Modified Kudryashov Method Research Articles

Effect of Space Fractional Parameter on Nonlinear Ion Acoustic Shock Wave Excitation in an Unmagnetized Relativistic Plasma

In this work, the model equation with space fractional-order (FO) is used to investigate the nonlinear ion acoustic shock wave excitations (NIASWEs) in an unmagnetized collisionless weakly relativistic plasma having inertial relativistic ions fluid with viscous effects, inertial-less non-thermal electrons and inertial-less Boltzmann positrons. To do it, the Korteweg-de Vries Burgers equation (KdVBE) is derived from the considered fluid model equations by implementing the standard reductive perturbation method. Accordingly, such equation is converted to space fractional KdVBE via Agrawal’s variational principle with the help of the beta fractional derivative and its properties. The exact analytical solutions of KdVBE with space FO are determined via the modified Kudryashov method. The influence of space fractional and other related plasma parameters on NIASWEs are investigated. The outcomes would be useful to understand the nature of shocks with the presence of non-local or local space in many astrophysical and space environments (especially in the relativistic wind of pulsar magnetosphere, polar regions of neutron stars, etc.) and further laboratory verification.

Abundant stable novel solutions of fractional-order epidemic model along with saturated treatment and disease transmission

Abstract This article proposes and analyzes a fractional-order susceptible, infectious, susceptible (SIS) epidemic model with saturated treatment and disease transmission by employing four recent analytical techniques along with a novel fractional operator. This model is computationally handled by extended simplest equation method, sech–tanh expansion method, modified Khater method, and modified Kudryashov method. The results’ stable characterization is investigated through the Hamiltonian system’s properties. The analytical solutions are demonstrated through several numerical simulations.

Exact solutions of the space time-fractional Klein-Gordon equation with cubic nonlinearities using some methods

Recently, finding exact solutions of nonlinear fractional differential equations has attracted great interest. In this work, the space time-fractional Klein-Gordon equation with cubic nonlinearities is examined. Firstly, suitable exact soliton solutions are formally extacted by using the solitary wave ansatz method. Some solutions are also illustrated by the computer simulations. Besides, the modified Kudryashov method is used to construct exact solutions of this equation.

Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation

This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modified Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation finite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation.

Solitary Wave Solutions of the Generalized (3+1)-Dimensional Shallow Water-Like Equation by using modified Kudryashov method

In this study, the generalized (3+1)-dimensional Shallow Water-Like (SWL) equation, which is one of the evolution equations, is taken into consideration. With the help of this evolution equation discussed, the modified Kudryashov method, traveling wave solutions are successfully obtained. In these solutions, graphs of solitary waves to be obtained by giving special values to arbitrary parameters are presented. At the same time, the effect of change of velocity parameter on the behavior on the solitary wave is examined in the solution obtained. The breaking of the wave is discussed. In this study, complex operations and graphic presentations are presented with the use of a ready-made package program.

Sub-10-fs-pulse propagation between analytical and numerical investigation

This paper investigates the analytical solutions of the well-known nonlinear Schrödinger (NLS) equation with the higher-order through three members of Kudryashov methods (the original Kudryashov method, modified Kudryashov method, and generalized Kudryashov method). The considered model is also known as the sub-10-fs-pulse propagation model used to describe these measurements’ implications for creating even shorter pulses. We also discuss the problem of validating these measurements. Previous measurements of such short pulses using techniques. This paper’s aim exceeds the idea of just finding the traveling wave solution of the considered model. Still, it researches to compare the used schemes’ accuracy by applying the quintic-B-Spline scheme and the convergence between three methods. Many distinct and novel solutions have been obtained and sketched, along with different techniques to show more details of the model’s dynamical behavior. Finally, the matching between analytical and numerical schemes has been shown through some tables and figures.

Some optical soliton solutions of space-time conformable fractional Schrödinger-type models

In this article, we introduce a family of nonlinear (1+1) dimensions Schrödinger-type models with space-time fractional evolution in the sense of a conformable fractional derivative. We apply the modified Kudryashov method in context of fractional complex transformation and seek a variety of optical soliton solutions for these equations. The modified Kudryashov method is efficient and consistent for solving nonlinear space-time fractional differential equations.

Novel explicit breath wave and numerical solutions of an Atangana conformable fractional Lotka–Volterra model

This research studies novel analytical solutions of an Atangana conformable fractional Lotka–Volterra (LV) model system arising in ecology by means of three systematic schemes (the extended simplest equation method, modified Kudryashov method, and the sech–tanh expansion method). Then we use the obtained solutions to evaluate the boundary and initial conditions that helps to apply the B-spline collection numerical schemes to evaluate the value of absolute error. The investigation aims also to explain the accuracy of the analytical solutions by evaluating absolute value of errors between exact and numerically obtained solutions. In order to better explain the obtained solutions, some relevant sketches are given in three different types. The novelty of the work done is attempted to explain by comparing our results with some available and relevant results. Further, relevant strengths, usefulness, practical applications, and the methods’ ability for applying in various nonlinear evolution equations have been explored.

A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons

The present paper explores a high-order nonlinear Schrödinger equation in a non-Kerr law media with the weak non-local nonlinearity describing solitons’ propagation through nonlinear optical fibers. To this end, the real and imaginary parts of the model are firstly extracted using a wave variable transformation. The modified Kudryashov method and symbolic computations are then adopted to successfully retrieve optical solitons of the model. The results presented in the current study demonstrate the great performance of the modified Kudryashov method in handling high-order nonlinear Schrödinger equations.

Genişletilerek Düzenlenmiş Kudryashov Yöntemi ile İki Fiziksel Modelinin Hareketli Dalga Çözümleri

In this paper, we propose the extended modified Kudryashov method (EMKM) for solving the Biswas-Milovic equation and Gerdjikov-Ivanov equation which are commonly special cases of Schrödinger equation in mathematical physics. We received many new extended traveling wave solutions when the special values of the parameters are taken for these equations which are pointed out by rational function, exponential function and hyperbolic function forms. The results show that EMKM is advantageous mathematical technique for solving nonlinear partial differential equations.

New optical soliton solutions for Fokas–Lenells dynamical equation via two various methods

In this paper, we examine the Fokas–Lenells equation (FLE) that depicts the promulgation of ultra-short pulsation in visual fibers while confirming the terms of the following asymptotic arrangement beyond those indispensable for the nonlinear Schrödinger equation. In addition the model includes both spatio–temporal dispersal and self-steepening terms. Then, we discuss deep visual solutions of the FLE via taking the modified Kudryashov method and the extended tanh expansion method.

A variety of solitons to the sixth-order dispersive [formula omitted]-dimensional nonlinear time-fractional Schrödinger equation with cubic-quintic-septic nonlinearities

This study examines the exact soliton solutions under the effect of cubic-quintic-septic nonlinearities for a sixth-order (3+1)-dimensional nonlinear time-fractional Schrödinger equation with fourth-order and sixth-order dispersive terms. We apply the generalized Riccati equation mapping method and the modified Kudryashov method to obtain the dark soliton, singular, periodic, and other soliton-type solutions. To determine the bright solition, other singular and periodic solutions, ansatz method is employed. The results obtained can help to understand the dynamics of waves in a cubic-quintic-septic nonlinear material.

Traveling wave solutions for the $$(2+1)$$-dimensional generalized Zakharov–Kuznetsov equation with variable coefficients

In this article, the unified method, the improved F-expansion method and the modified Kudryashov method are used to obtain the traveling wave solutions of the generalized Zakharov–Kuznetsov equation with variable coefficients. Different types of traveling wave solutions are derived including polynomial solutions and rational solutions by applying the unified method. The polynomial solutions include solitary wave solutions, soliton wave solutions and elliptic wave solutions, and the rational solutions contain periodic type rational solutions and soliton type rational solutions. By means of the improved F-expansion method with Riccati equation, the hyperbolic trigonometric solutions, trigonometric solutions and rational solutions are obtained containing several free parameters. The modified Kudryashov method is also applied to obtain new traveling wave solutions. In addition, the properties of several solutions are represented graphically with the appropriate parameters.

Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model

This article studies the generalized (2 + 1)-dimensional shallow water equation by applying two recent analytical schemes (the extended simplest equation method and the modified Kudryashov method) for constructing abundant novel solitary wave solutions. These solutions describe the bidirectional propagating water wave surface. Some obtained solutions are sketched in two- and three-dimensional and contour plots for demonstrating the dynamical behavior of these waves along shallow water. The accuracy of the obtained solutions and employed analytical schemes is investigated using the evaluated solutions to calculate the initial condition, and then the well-known variational iterational (VI) method is applied. The VI method is one of the most accurate semi-analytical solutions, and it can be applied for high derivative order. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

SOLITON SOLUTIONS FOR ANTI-CUBIC NONLINEARITY USING THREE ANALYTICAL APPROACHES

<abstract> In this article, three constructive techniques namely, Exp_a-function method, the modified Kudryashov method and the generalized tanh-method are adopted to analyze the nonlinear Schrödinger equation having anti-cubic nonlinearity. Nonlinear Schrödinger equation is a comprehensive model that governs wave behavior in optical fiber. Cubic-quintic nonlinear Schrödinger equation, additionally having anti-cubic nonlinear term is investigated to construct bright, dark, kink and singular soliton solutions. The graphical representations of the soliton propagation are also demonstrated by the solutions obtained using these three techniques. </abstract>

Numerical simulations of stochastic conformable space–time fractional Korteweg-de Vries and Benjamin–Bona–Mahony equations

Abstract In this paper, we investigate the effect of white noise on conformable time and space fractional KdV and BBM equations. For this purpose, we convert these equations with external noise to homogeneous conformable time and space fractional KdV and BBM equations with defined transformation and then we solve them by modified Kudryashov method. We bring our numerical results in some figures in the last section.

Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution

<p style='text-indent:20px;'>In this article, the oblique resonant traveling waves and dynamical behaviors of (2+1)-dimensional Nonlinear Schrödinger equation along with dual-power law nonlinearity, and fractal conformable temporal evolution are reported. The considered equation is converted to an ordinary differential equation by taking the traveling variable wave transform and properties of Khalil's conformable derivative into account. The modified Kudryashov method is implemented to divulge the oblique resonant traveling wave of such an equation. It is found that the obliqueness is only affected on width, but not on amplitude and phase patriots of resonant nonlinear propagating wave dynamics. The research outcomes are very helpful for analyzing the obliquely propagating nonlinear resonant wave phenomena and their dynamical behaviors in several nonlinear systems having Madelung fluids and optical bullets.

Computational schemes between the exact, analytical and numerical solution in present of time—fractional ecological model

This paper investigates the accuracy of three recent computational schemes (the extended simplest method (ESEM), sech—tanh expansion method (STEM), and modified Kudryashov method (MKM)) through calculating the absolute value of error between their solutions and numerical solutions. The computational schemes claim to obtain exact traveling wave solutions of the investigated models; therefore, it supposes the numerical study for any models that have been analytically investigated under any constructed computational solutions that will be matching, but our study shows a different fact. (Khater et al Soft Computing (Submitted)) has studied the computational solutions of the time-fractional Lotka—Volterra (LV) model through the above-mentioned computational schemes. Many solutions have been obtained in different mathematical formulas such as exponential, trigonometric, hyperbolic, etc. These solutions describe the interaction between the high -frequency Langmuir and the low-frequent ion-acoustic waves with many applications in electromagnetic waves, plasma physics, and signal processing through optical fibers, coastal engineering, and fluid dynamics. This manuscript applies the trigonometric quintic B—spline scheme to the fractional LV model along with the Caputo and Fabrizio fractional derivatives and computational obtained solutions for investigating the numerical solutions under each employed analytical scheme. The numerical solutions are simulated in two-dimensional sketches to explain the relation between exact and numerical solutions. This study proves the computational fact hypotheses for obtaining exact solutions, and they all obtain computational solutions.

Complex wave solutions described by a (3+1)-dimensional coupled nonlinear Schrödinger equation with variable coefficients

In this article, a (3+1)-dimensional variable coefficients coupled nonlinear Schrö-dinger equation is analysed with the unified method, the improved F-expansion method with Riccati equation and the modified Kudryashov method. The polynomial solutions are investigated and classified into three categories including complex solitary wave solutions, complex soliton wave solutions and complex elliptic wave solutions by applying the unified method. Besides, the physical insights of polynomial solutions are graphically discussed with suitable parameters. The complex wave solutions are derived by the improved F-expansion method with Riccati equation which contain the complex hyperbolic trigonometric solutions, complex trigonometric solutions and complex rational solutions. Lastly, the modified Kudryashov method is applied to obtain the complex wave solutions.

On the relations between some well-known methods and the projective Riccati equations

AbstractSolving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.