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This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.

In this paper, a nonlinear fractional emerging telecommunication model with higher–order dispersive cubic–quintic is studied by using two recent computational schemes. This kind of model is arising in many applications such as machine learning and deep learning, cloud computing, data science, dense sensor network, artificial intelligence convergence, integration of Internet of Things, self–service IT for business users, self-powered data centers, and dense sensor networks (DSNs) that is used in the turbine blades monitoring and health monitoring. Two practical algorithms (modified Khater method and sech–tanh functions method) are applied to higher–order dispersive cubic–quintic nonlinear complex fractional Schrodinger ( NLCFS ) equation. Many novel traveling wave solutions are constructed that do not exist earlier. These solutions are considered as the icon key in the emerging telecommunication field, were they able to explain the physical nature of the waves spread, especially in the dispersive medium. For more illustration, some attractive sketches are also depicted for the interpretation physically of the achieved solutions.

In this research, analytical and numerical solutions are studied of a two–dimensional discrete electrical lattice, which is mathematically represented by the modified Zakharov–Kuznetsov equation. Moreover, the stability property of the obtained analytical solutions is investigated based on the Hamiltonian system, and then it is used to evaluate the initial and boundary conditions that are used in the numerical investigation. Many kinds of analytical solutions are obtained, such as complex, exponential, hyperbolic, and trigonometric function solutions. The functioning of both schemes of both techniques is tested and investigated.

Novel explicit wave solutions are constructed for the fractional nonlinear model of the low–pass electrical transmission lines. A new fractional definition (Atangana–Baleanu derivative operator) is employed through the modified Khater method to get new wave solutions in distinct types of this model. The stability property of the obtained solutions is tested to show the ability of our obtained solutions in using through the physical experiments. Moreover, the obtained analytical solutions are used to evaluate the initial and boundary conditions that allows applying the cubic & septic B–spline schemes to investigate the numerical solutions of this model. The novelty and advantage of the proposed method are illustrated by applying to this model. Some sketches are plotted to show more about the dynamical behavior of this model.

In this paper, the fractional Broer–Kaup (BK) system is investigated by studying its novel computational wave solutions. These solutions are constructed by applying two recent analytical schemes (modified Khater method and sech–tanh function expansion method). The BK system simulates the bidirectional propagation of long waves in shallow water. Moreover, it is used to study the interaction between nonlinear and dispersive long gravity waves. A new fractional operator is used to convert the fractional form of the BK system to a nonlinear ordinary differential system with an integer order. Many novel traveling wave solutions are constructed that do not exist earlier. These solutions are considered the icon key in the inelastic interaction of slow ions and atoms, where they were able to explain the physical nature of the nuclear and electronic stopping processes. For more illustration, some attractive sketches are also depicted for the interpretation physically of the achieved solutions.

A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.

This paper explores accurate, stable, and novel soliton wave solutions of the nonlinear Qiao model. This model, which was derived in 2007, possesses a Lax representation and bi-Hamiltonian structure, where it is a second positive member in the utterly integrable hierarchy. The well-known generalized extended tanh-function method is employed to construct novel soliton wave solutions. The stable property of the obtained solutions is examined along with the Hamiltonian system’s characterizations. Furthermore, the accuracy of the obtained solutions is checked by comparing it with the model’s semi-analytical solutions that have been obtained by employing the variational iteration (VI) method. The obtained analytical and semi-analytical solutions are demonstrated through some distinct graphs to show more physical and dynamical behavior of the investigated model. The used analytical and semi-analytical schemes’ performance is checked to show if it is effective and powerful.

This study performs two recent methods on the nonlinear long-short wave interaction system (NLSWIS) to obtain novel formulas of solitary solutions representing an optical field that does not alter by multiplication a sensitive balance between the different terms impacts in the medium. Elastic medium is defined as the medium that can adjust in the figure due to a deforming strength and come back shape to its original form when the force eliminates. The wave is produced by vibrations that are a consequence of acoustic power. Also known as the sound wave, or acoustic wave. For examples of elastic mediums are air, water and so on. The improved $$\left( \frac{G'}{G}\right)$$ expansion method and novel $$\left( \frac{G'}{G}\right)$$ expansion method are implemented, which are considered as two general methods in this field. Applying these methods to many nonlinear partial differential equations gives diverse and different forms and configuration solutions that allow users of those two methods a large number of solutions that provide broader scope to use.

In this article, the modified Khater method is utilized to extract new analytical solutions to the cubic-quintic nonlinear non-paraxial pulse propagation model in a planar waveguide with Kerr nonlinearity, fifth-order nonlinearity, and spatiality dispersion due to non-paraxial effects. As a result, a variety of some new exact solutions are observed. Moreover, the three-dimensional shapes are visualized with the aid of the latest scientific tools.

In this survey, the ionic currents along the microtubules equation handled by applying two different techniques. It describes the ionic transport throughout the intracellular environment. This phenomenon explains the behavior of many applications in a biological nonlinear dispatch line for ionic currents. Using two different analytical techniques on this equation allows us to get new forms of analytical solitary traveling wave solutions. The obtained solutions support many researchers concerned with the discussion of the physical properties of the ionic currents along microtubules. Microtubules are one of the main components of the cytoskeleton, and function in many operations, comprehensive constitutional backing, intracellular transmit, and DNA division. The thing which makes us applies this property in many applications for this model. The earned results demonstrate that the proposed methods are significantly powerful, evangelist, relaxing, suitable, and convenient for solving many nonlinear models.

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

This article investigates the analytical wave solutions of the complex nonlinear Davey–Stewartson (CNLDS) equations by employing two novel analytical schemes. These equations are considered as the main icon in describing the water waves in the gravity-capillarity surface wave packets. These equations have been derived by Davey and Stewartson for the first time as a higher-general dimensional model of the well-known model Schrodinger equation. The extended exponential function (EEF) and the Khater II methods are employed to construct novel solutions. The obtained solutions are plotted in different techniques for illustrating the dynamical behavior of the water waves. Additionally, The obtained results’ stability is checked. The obtained results’ novelty is explained by comparing our solutions and published results in recent studies.

This research paper is connected with one of the most popular models in shallow water wave applications. The computational wave solutions of the Benjamin-Bona-Mahony-Peregrine (BBMP) equation with power-law nonlinearity are investigated via three new analytical schemes. This model is considered a natural extension of the well-known mathematical model called the Korteweg–de Vries (KdV) equation. The KdV has been used to study the shallow water wave behavior of long wavelengths and small amplitude. In contrast, the BBMP equation has been employed for the same purpose, but it concentrates on studying this kind of wave, especially in ocean beaches. Applying the modified auxiliary equation method (MAE) method, the direct algebraic (DA) method, and generalized Kudryashov (GK) method to the BBMP equation after harnessing suitable wave transformation gives plentiful unprecedented solitary wave solutions. Many novel solutions have been constructed through the suggested methods that have been built along with their 2D, 3D, and contour graphical configurations for clarity and exactitude.

The trigonometric quintic B-spline scheme is used in this research paper to research Zakharov’s (ZK) nonlinear dimensional equation’s numerical solution. The ZK model’s solutions explain the relationship between the high-frequency Langmuir and the low-frequency ion-acoustic waves with many applications in optical fiber, coastal engineering, and fluid mechanics of electromagnetic waves, plasma physics, and signal processing. Three recent computational schemes (the expanded [Formula: see text]-expansion method, generalized Kudryashov method, and modified Khater method) have recently been used to investigate this model’s moving wave solution. Many innovative solutions have been established in this paper to determine the original and boundary conditions that allow numerous numerical schemes to be implemented. Here, the trigonometric quintic B-spline method is used to analyze the precision of the collected analytical solutions. To illustrate the precision of the numerical and computational solutions, distinct drawings are depicted.

This research paper investigates novel explicit wave solutions of the fractional Korteweg–de Vries (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. These models are used as gravity models in water and an interaction model between the long waves. The Atangana–Baleanu ([Formula: see text]) fractional operator is utilized for the first time to convert the fractional form of both models into nonlinear partial differential equations with an integer order. The extended simplest equation method is employed to construct some distinct types of solitary wave solutions such as exponential, rational, hyperbolic and trigonometric functions. For more illustration of our obtained solutions, some figures for them are given. The power and practical properties of the used method are tested.

This paper studies the analytical, semi-analytical, and numerical solutions of the Cahn–Allen equation, which plays a vital role in describing the structure of the dynamics for phase separation inFe–Cr–X($X=Mo,Cu$X=Mo,Cu) ternary alloys. The modified Khater method, the Adomian decomposition method, and the quintic B-spline scheme are implemented on our suggested model to get distinct kinds of solutions. These solutions describe the dynamics of the phase separation in iron alloys and are also used in solidification and nucleation problems. The applications of this model arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The comparison between the obtained solutions is represented by using figures and tables to explain the value of the error between exact and numerical solutions. All solutions are verified by using Mathematica software.

In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical).

The analytical and numerical solutions of the (2+1) dimensional, Fisher-Kolmogorov-Petrovskii-Piskunov ((2+1) D-Fisher-KPP) model are investigated by employing the modified direct algebraic (MDA), modified Kudryashov (MKud.), and trigonometric-quantic B-spline (TQBS) schemes. This model, which arises in population genetics and nematic liquid crystals, describes the reaction–diffusion system by traveling waves in population genetics and the propagation of domain walls, pattern formation in bi-stable systems, and nematic liquid crystals. Many novel analytical solutions are constructed. These solutions are used to evaluate the requested numerical technique’s conditions. The numerical solutions of the considered model are studied, and the absolute value of error between analytical and numerical is calculated to demonstrate the matching between both solutions. Some figures are represented to explain the obtained analytical solutions and the match between analytical and numerical results. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

This paper investigates the analytical solutions of the well-known nonlinear Schrodinger (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.

The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model’s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions’ accuracy is checked by comparing the analytical and approximate solutions. The soliton waves’ characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper’s novelty is shown by comparing our obtained solutions with those previously published of the considered model.

This paper aims to determine some novel solitary wave solutions of the Chaffee–Infante equation, which have not yet been presented for this equation. This equation arises in various branches of science and technology, such as plasma physics, coastal engineering, fluid dynamics, signal processing through optical fibers, ion-acoustic waves in plasma, the sound waves, and the electromagnetic waves field. The $$(2+1)$$ -dimensional Chaffee–Infante equation describes the dynamical behavior of gas diffusion in a homogeneous medium. Soliton solutions are obtained for this equation using several computational schemes. Many physical significances are explained by sketching some two-dimensional and three-dimensional diagrams for the acquired solutions in three different types. These figures give us a better understanding of the behavior of these solutions. Moreover, the stability property is investigated based on the Hamiltonian system’s characterizations. The methods provide efficient way for the solving other equations that occur in other branches of science.

This manuscript handles the nonlinear Klein–Fock–Gordon ( KFG ) equation by applying two recent computational schemes (generalized exponential function (GEF) and generalized Riccati expansion (GRE) methods) to construct abundant novel wave solutions The considered model is the generalized form of the well-known nonlinear Schrodinger equation which is considered a quantized version of the relativistic energy-momentum relation. The accuracy of the employed analytical schemes by showing the matching between computational and approximate solutions and calculating the absolute value of error between these solutions. This matching is investigated by employing the variational iteration (VI) method to show the precision of the used schemes with the previously published solutions. The physical characterization of the evaluated solutions has explained through some distinct sketches in 2D, 3D, contour, polar, and spherical plots. The originality and novelty of our investigation have been checked by comparing our solution’s accuracy with previous other solutions’ accuracy.

This manuscript investigates the accuracy of the solitary wave solutions of the (2+1)-dimensional nonlinear Chiral Schrodinger ((2+1)-D CNLS) equation that are constructed by employing two recent analytical techniques (modified Khater (MKhat) and modified Jacobian expansion (MJE) methods). This investigation is based on evaluating the initial and boundary conditions through the obtained analytical solutions then employing the Adomian decomposition (AD) method to evaluate the approximate solutions of the (2+1)-D CNLS equation. This framework gives the ability to get large complex traveling wave solutions of the considered model and shows the superiority of the employed computational schemes by comparing the absolute error for each of them. The handled model describes the edge states of the fractional quantum hall effect. Many novel solutions are obtained with various formulas such as trigonometric, rational, and hyperbolic to the studied model. For more illustration of the results, some solutions are displayed in 2D, 3D, and density plots.

The computational solutions for the fractional mathematical system form of the HIV-1 infection of CD4+ T-cells are investigated by employing three recent analytical schemes along the Atangana–Baleanu fractional (ABF) derivative. This model is affected by antiviral drug therapy, making it an accurate mathematical model to predict the evolution of dynamic population systems involving virus particles. The modified Khater (MKhat), sech–tanh expansion (STE), extended simplest equation (ESE) methods are handled the fractional system and obtained many novel solutions. The Hamiltonian system’s characterizations are used to investigate the stability property of the obtained solutions. Additionally, the solutions are sketched in two-dimensional to demonstrate a visual representation of the relationship between variables.

In this survey, structure solutions for the longitudinal suspense equation within a magneto-electro-elastic (MEE) circular judgment are extracted via the implementation of two one-of-a-kind techniques that are viewed as the close generalized technique in its field. This paper describes the dynamics of the longitudinal suspense within a MEE round rod. Nilsson and Lindau provided the visual proof of the arrival concerning longitudinal waves within thin metal films. They recommended removal of anomalies between the ports concerning emaciated ([Formula: see text] Å) Ag layers preserved on amorphous silica because of being mildly [Formula: see text]-polarized at frequency tier according to the brawny plasma frequency. Anomalies have been due to confusion over the longitudinal waves mirrored utilizing the two borders. The wavelength is connected according to this wave’s property, which is an awful lot smaller than the wave over light. The wave is outstanding only for altogether superfine films. However, metallic movies defended amorphous substrates that are intermittent within the forward levels of their evolution. This made researchers aware of the possibility on getting ready altogether few layers with a desire to discussing the promulgation about longitudinal waves up to expectation colorful each among reverberation or transport on [Formula: see text]-polarized light. The mated options via the use of generalized Riccati equation mapping method or generalized Kudryashov technique show the rule and effectiveness regarding its methods then its ability because of applying one kind of forms over nonlinear incomplete differential equations.

Fractional derivative is nonlocal, which is more suitable to simulate physical phenomena and provides more accurate models of physical systems such as earthquake vibration and polymers. The present study suggested a new numerical approach for the fractional diffusion-wave equation (FDWE). The fractional-order derivative is in the Riemann-Liouville (R-L) sense. Discussed the theoretical analysis of stability, consistency, and convergence. The numerical examples demonstrate that the method is more workable and excellently holds the theoretical analysis, showing the scheme’s feasibility.

In this paper, we discuss the interaction of the rational function, hyperbolic as well as exponential function for the (3+1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation with the aid of Hirota’s bilinear approach. We find several families of lump-type solutions. This method is a powerful and advantageous mathematical tool for establishing abundant lump solutions of nonlinear partial differential equations. In order to illustrate their dynamic properties, some figures are plotted with determined parameters.

The accuracy of novel lump solutions of the potential form of the three&ndash, dimensional potential Yu&ndash, Toda&ndash, Sasa&ndash, Fukuyama (3-Dp-YTSF) equation is investigated. These solutions are obtained by employing the extended simplest equation (ESE) and modified Kudryashov (MKud) schemes to explore its lump and breather wave solutions that characterizes the dynamics of solitons and nonlinear waves in weakly dispersive media, plasma physics, and fluid dynamics. The accuracy of the obtained analytical solutions is investigated through the perspective of numerical and semi-analytical strategies (septic B-spline (SBS) and variational iteration (VI) techniques). Additionally, matching the analytical and numerical solutions is represented along with some distinct types of sketches. The superiority of the MKud is showed as the fourth research paper in our series that has been beginning by Mostafa M. A. Khater and Carlo Cattani with the title &ldquo, Accuracy of computational schemes&rdquo, The functioning of employed schemes appears their effectual and ability to apply to different nonlinear evolution equations.

Applying several latest theoretical techniques, the empirical description of the interaction between the high-frequency Langmuir and the low-frequent ion-acoustic waves, derived mathematically by Zakharov’s non-dimensional (ZE) equation. These interactions are described in electromagnetic waves, plasma physics, signal processing through optical fibers, coastal engineering, and fluid dynamics. Three modern computing methods are being used to construct several solutions: the extended exp( ϕ )-expansion process,the popular Kudryashov process, and the modified Khather method. Centered on the properties of the Hamilton system, the stability properties of the solutions are investigated. The physical proprieties are illustrated using 3D plots. The originality of the obtained solutions is explored. We demonstrate that we retrieve the old known solutions and we obtain new solutions that have never been found before.

In this paper, the Zakharov (Z.) equation in the dimensionless form is numerically investigated via (Cubic & Quantic & Septic) B-spline schemes to demonstrate the fidelity of the calculated computational solutions. The Z equation depicts the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves. This interaction is expounded in the prompts of the coastal engineering, electromagnetic field, signal handling in the optical fibres, plasma physics, and fluid dynamics. Three different computational schemes were applied to the Z equation for constructing many novel analytical solutions. In our paper, we try to check the accuracy of these solutions via the above-mentioned numerical schemes. Moreover, some separate sketches are given to indicate more physical features of this interaction. The originality of the obtained solutions is investigated by showing the similarities and differences between our obtained solutions and that was purchased in previously published papers.

In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model.

This research investigates the dynamical behavior of the transmission of nerve impulses of a nervous system (the neuron) by studying the computational solutions of the FitzHugh–Nagumo equation that is used as a model of the transmission of nerve impulses. For achieving our goal, we employ two recent computational schemes (the extended simplest equation method and Sinh–Cosh expansion method) to evaluate some novel computational solutions of these models. Moreover, we study the stability property of the obtained solutions to show the applicability of them in life. For more explanation of this transmission, some sketches are given for the analytical obtained solutions. A comparison between our results and that obtained in previous work is also represented and discussed in detail to show the novelty for our solutions. The performance of the two used methods shows power, practical and their ability to apply to other nonlinear partial differential equations.