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In engineering and applied sciences, several physical phenomena can be more precisely characterized by employing nonlinear fractional partial differential equations. The primary goal of this research is to examine the traveling wave solution in closed form for the nonlinear acoustic wave propagation model known as the time fractional simplified modified Camassa–Holm equation, which is used to explain the unidirectional propagation of shallow-water waves with non-hydrostatic pressure and explains the dispersion properties of numerous phenomena like fluid flow, control theory, liquid drop patterning in plasma, acoustics, fusion, and fission processes, etc. The utmost potential approach, namely the new auxiliary equation technique, is applied for analyzing the time nonlinear fractional simplified modified Camassa-Holm equation in the logic of the newest established truncated M-fractional derivative. The fractional partial differential equations have been transformed to the ordinary differential equation using the complex wave transformation in the sense of truncated M-fractional derivative. A variety of soliton solutions, including anti-kink, single soliton, anti-bell, bell, kink, multiple soliton, double soliton, singular-kink, compacton shape, periodic shape, and so many, are displayed in the diagram of 3D and contour plots by taking into account a number of various parameters. It is essential to point out that all derived outcomes are directly compared to the original solutions to certify their exactness. Results show that the used scheme is capable, simple, and straightforward and can be useful to a variety of complex phenomena. The acquired results are unique for the model equation and could be applied to the analysis of several nonlinear study fields.

Nonlinear fractional partial differential equations can explain a vast scope of engineering and science, like atomic physics, wireless transmission, nonlinear optics, acoustics, economics, materials science, control theory, plasma physics, quantum plasma in mechanics, biological systems of nonlinear case and so on. In this study, the new generalized (G′/G)-expansion technique have been studied for examine the nonlinear fractional equations and built exact analytical traveling as well as solitary wave solutions namely space–time one-dimensional fractional wave equation used to simulate wave transmission in a nonlinear medium with blending and space–time fractional regularized long wave equation enables bore expansion and wave generating in nonlinear model of solitary waves, ion acoustic plasma waves, shallow waves in water, and nonlinear dispersive waves with the help of conformable derivatives. The creatable equations are transformed into ordinary differential equations by fractional complex transformation. Some dynamical wave shapes of multiple solitons, single soliton, periodic anti-kink, anti-kink, flat-kink type solitary wave models have been created, and 3D, and contour have been built to represent these solutions. The results are expressed using hyperbolic, rational functions, and trigonometric and Maple or Mathematica utilized to graphically represent the obtained solutions. It is essential to point out that all resultant solutions are directly compared to the original solutions to ensure their exactness. The new generalized (G′/G)-expansion method is an operative, compatible, and fruitful approach for analyzing nonlinear fractional traveling waves.

The space–time fractional Landau-Ginzburg-Higgs equation and coupled Boussinesq-Burger equation describe the behavior of nonlinear waves in the tropical and mid-latitude troposphere, exhibiting weak scattering, extended connections, arising from the interactions between equatorial and mid-latitude Rossby waves, fluid flow in dynamic systems, and depicting wave propagation in shallow water. The improved Bernoulli sub-equation function method has been used to achieve new and wide-ranging closed-form solitary wave solutions to the mentioned nonlinear fractional partial differential equations through beta-derivative. A wave transformation is applied to renovate the fractional-order equation into an ordinary differential equation. Some standard wave shapes of multiple soliton type, single soliton, kink shape, double soliton shape type, triple soliton shape, anti-kink shape, and other types of solitons have been established. The more updated software Python is used to display the solutions by using 3D and contour plotlines to describe the physical significances of attained solutions more clearly. The findings of this study are straightforward, adaptable, and quicker to simulate. It has been notable that the improved Bernoulli sub-equation function method is practical, effective, and offers more sophisticated solutions that can help to generate a large number of wave solutions for various models.

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

Abstract The space–time fractional nonlinear Klein-Gordon and modified regularized long-wave equations explain the dynamics of spinless ions and relativistic electrons in atom theory, long-wave dynamics in the ocean, like tsunamis and tidal waves, shallow water waves in coastal sea areas, and also modeling several nonlinear optical phenomena. In this study, the improved Bernoulli sub-equation function method has been used to generate some new and more universal closed-form traveling wave solutions of those equations in the sense of beta-derivative. Using the fractional complex wave transformation, the equations are converted into nonlinear differential equations. The achieved outcomes are further inclusive of successfully dealing with the aforementioned models. Some projecting solitons waveforms, including, kink, singular soliton, bell shape, anti-bell shape, and other types of solutions are displayed through a three-dimensional plotline, a plot of contour, and a 2D plot for definite parametric values. It is significant to note that all obtained solutions are verified as accurate by substituting the original equation in each case using the computational software, Maple. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed technique is effective, computationally attractive, and trustworthy to establish more generalized wave solutions.

The time-fractional unstable nonlinear Schrödinger (NLS) equations capture the time evolution of disturbances within media, tailored for describing phenomena in unstable media to help model and understand the intricate dynamics of systems prone to instability, the behavior of disturbances in complex and unstable environments and many more. In this study, we investigate the exact and analytical solutions of time-fractional unstable nonlinear Schrödinger (NLS) equations with the time-fractional β-time derivative, a powerful technique known as the sine-Gordon expansion (SGE) method is used. Non-locality behaviors that cannot be modeled using classical calculus are included in this equation. It discusses the relationships between waves of different frequencies or the same frequency but with distinct polarizations having a diverse set of key applications in nonlinear optics. We begin by using a fixed-point argument to prove the existence and uniqueness of solutions to the time-fractional problem under consideration. The partial differential equation of fractional order is transformed into an ordinary differential equation using the complex transformation. Numerous exact solutions are achieved with all the arbitrary parameters. The specific values of the obtained solutions' associated parameters are analyzed graphically with 3D plots as multiple soliton types of different wave solutions with the help of symbolic software. Understanding that the features of the solutions are far from intuitive because it is based on the parameter selection from the figures is difficult. The main objective of this paper is to investigate some fresh and further general solitary wave solutions to the suggested equation that properly explain the above-stated phenomena.

In a range of nonlinear fields, for example molecular biology, physics in plasma, quantum mechanics, elastic media, nonlinear optics, the surface of water waves, and others, many complicated nonlinear behaviors can be pronounced using nonlinear fractional partial differential equations. The nonlinear traveling waves, including nonlinear space-time fractional order differential equations such as the time fractional Sharma-Tasso-Olver (STO) and the space-time fractional Kowerteg-de Vries-Burgers (KdV-Burgers) equations, are analysis through a resourceful approach, namely the extended tanh-function by the definition of the conformable derivative. The stated equations are prevalently used in marine and coastal infrastructure, such as fluid flow, plasma engineering, fiber optics, fusion and fission phenomena, and so on. By renovating fractional order differential equations to ordinary differential equations, fractional order differential transformation simplifies the solution process. Several types of solutions, such as soliton types, bell types, kink types, and some other kinds of results, are developed and portrayed for definite values of free parameters that are plotted in terms of 3D, contour, and spherical shapes. The method is an algebraic approach that is both direct and efficient for dealing with nonlinear equations and yield a distinct category of solutions known as solitons. The suggested method is being used in this study to generate attractive, and effective results that are flexible, easy, and quicker to simulate using general mathematical techniques, namely Maple. It is worth declaring that all resultant solutions are verified for accuracy by exchanging them directly within the original equation through the software package and originating them accurately.MSC: 35C25, 35C07, 35C08, 35Q20, 76B25

In this study, the closed-form wave solutions has been examined to the space–time fractional simplified Camassa-Holm equation through two potential techniques, namely the sine-Gordon expansion approach and the extended tanh function scheme. The equation explains the dispersion effects of several phenomena, including liquid drop patterning in plasma, fluid flow, fission and fusion processes, acoustics, control theory, etc. The fractional-order equation is transformed into a nonlinear equation through a complex transformation. Diverse solutions are determined, and for specified parametric values, the solutions are graphically assessed by depicting 3D and contour plots that provide kinks, singular kinks, plane-shaped, bell-shaped, and other types of solitons. All derived solutions are substituted into the original equation to ensure their accuracy, and the obtained solutions are compared with existing solutions in the literature to demonstrate their novelty. It is noteworthy that the suggested techniques are reliable, competent, and potential mathematical tools to establish closed-form wave solutions.

In this study, the extended tanh-function method has been used to find further general travelling wave solutions for space-time fractional nonlinear partial differential equations, namely, the time fractional nonlinear Sine-Gordon equation and Klein–Gordon equation. The mentioned equations are useful for explaining a series of experimental seismic data, modelling of strain waves, and structures correlated to fault dynamics and the subduction slab, as well as slow earthquakes, episodic tremor, slow slip events, and tremor migration patterns. This method is used to attain exact solutions to a variety of nonlinear partial differential equations that were spatially temporal fractional. To get analytical solutions of the travelling wave type of certain nonlinear evolution equations, a power series in tanh was first utilized as an ansatz. These nonlinear fractional partial differential equations (NLFPDEs) are solved on a variety of non-rectangular domains. By using the fractional complex transform and the properties of the confirmable derivative, the two equations are reduced to ordinary differential equations. The solutions are sketched in 3D, and contour patterns, including king type, single soliton, double solitons, multiple solitons, bell shape, and other sorts of solutions. The achieved solutions are very much effective for explaining the above-stated phenomena.

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.

Shallow water waves are one of the most prominent and widely used waves in coastal areas. The anomalous bi-directional transmission of long waves over shallow water is described by the nonlinear fractional partial differential equations (NLFPDEs), namely space-time-fractional coupled Whitham-Broer-Kaup (WBK) and coupled approximate long water (ALW) wave equations. To achieve new analytical solutions, the enhanced extended tanh-function technique for the mentioned fractional equations in the sense of conformable derivative is used. Some well-known wave shapes of solitons, namely, kink, bell-shape, and other types, through the suggested technique are achieved, and the 3D, contour, list point plots, and vector plots of the solutions are sketched to further illustrate the wave profile. The results achieved in this study have been validated using the computational software Maple and found to be accurate. It is observed that the method is realistic, and dependable to investigate more generalized analytical wave solutions.

Nonlinear fractional evolution equations play a crucial role in characterizing assorted complex nonlinear phenomena observed in different scientific fields, including plasma physics, quantum mechanics, elastic media, nonlinear optics, surface water waves, nonlinear dynamics, molecular biology, and some other domains. In this study, we investigate diverse solitary wave solutions to the space–time fractional modified Equal Width equation, focusing on their significance for wave propagation behaviors in plasma and optical fiber systems. Two effective approaches, namely the improved Bernoulli sub-equation function and the new generalized (G′/G)-expansion methods are exploited. A fractional wave transformation technique is employed to convert the fractional-order equation into an ordinary differential equation. The solitary wave solutions obtained in terms of exponential, hyperbolic, rational, and trigonometric functions, have wide applications in various nonlinear phenomena. The 3D, spherical, contour, and vector plots are presented to elucidate the physical implications of the obtained solutions for specific parameter values. The soliton structures reveal various wave types, including bell-shaped, multi-soliton, kink, single soliton, anti-bell-shaped, periodic soliton, compacton, and other types for definite values of the free parameters. We compare the attained solutions with the solutions available in the literature to demonstrate the feasibility of the solutions. It is observed that both approaches are efficient, straightforward, and significant for investigating diverse nonlinear models that modulate complex phenomena.

The (2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis (CD) and fractional potential Kadomstev-Petviashvili (PKP) equation. It can be modeled according to the Hamiltonian structure, the lax pair with the non-isospectral problem, and the pain level property. The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves, demonstrate the propagation of waves in dissipative and nonlinear media, and reveal the propagation of waves in dissipative and nonlinear media. In this paper, we have established further exact solutions to the nonlinear fractional partial differential equation (NLFPDEs), namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable (G′/G,1/G)-expansion method. As far as trigonometric, hyperbolic, and rational function solutions containing parameters are concerned, solutions are acquired when unique characteristics are assigned to the parameters. Subsequently, the solitary wave solutions are generated from the solutions of the traveling wave. It is important to observe that this method is a realistic, convenient, well-organized, and ground-breaking strategy for solving various types of NLFPDEs.

The space–time fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equations are used to explain the transmission of shallow water waves inside a water channel of uniform speed and constant depth. It moreover simulates water waves in harbors and shallow water, which is useful for oceanography. These equations are significant nonlinear model equations to illustrate numerous physical structures namely in marine and coastal science, water wave mechanics, control theory, plasma engineering, fluid flow, optical fibers, phenomena of fission and fusion, and so on. In this study, we employed the extended tanh-function approach to generate some fresh and further general closed-form traveling waves solutions of those equations in the lite of conformal derivatives. The achieve results are more applicable to resolve the above mentioned phenomena properly. The fractional differential transform simplifies generate ordinary differential equations from fractional order differential equations. We discovered many types of solutions using the maple, including solitons, kink types, bell types, and other types of solutions that are illustrated using 3D and contour plots. It is important to note that all derived solutions are checked for accuracy by being directly replaced with the original equation. We proposed that the technique be revised to be more realistic, effective, and trustworthy and that we explore more generalized exact results of traveling waves, like solitary wave solutions.

This work aims to construct exact solutions for the space-time fractional (2 + 1)- dimensional dispersive longwave (DLW) equation and approximate long water wave equation (ALW) utilizing the two-variable (G′/G,1/G)-expansion method and the modified Riemann–Liouville fractional derivative. The recommended equations play a significant role to describe the travel of the shallow water wave. The fractional complex transform is used to convert fractional differential equations into ordinary differential equations. Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package. The Maple package program was used to set up and validate all of the computations in this investigation. By choosing particular values of the embedded parameters, we produce multiple periodic solutions, periodic wave solutions, single soliton solutions, kink wave solutions, and more forms of soliton solutions. The achieved solutions might be useful to comprehend nonlinear phenomena. It is worth noting that the implemented method for solving nonlinear fractional partial differential equations (NLFPDEs) is efficient, and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

The space–time fractional equal width (EW) and the space–time fractional generalized equal width (GEW) equations are two important models that represent nonlinear dispersive waves, namely, waves ensuing in the shallow water channel, 1-D wave generation ascending in the nonlinear dispersive medium estimation, cold plasma hydro-magnetic waves, chemical kinematics, electromagnetic interaction, etc. In this article, we search advanced and broad-spectrum wave solutions of the formerly indicated models in diverse families in conjunction with the Riemann-Liouville fractional derivative via the double (G′/G,1/G)-expansion approach. The nonlinear fractional differential equations (NLFDEs) are transformed into ODEs by the composite function derivative and the chain rule putting together with the wave transformation. We acquire kink wave solution, multiple periodic solutions, single soliton solution, periodic wave solution and other types of soliton solutions by setting particular values of the embodied parameters. The suggested technique is functional, convenient, powerful, and computationally feasible to examine scores of NLFDEs.

In this work, a (2 + 1)-dimensional Ito equation is investigated, which represents the generalization of the bilinear KdV equation. Abundant double-periodic soliton solutions to the (2 + 1)-dimensional Ito equation are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. The dynamic properties are described through some 3D graphics and contour graphics.

The general time fractional Burger- Fisher (TF-BF) and the space–time regularized long-wave (STF-RLW) equations are considered as examples of gravitational water waves in cold plasma as well as so many areas. The above equations are used in nonlinear science and engineering to study long waves in seas and harbors that travel in just one direction. First, the two equations are transformed to ODEs by applying a fractional complex transform along with characteristics of confirmable fractional derivative (CFD). Then, the extended tanh-function (ET-F) approach is investigated to find a variety of analytical solutions with different geometrical wave structures the mentioned models. The results are in the form of kink, one-, two-, multiple-solitons solutions, and other types sketched in 2D, 3D, and contour patterns.

Nonlinear fractional partial differential equations (NLFPDEs) are well suited for describing a broad range of factors in engineering and science, including plasma physics, optical fiber, acoustics, finance, turbulence, mechanical engineering, control theory, nonlinear biological systems, and so on. Through this memorandum, a novel technique is used for solving the class of NLFPDEs. We opted to construct a traveling wave solution to the nonlinear space-time fractional Phi-4 and Allen-Cahn (AC) equations, which are often used as frameworks for a variety of phenomena such as nano fluids, reaction–diffusion model and used to investigate the phase separation process in a variety of components, order–disorder interchange in the bass system and, so on. The subsidiary new generalized (G'/G)-expansion technique for the mentioned equations is used to obtain new precise solutions with the help of conformable derivatives. This strategy is utilized in this article to obtain some new, appealing, and more general outcomes that are simple, versatile, and faster to simulate. Using the proposed technique, some dynamical wave shapes of compaction types, periodic compaction type, single soliton type, kink types, singular kink types, and some other types are accomplished and we used Mathematica to describe the solutions through 3D, contour, 3D list point plot, and vector plots to describe the physical sketch more clearly.

In this article, a (2+1)-dimensional Korteweg-de Vries equation is investigated. Abundant periodic wave solutions are obtained based on the Hirota’s bilinear form and a direct test function. Meanwhile, the interaction solutions between lump and periodic waves are presented. What is more, we derive the interaction solutions among lump, periodic, and solitary waves. Based on the extended homoclinic test technique, some new double periodic-soliton solutions are presented. Finally, some 3D and density plots are simulated and displayed to respond the dynamic behavior of these obtained solutions.

We utilize the fractional modified Riemann-Liouville derivative in the sense to develop careful arrangements of space–time fractional coupled Boussinesq equation which emerges in genuine applications, for instance, nonlinear framework waves iron sound waves in plasma and in vibrations in nonlinear string and space–time fractional-coupled Boussinesq Burger equation that emerges in the investigation of liquids stream in a dynamic framework and depicts engendering of shallow-water waves. A decent comprehension of its solutions is exceptionally useful for beachfront and engineers to apply the nonlinear water wave model to the harbor and seaside plans. A summed-up partial complex transformation is correctly used to change this equation to a standard differential equation thus, many precise logical arrangements are acquired with all the free parameters. At this point, the traveling wave arrangements are articulated by hyperbolic functions, trigonometric functions, and rational functions, if these free parameters are considered as specific values. We obtain kink wave solution, periodic solutions, singular kink type solution, and anti-kink type solutions which are shown in 3D and contour plots. The presentation of the method is dependable and important and gives even more new broad accurate arrangements.

The nonlinear Riemann wave equations (RWEs) and the Landau-Ginsburg-Higgs (LGH) equation are related to plasma electrostatic waves, ion-cyclotron wave electrostatic potential, superconductivity, and drift coherent ion-cyclotron waves in centrifugally inhomogeneous plasma. In this article, the interactions between the maximum order linear and nonlinear factors are balanced to compute realistic soliton solutions to the formerly stated equations in terms of hyperbolic functions. The linear and nonlinear effects rheostat the structure of the wave profiles, which vary in response to changes in the subjective parameters combined with the solutions. The established solutions to the aforementioned models using the extended tanh scheme are descriptive, typical, and consistent, and include standard soliton shapes such as bright soliton, dark soliton, compacton, peakon, periodic, and others that can be used to analyze in ion-acoustic and magneto-sound waves in plasma, homogeneous, and stationary media, particularly in the propagation of tidal and tsunami waves.

The space time fractional modified regularized long wave equation is a model equation to the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, the hydro-magnetic waves in cold plasma, the phonetic waves in dissident quartz and phonetic gravitational waves in contractible liquids. In nonlinear science and engineering, the mentioned equation is applied to analyze the one way tract of long waves in seas and harbors. In this study, the closed form traveling wave solutions to the above equation are evaluated due to conformable fractional derivatives through double (G'⁄G,1⁄G)-expansion method and the Exp-function method. The existence of chain rule and the derivative of composite function permit the nonlinear fractional differential equations (NLFDEs) converted into ODEs using wave transformation. The obtain solutions are very much effective to analyze the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, one way tract of long waves in seas and harbors. These two methods are efficient, convenient, and computationally attractive.

The exact wave solutions to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) and space time fractional Zakharov-Kuznetsov Benjamin-Bona-Mahony (ZKBBM) equations are studied in the sense of conformable derivative. The existence of chain rule and the derivative of composite functions permit the nonlinear fractional differential equations (NLFDEs) to convert into the ordinary differential equation using wave transformation. The wave solutions of these equations are examined by means of the expanding and effective two variable (G'/G,1/G)-expansion method. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The method is efficient, convenient, accessible and is the generalization of the original (G'/G)-expansion method.

The space time fractional Sharma–Tasso–Olver (STO) equation and the time fractional Cahn–Allen (CA) equation are well-designated to fission and fusion phenomena including those for solitons, electromagnetic interactions, quantum relativistic atom theory, phase isolation in several components bass system, the relativistic energy-momentum relation. In this article, we have exerted the exact solution to STO and CA equation in the light of fractional derivative (e.g. modified Riemann–Liouville derivative). By using wave transformation the ODE of integer order is generated by the fractional partial differential equation (FPDE). We investigated the travelling wave solution to these equations through the recently established double-expansion method. The results obtained in view of hyperbolic, trigonometric and rational functions containing parameters. We have demonstrated that this method is convenient, effective and powerful tools for solving nonlinear fractional differential equations (NLFDEs).