Exact Traveling Wave Solutions Research Articles

Some New Traveling Wave Solutions of Nonlinear Fluid Models via the MSE Method

In this study, some new exact wave solutions of nonlinear partial differential equations are investigated by the modified simple equation method. This method is applied to the $(2+1)$-dimensional Calogero-Bogoyavlenskii-Schiff equation and the $(3+1)$-dimensional Jimbo-Miwa equation. Our applications reveal how to use the proposed method to solve nonlinear partial differential equations with the balance number equal to two. Consequently, some new exact traveling wave solutions of these equations are achieved, and types of waves are determined. To verify our results and draw the graphs of the solutions, we use the Mathematica package program.

Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations

Recently, non-linear fractional partial differential equations are used to model many phenomena in applied sciences and engineering. In this study, the modified simple equation scheme is implemented to obtain some new traveling wave solutions of the non-linear conformable time-fractional approximate long water wave equation and the non-linear conformable coupled time-fractional Boussinesq-Burger equation, which are used in the expression of shallow-water waves. The time- fractional derivatives are described in terms of conformable fractional derivative sense. Consequently, new exact traveling wave solutions of both equations are achieved. The graphics and correctness of the wave solutions are obtained with the Mathematica package program.

The Modified Exp (-vartheta(sigma)) -Expansion Function Method For Exact Solutions Of The Simplified MCH Equation And The Getmanou Equation

The present research explores, the modified exp -expansion function method (MEFM) is tested by applying it to obtain the exact travelling wave solutions for the simplified MCH equation and the Getmanou equation. Dark optical soliton solutions and dark-bright optical soliton solutions of the simplified MCH equation and the Getmanou equation are successfully constructed by using this method. We carry out all the computations and draw the 2D and 3D graph in this paper by Wolfram Mathematica 9. Besides, the graphical representation obviously shows forcefulness of this method.

Optical soliton perturbation with Kudryashov’s generalized law of refractive index and generalized nonlocal laws by improved modified extended tanh method

In this work, the improved modified extended tanh scheme is implemented to extract exact travelling wave solutions for perturbed nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index and dual form of generalized nonlocal nonlinearity. Various types of solutions are extracted such as bright solitons, singular solitons, dark solitons, singular periodic wave solutions, periodic wave solutions, Jacobi elliptic functions, plane wave and hyperbolic wave solutions. Moreover, 3D and contour plots of some solutions are illustrated to show the physical nature of obtained solutions.

Two Reliable Techniques for Solving Conformable Space-Time Fractional PHI-4 Model Arising in Nuclear Physics via β-derivative

Nowadays, nonlinear fractional partial differential equations have been highly using for modelling of physical phenomena. Therefore, it is very important to achieve exact solutions of fractional differential equations for understanding complex phenomena in mathematical physics. In this study, new exact traveling wave solutions are reached of space-time fractional Phi-4 equation indicated by Atangana’s conformable derivative using two powerful different techniques. These are the functional variable method and the first integral method. Obtaining new solutions of this equation show that method is effective to understanding other nonlinear complex problems in particle and nuclear physics.

Kinetics of phase separation in Fe–Cr–X (X =Mo, Cu) ternary alloys — a dynamical wave study

Ternary alloys of Fe are very important materials having good corrosion resistance and are also famous for several high-temperature applications. The dynamical behavior of exact traveling waves for the kinetics of phase separation in Fe–Cr–X (X[Formula: see text]Mo, Cu) ternary alloys is modelled by convective-diffusive Cahn–Hilliard (CH) equation. A variety of nonlinear dynamical exact and solitary wave structures are extracted in several forms like rational, hyperbolic, trigonometric function solutions by the utilization of a sound computational integration tool, i.e., [Formula: see text]-model expansion method. Besides, we also secure mixed combined solitons and singular periodic wave solutions with unknown parameters. Moreover, the constraint conditions observed during derivation lead to substantial solutions. The findings elucidate that the governing model theoretically possesses significantly rich structures of exact traveling wave solutions. Hence, our technique via fortification of symbolic computations provides an active and potent mathematical implementation for solving diverse benevolent nonlinear wave problems.

Exact traveling wave solutions for two prolific conformable M-Fractional differential equations via three diverse approaches

In this paper, we obtain the exact traveling solutions of the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation by using three fertile, G′/G,1/G, modified G′/G2 and 1/G′-expansion methods. These methods contribute a variety of exact traveling wave solutions to the scientific literature. The obtained solutions are also verified for the aforesaid equations through symbolic soft computations. Furthermore, some results are explained through numerical simulations that show the novelty of our work. Moreover, we observe that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

Dynamical behaviour of travelling wave solutions to the conformable time-fractional modified Liouville and mRLW equations in water wave mechanics

In this current study, we described a modified extended tanh-function (mETF) method to find the new and efficient exact travelling and solitary wave solutions to the modified Liouville equation and modified regularized long wave (mRLW) equation in water wave mechanics. Travelling wave transformation decreases the leading equation to traditional ordinary differential equations (ODEs). The standardized balance technique provides the instruction of the portended polynomial related result stimulated from the mETF method. The substitution of this result follows the preceding step. Balancing the coefficients of the like powers of the portended solution leads to a system of algebraic equations (SAE). The solution of that SAE for coefficients provides the essential connection between the coefficients and the parameters to build the exact solution. Here the acquired solutions are hyperbolic, rational, and trigonometric function solutions. Our mentioned method is straightforward, succinct, efficient, and powerful and can be emphasized to establish the new exact solutions of different types of nonlinear conformable fractional equations in engineering and further nonlinear treatments.

Exact travelling wave solutions for nonlinear system of spatiotemporal fractional quantum mechanics equations

Schrödinger equation is an indispensable model for quantum mechanics, used for modelling several fascinating complex nonlinear physical systems, such as quantum condensates, nonlinear optics, hydrodynamics, shallow-water waves, and the harmonic oscillator. The objective of this paper is to investigate and study the exact travelling wave solutions of nonlinear triple fractional Schrödinger equations involving a modified Riemann–Liouville fractional derivative. Using the Riccati-Bernoulli Sub-ODE technique, the Bäcklund transformation is employed to handle the posed system. The traveling wave solutions methodology lies in converting the fractional Schrödinger equations into a nonlinear system of fractional ODEs. An infinite sequence of solutions to the fractional partial differential equations can be obtained directly through solving the resulting nonlinear fractional system. Some graphical representations of the obtained solutions after selecting suitable values for fractional values and parameters are illustrated to test accuracy and verify the power, and effectiveness of the proposed method.

Application of extended rational trigonometric techniques to investigate solitary wave solutions

In this paper, a variety of novel exact traveling wave solutions are constructed for the $$(2+1)$$ -dimensional Boiti-Leon-Manna-Pempinelli equation via analytical techniques, namely, extended rational sine-cosine method and extended rational sinh-cosh method. The physical meaning of the geometrical structures for some of these solutions is discussed. Obtained solutions are expressed in terms of singular periodic wave, solitary waves, bright solitons, dark solitons, periodic wave and kink wave solutions with specific values of parameters. For the observation of physical activities of the problem, achieved exact solutions are vital. Moreover, to find analytical solutions of the proposed equation many methods have been used but given methodologies are effective, reliable and gave more and novel exact solutions.

Abundant traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line

The main idea of this paper is to search for all traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line, which plays a essential role in obtaining the new insights in nonlinear voltage dynamics. We first give a brief introduction how to transform the discrete system into a continuous one, which is described by a fractional-order partial differential equation. After that, this equation is handled by conformable fractional transformation and the complete discrimination system for polynomial method (CDSPM). By applying the advanced method, the whole of the exact traveling wave solutions emerged in existing articles are obtained, especially we obtain the solitary wave solutions and elliptic functions solutions which are hardly founded by other methods. Notably, the elliptic functions solutions in rational form are discovered for the first time. Finally, the electrical characteristics and the fractional nature are revealed via graphical represents. By the depicted graphs, we intuitively observe the existence of the phenomena for periodic wave and solitary wave, and the time-fractional derivative is proved do has important influence on the behaviors of the solutions. Considering the significance of the nonlinear electrical transmission line, the acquired results would have wide application in electrical engineering and nonlinear voltage dynamics, liking analyzing and predicting the complex voltage wave propagation phenomenon in realistic electrical transmission system.

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.

Travelling Wave Solutions for Fisher’s Equation Using the Extended Homogeneous Balance Method

In this work, the extended homogeneous balance method is used to derive exact solutions of nonlinear evolution equations. With the aid of symbolic computation, many new exact travelling wave solutions have been obtained for Fisher’s equation and Burgers-Fisher equation. Fisher’s equation has been widely used in studying the population for various systems, especially in biology, while Burgers-Fisher equation has many physical applications such as in gas dynamics and fluid mechanics. The method used can be applied to obtain multiple travelling wave solutions for nonlinear partial differential equations.

Analytical optical soliton solutions of the Schrödinger-Poisson dynamical system

The article studies the exact traveling wave solutions to the Schrödinger-Poisson system which has applications in gravity’s role of quantum state and approximate the coupling between quantum mechanics with gravitation. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtain using two norms of integration. For this sake modified extended direct algebraic (MEDA) and (G′/G)-expansion techniques are used. The 3D plots and their corresponding contour graphs are also depicted. The constraints conditions for the exact of solutions are also emerged during the derivation of solution.

Nonlinear dispersion in parabolic law medium and its optical solitons

This paper studies the optical soliton solutions of a nonlinear Schrödinger equation (NLSE) involving parabolic law of nonlinearity with the presence of nonlinear dispersion by using the generalized auxiliary equation technique. As a result, new varieties of exact traveling wave solutions have been uncovered, comprising of the hyperbolic trigonometric, trigonometric, exponential, and rational. Interestingly, we obtain the bright, dark, periodic, singular, and other soliton solutions to the nonlinear model. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behaviour. Furthermore, the findings discussed in this present investigation may be useful in explaining the propagation of optical solitons in a weakly nonlocal parabolic law medium.

Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach

This study investigates exact analytical solutions of some nonlinear partial differential equations arising in mathematical physics. To this reason, the Kudryashov-Sinelshchikov equation, the ZK-BBM equation and the Gardner equation have been considered. With the implementation of the trial solution algorithm, solitary wave, bright, dark and periodic exact traveling wave solutions of the considered equations have been attained. The solutions have been checked and graphs have been given via package programs to see the behavior of the waves.

Constructions of new abundant traveling wave solutions for system of the ion sound and Langmuir waves by the variational direct method

In this paper, a novel and simple approach called the variational direct method (VDM) is applied to construct the exact traveling wave solutions of the system of ion sound and Langmuir waves (SEISLWs), which is under the influence of the ponderomotive force caused by a nonlinear force experienced by a charged particle in an inhomogeneous oscillating electromagnetic field due to high-frequency field. These exact solutions include the bright soliton, bright-like soliton, kinky-bright soliton, bright-dark soliton and periodic wave solutions, which are all presented through the numerical results in the form of 3-D plots. Obtained results show that the VDM is simple, straightforward and powerful, which is expected to open new perspectives for the traveling wave theory.

General Traveling Wave Solutions of Nonlinear Conformable Fractional Sharma-Tasso-Olever Equations and Discussing the Effects of the Fractional Derivatives

The exact traveling wave solution of the fractional Sharma-Tasso-Olever equation can be obtained by using the function expansion method, but the general traveling wave solution cannot be obtained. After transforming it into the Sharma-Tasso-Olever equation of the integer order by the fractional complex transformation, the general solution of its traveling wave is obtained by a specific function transformation. Through parameter setting, the solution of the kinked solitary wave is found from the general solution of the traveling wave, and it is found that when the two fractional derivatives become smaller synchronically, the waveform becomes more smooth, but the position is basically unchanged. The reason for this phenomenon is that the kink solitary wave reaches equilibrium in the counterclockwise and clockwise rotation, and the stretching phenomenon is accompanied in the process of reaching equilibrium. This is a further development of our previous work, and this kind of detailed causative analysis is rare in previous papers.

Computational solutions of the generalized Ito equation in nonlinear dispersive systems

This papers presents new exact analytical solutions of a generalized Ito equation having three nonlinear terms, third- and fifth-order derivative forms that model the dynamics of traveling waves in nonlinear dispersive systems. With the help of Riccati equation method, we obtain different kinds of exact traveling wave solutions containing dark, singular, trigonometric, rational and other form of waves solutions that are more general than classical ones existing in the literature. Despite the originality of the new results obtained, the method used here is very efficient, powerful and can be extended to other types of nonlinear equations and more. Moreover, the behaviors of traveling waves solutions are portrayed graphically by selecting suitable values for the physical parameters.