bright, singular or combined singular and optical soliton solutions or wave features are derived from the equations. All solitons are verified through the corresponding equations with the help of Maple package program. The three-dimensional and two-dimensional waves are generated with the help of MATLAB for investigating the real significance of the proposed scheme to draw all solutions.

New wave form solutions of nonlinear conformable time-fractional Zoomeron equation in (2 + 1)-dimensions

Under investigation in the current paper is the nonlinear conformable time-fractional Zoomeron equation in (2 + 1)-dimensions, which is a model to display the novel phenomena associated with boomerons and trappons. The well-designed techniques, -expansion approach and modified Kudryashov method are formally utilized to produce a variety of wave form solutions such as kink, singular kink, and periodic wave solutions for the governing model. Results confirm the effectiveness of the methods for extracting different wave form solutions of nonlinear time-fractional differential equations.

Analytical study of exact traveling wave solutions for time-fractional nonlinear Schrödinger equations

The present paper deals with the time-fractional unstable nonlinear Schrodinger equation and the time fractional modified unstable nonlinear Schrodinger equation in the conformable context. For this aim, the modified Kudryashov method and the sine–Gordon expansion approach have been applied to retrieve a series of exact solutions for the previously mentioned equations. The potential of the schemes in producing exact traveling wave solutions of nonlinear conformable time-fractional equations is demonstrated.

Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems yields the analytical solution. By this way, various exact solutions including complex structures are found, and their behavior is drawn in the 2D plane by Maple to compare the uniqueness and wave traveling of the solutions.

The generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Burgers type

The nonlinear space–time fractional differential equations (FDE) of Burgers’ type play an important role for describing many phenomena in applied sciences. This paper develops a generalized version of the modified Kudryashov method to obtain the exact solutions for FDE of Burgers’ type. The FDE are firstly reduced to a set of ordinary differential equations by means of a fractional complex transformation and, afterward, they are solved by applying the generalized Kudryashov method (GKM). Several kinds of traveling wave solutions such as kink and singular kink waves are illustrated with various figures and discussed. The results demonstrate the efficiency of the GKM in tackling space–time partial FDE.

Multiple soliton solutions of the nonlinear partial differential equations describing the wave propagation in nonlinear low–pass electrical transmission lines

This paper focuses on investigating soliton and other solutions using three integration schemes to integrate a nonlinear partial differential equation describing the wave propagation in nonlinear low–pass electrical transmission lines. By applying the Kirchhoff’s laws and complex transformation, the nonlinear low–pass electrical transmission lines are converted into an equation wave propagation in nonlinear ODE low–pass electrical transmission lines. Later on, mentioned integration schemes viz modified Kudryashov method, sine–Gordon equation expansion method and extended sinh–Gordon equation expansion method are used to carry out new hyperbolic and trigonometric solutions which shows the consistency via computerized symbolic computation package maple. Various types of solitary wave solutions are derived including kink, anti-kink, dark, bright, dark-bright, singular, combined singular, and periodic singular wave soliton solutions. The corresponding three integration schemes are robust and effective for acquiring the new dark, bright, dark–bright, singular or combined singular and optical soliton solutions of the wave propagation in nonlinear low–pass electrical transmission lines. To show the real physical significance of the studied equation, some three dimensional (3D) and two dimensional (2D) figures of obtained solutions are plotted with the use of the Matlab software under the proper choice of arbitrary parameters. Moreover, all derived solutions were verified back into its corresponding equation with the aid of maple program.

Oblique closed form solutions of some important fractional evolution equations via the modified Kudryashov method arising in physical problems

The paper deals with the obliquely propagating wave solutions of fractional nonlinear evolution equations (NLEEs) arising in science and engineering. The conformable time fractional (2 + 1)-dimensional extended Zakharov-Kuzetsov equation (EZKE), coupled space-time fractional (2 + 1)-dimensional dispersive long wave equation (DLWE) and space-time fractional (2 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation are considered to investigate such physical phenomena. The modified Kudryashov method along with the properties of conformable and modified Riemann-Liouville derivatives is employed to construct the oblique wave solutions of the considered equations. The obtained results may be useful for better understanding the nature of internal oblique propagating wave dynamics in ocean engineering.

Application of the modified Kudryashov method to the generalized Schrödinger–Boussinesq equations

In the paper, the modified Kudryashov method is applied to find new exact solutions for the generalized Schrodinger–Boussinesq equation with the help of symbolic computation package Maple through the complex transform. The obtained solutions have been checked by substituting back into its corresponding equation with the aid of Maple package program.

Symmetrical Fibonacci and Lucas Wave Solutions for Some Nonlinear Equations in Higher Dimensions

We consider some nonlinear partial differential equations in higher dimensions, the negative order of the Calogero-Bogoyavelnskii-Schiff (nCBS) equationin (2+1) dimensions, the combined of the Calogero-Bogoyavelnskii-Schiff equation and the negative order of the Calogero-Bogoyavelnskii-Schiff equation (CBS-nCBS) in (2+1) dimensions, and two models of the negative order Korteweg de Vries (nKdV) equations in (3+1) dimensions. We show that these equations can be reduced to the same class of ordinary differential equations via wave reduction variable. Solutions in terms of symmetrical Fibonacci and Lucas functions are presented by implementation of the modified Kudryashov method.