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In this work, the Date–Jimbo–Kashiwara–Miwa (DJKM) equation include time-dependent in (2+1)-dimensions that characterize the propagation of nonlinear dispersive waves in inhomogeneous media is stud...

In this paper, the modified auxiliary expansion method is used to construct some new soliton solutions of coupled Schrödinger-Boussinesq system that includes beta derivative. The new exact solution is obtained have a hyperbolic function, trigonometric function, exponential function, and rational function. These solutions might appreciate in laser and plasma sciences. It is shown that this method, provides a straightforward and powerful mathematical tool for solving the nonlinear problems. Moreover, the linear stability of this nonlinear system is analyzed.

In this work, we attempt to construct some novel solutions of nematicons within liquid crystals including three types of nonlinearity namely Kerr, parabolic, and power law, using the generalized exponential rational function method. The investigation of nematic liquid crystals, using the proposed method, shows that there is diversity between the solutions gained via this method with those obtained via different methods. Further, we use the constraint conditions to guarantee the existence of the solutions. The W-shaped surfaces, dark soliton, bright soliton, singular soliton, period singular soliton, periodic waves, and complex solutions of the studied equations are successfully constructed. Moreover, some obtained solutions are drawn to a better understanding of the characteristics of nematicons in liquid crystals.

In this research paper, we investigate some novel soliton solutions to the perturbed Fokas-Lenells equation by using the (m + 1/G') expansion method. Some new solutions are obtained and they are plotted in two and three dimensions. This technique appears as a suitable, applicable, and efficient method to search for the exact solutions of nonlinear partial differential equations in a wide range. All gained optical soliton solutions are substituted into the FokasLenells equation and they verify it. The constraint conditions are also given.

This study investigates the Gilson–Pickering equation by using the sine-Gordon expansion method. Sine-Gordon expansion method is one of the most powerful methods for solving the nonlinear partial differential equations. We successfully construct various exact solitary wave solutions to the governing equation, such as shock wave, topological, non-topological, compound topological, and non-topological soliton wave solutions. In addition, the stability of the studied nonlinear equation is analyzed via the linear stability analysis. The 2D, 3D, and contour surfaces are also plotted for all obtained solutions.

In this manuscript, the application of the extended sinh-Gordon equation expansion method to the Davey-Stewartson equation and the (2+1)-dimensional nonlinear complex coupled Maccari system is presented. The Davey-Stewartson equation arises as a result of multiple-scale analysis of modulated nonlinear surface gravity waves propagating over a horizontal seabed and the (2+1)-dimensional nonlinear complex coupled Maccari equation describes the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics. We successfully construct some soliton, singular soliton and singular periodic wave solutions to these two nonlinear complex models. The 2D, 3D and contour graphs to some of the obtained solutions are presented.

In this paper, we study the space-time fractional perturbed nonlinear Schr \begin{document}$ \bf{\ddot o} $\end{document} dinger equation under the Kerr law nonlinearity by using the extended sinh-Gordon equation expansion method. The perturbed nonlinear Schr \begin{document}$ \bf{\ddot o} $\end{document} dinger equation is a nonlinear model which arises in nano-fibers. Some family of optical solitons and singular periodic wave solutions are successfully revealed. The parametric conditions for the existence of valid solitons are stated. Under the choice of suitable values of the parameters, the 3-dimensional and 2-dimensional graphs to some of the reported solutions are plotted.

The studies of the dynamic behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. In this paper, an efficient mathematical technique, namely; the sine-Gordon expansion method is employed to construct the traveling wave solutions to the (3 + 1)-dimensional Kadomtsev–Petviashvili and (3 + 1)-dimensional nonlinear Schrödinger equations. Using suitable values of the parameters, the two- and three-dimensional figures of the obtained solutions are plotted. Keywords: Sine-Gordon expansion method, Kadomtsev–Petviashvili equation, Schrödinger equation, wave solutions

The studies of the dynamics behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. This study investigates the nonlinear fractional modified Boussinesq equation and the fractional bad Boussinesq equation using the extended sinh-Gordon equation expansion method. Several travelling wave solutions are successfully constructed. By choosing suitable values of parameters, the 2D and 3D graphs of the reported solutions are successfully plotted. The convergence analysis of the applied method is also discussed. Keywords: The Sinh-Gordon equation, Boussinesq equations, Solitons, M-fractional derivative, Convergence analysis

This study presents the applications of the extended rational sine-cosine/sinh-cosh schemes to the Klein-Gordon-Zakharov equations and the (2+1)-dimensional Maccari system. Various wave solutions such as singular periodic, periodic wave, topological, topological kink-type, dark and singular soliton solutions are successfully revealed. To display the physical features of the reported solutions, we use some appropriate choice of parameters in plotting the 3D, 2D, and contour graphs of some attained solutions.

In this study, the applications of the extended sinh-Gordon equation expansion method to a nonlinear Schrodinger equation that describes the nonlinear spin dynamics of (2 + 1)-dimensional Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semiclassical limit is presented. We successfully construct dark, bright, combined dark-bright, singular and combined singular soliton solutions to this equation. We give the parametric conditions for the existence of valid soliton to each of the obtained solutions. We plot the 2D and 3D graphics to some of the obtained solutions. The reported results of this study may be helpful in explaining the physical meaning of the studied model.

The studies of the dynamic behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. In this study, we investigate the coupled Boussinesq equation which arises in the shallow water waves for two-layered fluid flow. The modified exp $$(-\varphi (\zeta ))$$ -expansion function method is utilized in reaching the solutions to this equation such as the topological kink-type soliton and singular soliton solutions. The interesting 2D and 3D graphics of the obtained analytical solutions in this study are presented. Via one of the reported analytical solutions, the finite forward difference method is used in obtaining the approximate numerical and exact solutions to this equation. The Fourier–Von Neumann analysis is used in checking the stability of the used numerical method with the studied model. The $$L_{2}$$ and $$L_{\infty }$$ error norms are computed. We finally present a comprehensive conclusion to this study.

This study uses the extended sinh-Gordon equation expansion method in constructing various optical soliton solutions to the resonant nonlinear Schrodinger equation with both spatio-temporal and inter-modal dispersions. Resonant nonlinear Schrodinger equation expresses the propagation dynamics of optical solitons and Madelung fluids. Dark, bright, combined dark–bright and singular optical solitons are successfully constructed. Under the choice of suitable values of parameters, the two-dimensional, three-dimensional and the contour graphs to some of the acquired results are plotted. The reported results may be useful in explaining the physical meaning of the studied model.

In this study, the numerical solutions to the KdV equation with dual power nonlinearity by using the finite difference method are obtained. Discretize equation is presented in the form of finite difference operators. The numerical solutions are secured via the analytical solution to the KdV equation with dual power nonlinearity which is present in the literature. Through the Fourier–Von Neumann technique and linear stable, we have seen that the FDM is stable. Accuracy of the method is analyzed via the $${L}_{2}$$ and $${L}_{\infty }$$ norm errors. The numerical, exact approximations and absolute error are presented in tables. We compare the numerical solutions with the exact solutions and this comparison is supported with the graphic plots. Under the choice of suitable values of parameters, the 2D and 3D surfaces for the used analytical solution are plotted.

This study uses two mathematical approaches in constructing dark, bright, kink-type and singular soliton solutions to the Heisenberg ferromagnetic spin chain equation. The (2+1)-dimensional Heisenberg ferromagnetic spin chain equation describes nonlinear dynamics of magnets. The acquired results in this study may help in explaining some physical meanings of some nonlinear physical models arising in electromagnetic waves. For instance, the hyperbolic tangent arises in the calculation of magnetic moment and rapidity of special relativity, the hyperbolic cotangent arises in the Langevin function for magnetic polarization. The 2-, 3-dimensional and the contour plots of all the acquired solutions are presented.

In this study, the $$(1+1)$$ - and $$(2+1)$$ -dimensional Chiral nonlinear Schrodinger equations are investigated. These equations describe the edge states of the fractional quantum hall effect. We successfully acquired dark and bright soliton solutions to these equations by using the sine-Gordon expansion method. The constraint conditions for the existence of valid soliton are given. We present the numerical simulations to some of the obtained solutions under the choice of suitable parameters.

In this study, with the aid of Wolfram Mathematica 11, the modified exp ( − Ω ( η ) ) -expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well-known nonlinear evolutionary equation, namely; the two-component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two-component second order KdV evolutionary system with the finite forward difference method by using the Fourier-Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of L 2 and L ∞ norm error. We present the comparison between the exact and numerical solutions of the two-component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp ( − Ω ( η ) ) -expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models.

In this paper, we investigate and use the new modified exp ( − Ω ( ξ ) ) -expansion method, (MEM). We apply the new MEM to nonlinear long–short-wave interaction systems (NLSWIS). Among our findings are sets of solutions including, but not limited to, new hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease.

In this paper, we seek for the new soliton solutions to the coupled nonlinear Maccari's system by using sine-Gordon expansion method. Sine-Gordon expansion method is based on sine-Gordon Eq. (2.1) and a travelling wave transformation. We obtain new complex hyperbolic functions solutions to this system with help of Wolfram Mathematica 9, we also plot 3-dimensional and 2-dimensional graphics to some obtained solution using the same program in Wolfram Mathematica 9. Finally, we submit a comprehensive conclusions.

The purpose of this study is to find some novel soliton solutions of Fokas–Lenells (FL) equation where the perturbation terms are taken into account with nonlinearity. The sine-Gordon expansion method (SGEM) and the $$(m+({G'}/{G}))$$ -expansion method are used in this context. The dark, bright, dark–bright and singular optical soliton solutions are successfully obtained. Moreover, the constraint conditions for guaranteeing the existence of solutions are also given.

In this paper, the cubic-quartic nonlinear Schr&ouml, dinger and resonant nonlinear Schr&ouml, dinger equation in parabolic law media are investigated to obtain the dark, singular, bright-singular combo and periodic soliton solutions. Two powerful methods, the m + G &prime, G improved expansion method and the exp &minus, &phi, &xi, expansion method are utilized to construct some novel solutions of the governing equations. The obtained optical soliton solutions are presented graphically to clarify their physical parameters. Moreover, to verify the existence solutions, the constraint conditions are utilized.

This research communication aims to obtain the periodic wave solutions of the double dispersive equation of the wave propagation in a nonlinear elastic inhomogeneous Murnaghan's rod by using the F -expansion technique. This method first converts the partial differential equation into an ordinary differential equation under the wave transformation, then the assumed solution converts the problem under study into systems of algebraic equations. Once these algebraic systems are solved for the unknowns and are shifted into the assumed solution, the exact solutions of the double dispersive equation is obtained. Next by making the modulus of Jacobi elliptic functions into either 0 (or) 1, non-topological, singular and their compound solitons are gleaned. The two- and three-dimensional plots are given to show the roving properties of the solutions.

This manuscript aims to construct a family of travelling wave solutions to the high nonlinearity differential equation for obtaining new hyperbolic function solutions by using an analytical method, which is based on the exponential function. A number of new hyperbolic solutions for the model have been newly derived by using the method. We take advantage of some computer programming for all numerical calculations, and surfaces of solutions obtained in this paper.

Under investigation in this paper is the ( 2 + 1 ) -dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. Based on bilinear method, the multiple rogue wave solutions and the novel multiple soliton solutions are constructed by giving some specific activation functions in the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained, via Maple 18. By utilizing improved tan ( Φ ( ρ ) ∕ 2 ) -expansion technique the series of novel exact solutions in terms of rational, periodic and hyperbolic functions for the fractional cases are derived. Also, the semi-inverse variational principle is offered to get the solitary solutions. We construct the exact lump and rogue wave solutions, by solving the under-determined nonlinear system of algebraic equations for the specified parameters. Via various three-dimensional plots, curve plots, density plots and contour plots, dynamical characteristics of these waves are represented.

Obtaining new and important travelling wave solutions of wave propagation modelling of waves such as nonlinear optics models, propagation and transmission models of waves by using different methods plays an important role in maritime engineering, ocean, beach science and floating structures, and also for understanding new physical meanings of coastal structural properties. New complex travelling wave solutions obtained by using different newly improved methods explain new and general properties of nonlinear optic structures, wave propagations and motions, major structures of the impact of environmental factors on the beach like tsunami, the impact of the waves on the vessel, the power of the effects of the waves on wave distribution panels. These solutions structures may be trigonometric, complex function, hyperbolic function, exponential and rational function. In this study, we apply two effective methods to the nonlinear evolution equation used to describe the new versions of different mathematical models for wave motion and propagations. The first is improved Bernoulli sub-equation function method (IBSEFM), the latter is modified exp ( - Ω ( ξ ) ) -expansion function method (MEFM). We obtain some new travelling wave structures such as complex function, hyperbolic function and rational function, exponential function solutions. We observe that all travelling wave solutions have been verified the nonlinear partial differential equation by using Wolfram Mathematica 9. Then, we plot the two and three dimensional surfaces for all travelling wave structures obtained in this paper by the same computer program.

This manuscript focuses on the new wave behaviors of the system of equations for the ion sound wave under the action of the ponderomotive force which results from a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field due to high-frequency field. The sine-Gordon expansion method which is one of the powerful methods has been considered for finding traveling wave solutions of the system of equations for the ion sound wave under the pressure of the Langmuir wave. This algorithm yields new complex hyperbolic function solutions to the system considered in this paper. Wolfram Mathematica 9 has been successfully used throughout the paper for mathematical calculations.

In this paper, three images of nonlinearity to the fractional Lakshmanan Porsezian Daniel model in birefringent fibers are investigated. The new bright, periodic wave and singular optical soliton solutions are constructed via the \begin{document}$ \left( m+\frac{G'}{G} \right) $\end{document} expansion method, which are applicable to the dynamics within the optical fibers. All solutions are novel compared with solutions obtained via different methods. All solutions verify the conformable Lakshmanan-Porsezian-Daniel model and also, for the existence the constraint conditions are utilized. Moreover, 2D and 3D for all solutions are plotted to more understand its physical characteristics.

In the current study, we instigate the four-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation. The NLmZK equation guides the attitude of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Two different methods are used, namely the sine-Gordon expansion method (SGEM) and the [Formula: see text]-expansion method to the proposed model. We have successfully constructed some topological, non-topological, and wave solutions. In addition, the 2D, 3D, and contour graphs of the solutions are also plotted under the choice of appropriate values of the parameters.

In this paper, we used the modied auxiliary expansion method to nd some new family solution of the paraxial nonlinear Schrodinger equation. The solutions have a hyperbolic function, trigonometric function, exponential function, and rational function forms. The linear stability analysis of paraxial NLSE is also studied. Two cases when the instability modulation becomes to occur are investigated. All solutions are new and veried the main equation of the paraxial wave equation. Moreover, the constraint conditions for the existence of soliton solutions are also showed.

This study reaches the dark, bright, mixed dark-bright, singular, mixed singular optical solitons and singular periodic wave solutions to the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. The parametric conditions that guarantee the existence of valid solitons and other solutions are stated. By choosing some suitable values of parameters, the 2- and 3-dimensional surfaces to some of the reported solutions are plotted. The reported solutions may be useful in expalining the physical meaning of the Radhakrishnan–Kundu–Lakshmanan equation and other related nonlinear models arising in nonlinear sciences.

This study reveals the dark, bright, combined dark–bright, singular, combined singular optical solitons and singular periodic solutions to the conformable space–time fractional complex Ginzburg–Landau equation. We reach such solutions via the powerful extended sinh-Gordon equation expansion method (ShGEEM). Constraint conditions that guarantee the existence of valid solitary wave solutions are given. Under suitable choice of the parameter values, interesting three-dimensional graphs of some of the obtained solutions are plotted.

In this study, the modified exp $$ ( - \Phi (\eta )) $$ -expansion function method is used in constructing some solitary wave solutions to the Oskolkov–Benjamin–Bona–Mahony–Burgers, one-dimensional Oskolkov equations and the Dodd–Bullough–Mikhailov equation. We successfully construct some singular solitons and singular periodic waves solutions with the hyperbolic, trigonometric and exponential function structures to these three nonlinear models. Under the choice of some suitable values of the parameters involved, we plot the 2D and 3D graphics to some of the obtained solutions in this study. All the obtained solutions in this study verify their corresponding equation. We perform all the computations in this study with the help of the Wolfram Mathematica software. The obtained solutions in this study may be helpful in explaining some practical physical problems.

In this paper, we construct a family of wave solutions to the doubly dispersive equation, such as topological, non-topological, singular, compound topological-non-topological bell-type and compound singular, soliton-like, singular periodic wave and exponential function solutions. These analytical solutions are obtained by using the extended sinh-Gordon equation expansion method and the modified $\exp(-\varphi(\zeta))$ -expansion function method. The doubly dispersive equation is an important nonlinear physical model describing the nonlinear wave propagation in the elastic inhomogeneous Murnaghan’s rod. Under a suitable choice of parameters, the 2D, 3D and contour graphics to the reported results are also plotted.

The dynamical systems of soliton propagation through optical fibers for trans-continental and trans-oceanic distances is one of the most interesting areas of study. Optical solitons are restrained electromagnetic waves that stretch in nonlinear dispersive media and allow the intensity to remain unchanged due to the balance between dispersion and nonlinearity effects. In this study, we successfully acquire dark, bright, combined dark–bright, singular and combined singular soliton solutions to the decoupled nonlinear Schrodinger equation arising in dual-core optical fibers by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid soliton solutions are given. We discuss how change in parameters affect the solitons transmission. We present the 2D, 3D and the contour graphs to some of the reported solutions.

This study investigates the longitudinal wave equation in a magneto-electro-elastic circular rod by using the extended sinh-Gordon equation expansion method. Topological, non-topological and singular soliton solutions are extracted. To illustrate the physical appearance of the obtained solutions, 2D, 3D and the contour graphs to some of the obtained solutions are plotted. The reported results may be useful in explaining the physical meaning of the studied models and other nonlinear physical models arising in nonlinear sciences.

In this study, we obtain some new complex analytical solutions to the Kundu–Eckhaus equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics using improved Bernoulli sub-equation function method. After we have mentioned the general structure of improved Bernoulli sub-equation function method, we have successfully applied this method and then obtained some new complex hyperbolic and complex trigonometric function solutions. Two- and three-dimensional surfaces of analytical solutions have been plotted via wolfram Mathematica 9 version. At the end of this article, a conclusion has been submitted by mentioning important points founded in this study.

In this study, the modified exp $$(-\phi (\eta ))$$ -expansion function method is utilized in acquiring some new results to the coupled nonlinear Maccari’s system. The Maccari’s system is a nonlinear model that describes the dynamics of isolated waves, confined in a small part of space, in various fields such as hydrodynamic, plasma physics and nonlinear optics. We construct some new results with a complex structure to this model, such as; the trigonometric and hyperbolic function solutions. Under the suitable choice of the values of parameters, we plot the 2D, 3D and the contour graphs to some of the obtained solutions in this study. We observed that our results may be helpful in detecting the movement of an isolated wave in a small space to some practical physical problems.

A newly propose mathematical approach is presented in this study. We utilize the new approach in investigating the solutions of the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation. The new analytical technique is based on the popularly known sinh-Gordon equation and a wave transformation. In developing this new technique at each every steps involving integration, the integration constants are considered to not be zero which gives rise to new form of travelling wave solutions. The (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony is used in modelling an approximation for surface long waves in nonlinear dispersive media. We construct some new trigonometric function solution to this equation. Moreover, the finite forward difference method is utilized in investigating the numerical behavior of this equation by taking one of the obtained analytical solutions into consideration. We finally, give a comprehensive conclusions.

In this study, a nonlinear model is investigated, namely; the time regularized long wave equation. Various solitary wave solutions are constructed such as the non-topological, compound topological-non-topological bell-type, singular and compound singular soliton solutions. Under the choice of suitable parameters values, the 2D and 3D graphs to all the obtained solutions are plotted. The reported results in this study may be helpful in explaining the physical meanings of some important nonlinear models arising in the field of nonlinear science.

Some new travelling wave transform methods are very importantfor obtaining analytical solutions of special type of nonlinear partial differentialequations (NLPDEs). Some of these solutions of NLPDEs may be inthe different forms such as rational function solutions, trigonometric functionsolutions, hyperbolic function solutions, exponential function solutions andJacobi elliptic function solutions. These forms tell us the various propertiesof the NLPDEs from scientifical applications to engineering.In this research, we have studied to obtain the analytical solution ofthe nonlinear (2+1)-dimensional Burgers equation which is named from JohannesMartinus Burgers and the nonlinear special type of the Dodd-Bullough-Mikhailov equation introduced to the literature by Roger Dodd, Robin Bullough,and Alexander Mikhailov.

This study acquires the wave solutions of the two well-known nonlinear models, namely; the modified Benjamin–Bona–Mahony and the coupled Klein–Gordon equations. The modified Benjamin–Bona–Mahony is a nonlinear model that describes the long surface gravity waves of small amplitude and the coupled Klein–Gordon equation describes the quantized version of the relativistic energy–momentum relation. We successfully acquire some new solutions to these models such as kink-type and soliton solutions in complex hyperbolic functions form. We plot the 3D and 2D surface of the all the obtained solutions in this study. The mathematical approach used in this study is the sine-Gordon expansion method.

In this study, with the aid of the Wolfram Mathematica software, the powerful sine-Gordon expansion method is utilized to search for the solutions to some important nonlinear mathematical models arising in nonlinear sciences, namely, the (2 + 1) -dimensional Zakharov-Kuznetsov modified equal width equation, the cubic Boussinesq equation and the modified regularized long wave equation. We successfully obtain some new soliton, singular soliton, singular periodic waves and kink-type solutions with complex hyperbolic structures to these equations. We also present the two- and three-dimensional shapes of all the solutions obtained in this study. We further give the physical meaning of all the obtained solutions. We compare our results with the existing results in the literature.

In this study, with the help of Wolfram Mathematica 9 software, the powerful sine-Gordon expansion method is used in constructing new hyperbolic function solutions to the two well known nonlinear differential equations that arise in the field of nonlinear sciences, namely; the modified Zakharov–Kuznetsov and the (2+1)-dimensional cubic Klein–Gordon equations. We also plot the two- and three-dimensional graphics of all the obtained solutions in this paper by utilizing the same program in the Wolfram Mathematica 9 software.

With the aid of the Wolfram Mathematica software, the powerful sine-Gordon expansion method is used in constructing some new solutions to the two well-known nonlinear models, namely, the simplified modified Camassa-Holm and symmetric regularized long-wave equations. We obtain some novel complex, trigonometric and hyperbolic function solutions to these two equations. We also plot the three-dimensional and two-dimensional graphics for each solutions obtained using the same programming code in Wolfram Mathematica software. Finally, we submit a comprehensive conclusions.

This study is related to new soliton solutions of Davey–Stewartson equation (DSE) with power-law nonlinearity. The generalized Kudryashov method which is one of the analytical methods has been used for finding exact solutions of this equation. By using this method, dark soliton solutions of DSE have been found. Also, by using Mathematica Release 9, some graphical representations have been done to analyze the motion of these solutions.

In this paper, we have executed the Homotopy Analysis Method and Modified Trial Equation Method which has newly been submitted to the literature for obtaining analytical solution of the nonlinear time-fractional generalized Burgers equation occuring in various areas of physics, chemistry, applied sciences, applied mathematics such as modeling of gas dynamics and traffic flow. Then, we have formed a table which includes numerical conclusions for time-fractional generalized Burgers equation. Finally, we have obtained the 2D and 3D surfaces by means of programming language Mathematica 9 in order to interpret in the sense of physical phenomena for analytical solution and approximate solution which have been obtained.

In this study, we construct several wave solutions to the nonlinear fractional Hirota–Maccari equation with a truncated M-fractional derivative via the extended sinh-Gordon equation expansion method. The constraint conditions that guarantee the existence of valid solutions are stated. We use suitable values of parameters in plotting the 2- and 3-dimensional graphs of the reported solutions.