Improved Bernoulli Sub-equation Function Method Research Articles

A study of the wave dynamics of the space–time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach

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.

Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics

Abstract In this article, we investigated the Landau–Ginzburg–Higgs (LGH) equation, focusing on the analysis of isolated soliton solutions and their stability. To compute the isolated soliton solutions, we used the advanced auxiliary equation (AAE) approach, which has proven to be a powerful and efficient method for extracting soliton solutions in various nonlinear partial differential equations (NLPDEs). We provided a detailed explanation, both graphically and physically, of the obtained soliton solutions in this article. Furthermore, we used the linear stability technique to conduct a stability analysis of the LGH equation. Additionally, we studied the bifurcation and stability of the equilibria and performed phase plane analysis of the model. We also provided a discussion on the comparisons between the AAE method and two other well-known approaches: the generalized Kudryashov method and the improved Bernoulli sub-equation function method. The application of the AAE approach in this study demonstrates its effectiveness and capability in analysing and extracting soliton solutions in NLPDEs.

Exact Solutions of Nonlinear Time Fractional Schrödinger Equation with Beta-Derivative

This article consists of Improved Bernoulli Sub-Equation Function Method (IBSEFM) to get the new solutions of nonlinear fractional Schrödinger equation described by beta-derivative. Foremost, it is dealt with derivative of Atangana. Secondly, basic properties of the IBSEFM are given. Finally, the proposed method has been applicated to the considered equation to get its new solutions. Moreover, the graphs of the obtained solutions are plotted via Mathematica. It is inferred from the results that IBSEFM is effectual technique for new solutions of nonlinear equations containing conformable derivatives.

Propagation of shock wave of the time‐fractional Gardner Burger equation in a multicomponent plasma using novel analytical method

The time‐fractional Gardner Burger (TFGB) equation is an efficient tool for studying nonlinear fluctuations of different types of the wave profiles, such as the gravity solitary waves in the ocean and the dust ion acoustic wave (DIAW) in a plasma environment. Here, to create an example for the existence of the classical Gardner Burger (GB) equation, a multicomponent plasma environment is considered and a classical GB equation has been derived using the basic governing equation by employing reductive perturbation technique (RPT). In addition, employing Agrawal's technique, the classical GB equation is transformed to the TFGB equation by applying the Riesz fractional derivative to the time‐fractional term. To solve TFGB equations, a new approach, utilizing the improved Bernoulli subequation function method (IBSEFM), is adopted. The physical structures of the solution are explored through some two‐ and three‐dimensional figures, and the significant impact of Burgers term and fractional order of the TFGB equation are demonstrated numerically. Finally, the wave dynamics of the DIAW in the parametric space for the present plasma environment are described from the numerical understanding.

New exact solutions of space-time fractional Schr¨odinger-Hirota equation

In this study, improved Bernoulli sub-equation function method (IBSEFM) is presented to construct the exact solutions of the nonlinear conformable fractional Schr¨odinger-Hirota equation (FSHE). By using the traveling wave transformation FSHE turns into the ordinary differential equation (ODE) and by the aid of symbolic calculation software, new exact solutions are obtained. 2D, 3D figures and contour surfaces acquired from the values of the solutions are plotted. The results show that the proposed method is powerful, effective and straightforward for formulating new solutions to various types of nonlinear fractional partial differential equations in applied sciences.

Analysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches

Abstract The variant Boussinesq equation has significant application in propagating long waves on the surface of the liquid layer under gravity action. In this article, the improved Bernoulli subequation function (IBSEF) method and the new auxiliary equation (NAE) technique are introduced to establish general solutions, some fundamental soliton solutions accessible in the literature, and some archetypal solitary wave solutions that are extracted from the broad-ranging solution to the variant Boussinesq wave equation. The established soliton solutions are knowledgeable and obtained as a combination of hyperbolic, exponential, rational, and trigonometric functions, and the physical significance of the attained solutions is speculated for the definite values of the included parameters by depicting the 3D profiles and interpreting the physical incidents. The wave profile represents different types of waves associated with the free parameters that are related to the wave number and velocity of the solutions. The obtained solutions and graphical representations visualize the dynamics of the phenomena and build up the mathematical foundation of the wave process in dissipative and dispersive media. It turns out that the IBSEF method and the NAE are powerful and might be used in further works to find novel solutions for other types of nonlinear evolution equations ascending in physical sciences and engineering.

Computational simulations; propagation behavior of the Riemann wave interacting with the long wave

The analytical solutions to the (2+1)-dimensional integrable Schwarz-Korteweg-de Vries (SKdV) problem are investigated in this article. To get solitary wave formulae such as complex, exponential, hyperbolic, and trigonometric function solutions, the modified Khater technique and the improved Bernoulli sub-equation function method (IBSEFM) are used. Furthermore, their stability characteristics are investigated. The solutions produced using the two methods are compared in order to uncover relationships between them and their features as indicated by the appropriate parameter values.

An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional equation

The nonlinear conformable time-fractional modified Camassa-Holm (MCH) equation plays an important role in physics. It is an interesting model to define change waves with weak nonlinearity. The aim of this study is to present the new exact solutions of conformable time-fractional MCH equation. For this purpose, an effective method which is the Improved Bernoulli Sub-Equation Function Method (IBSEFM) has been used. The 2D and 3D graphs and contour surfaces acquired from the values of the solutions are plotted by the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable time-fractional partial differential equations arising in mathematical physics.

О ТОЧНОМ РЕШЕНИИ СОГЛАСОВАННОГО РАВНОМОЩНОГО ВОЛНОВОГО УРАВНЕНИЯ С ПОМОЩЬЮ УСОВЕРШЕНСТВОВАННОГО ФУНКЦИОНАЛЬНОГО МЕТОДА ПОД-УРАВНЕНИЯ БЕРНУЛЛИ

In this paper, we consider conformable equal width wave (EW) equation in order to construct its exact solutions. This equation plays an important role in physics and gives an interesting model to define change waves with weak nonlinearity. The aim of this paper is to present new exact solutions to conformable EW equation. For this purpose, we use an effective method called Improved Bernoulli Sub-Equation Function Method (IBSEFM). Based on the values of the solutions, the 2D and 3D graphs and contour surfaces are plotted with the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable partial equations arising in mathematical physics.

On an application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional regularized long wave equation

On an application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional regularized long wave equation

Search for interactions of phenomena described by the coupled Higgs field equation through analytical solutions

The coupled Higgs field equation is significant for describing a system of conserved scalar nucleons interacting with the neutral scalar meson. The improved Bernoulli sub-equation function method and the sine–Gordon expansion method have been put in used in the current article, to interpret the dynamism of the particles through analytical solutions modelled by the coupled Higgs field equation. The interaction among the electrons and mesons has been found out and it is observed that the particle moves along the smooth path from one state to another. It is also vital to mark that the wave profile of the waves are velocity dependent and the incident is shown in the figures. Different types of soliton shape like, bell shape soliton, kink soliton, consolidated bell shape; anti-bell shape soliton, compacton, singular soliton, flat kink soliton etc. are ascertained by assigning individual quantities of the unidentified parameters whose propagations are portrayed in 3D and 2D surfaces. The solutions established in this study are further comprehensive, broad–ranging and some are new which have not been found in earlier research. The executed schemes are effective, compatible to computer algebra and deliver ample wide–ranging wave solutions.

New exact solutions for the doubly dispersive equation using the improved Bernoulli sub-equation function method

In this study, the doubly dispersive equation is presented by the application of the improved Bernoulli sub-equation function method (IBSEFM). The doubly dispersive equation which is a nonlinear partial differential equation is transformed into nonlinear ordinary differential equation using a wave transformation and then is solved by IBSEFM. Some new solutions are successfully constructed. All the obtained solutions in this study have been satisfied the doubly dispersive equation. In the present study, we have used Wolfram Mathematica 9 software for all of the computations and graphic plottings.

On the complex solutions to the (3+1)-dimensional conformable fractional modified KdV–Zakharov–Kuznetsov equation

This paper presents some exponential function solutions of the (3[Formula: see text]+[Formula: see text]1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK). The improved Bernoulli sub-equation function method (IBSEFM) is used in carrying out this computation. Choosing the suitable parameters, the 2D and 3D surfaces of solutions are plotted. The constructed results may be useful in explaining the physical meaning of the studied model.

Stable wave solutions to the Landau-Ginzburg-Higgs equation and the modified equal width wave equation using the IBSEF method

The Landau-Ginzburg-Higgs equation and the modified equal width wave equation (MEWE) underscore to describe superconductivity and unidirectional wave propagation in nonlinear media with dispersion systems. In the present study, the improved Bernoulli sub-equation function method (IBSEFM) has been introduced to accomplish applicable soliton solutions to the above stated wave equations. We ascertain adequate soliton solutions, videlicet the combination of hyperbolic function, exponential function etc. and speculate the physical significance of the obtained solutions for the definite values of the included parameters through depicting graphs and interpreted the physical phenomena. It is shown that the IBSEFM is powerful, suitable, direct and provide general wave solutions to NLEEs in mathematical physics.

An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation

The nonlinear conformable time-fractional Symmetric Regularized Long Wave (SRLW) equation plays an important role in physics. This equation is an interesting model to describe ion-acoustic and space change waves with weak nonlinearity. In this paper, we solve the SRLW equation via the Improved Bernoulli Sub-Equation Function Method (IBSEFM). New exact solutions are constructed and by the aid of software mathematics programme, 2D and 3D graphs of the solutions according to the parameters are plotted. The results show that IBSEFM is a powerful mathematical tool to solve nonlinear conformable time-fractional equations arising in mathematical physics.

New Exact Solutions for Generalized (3+1) Shallow Water-Like (SWL) Equation

Abstract In this study, we use the improved Bernoulli sub-equation function method for exact solutions to the generalized (3+1) shallow water-like (SWL) equation. Some new solutions are successfully constructed. We carried out all the computations and the graphics plot in this paper by Wolfram Mathematica.

Bir sinirsel iletişim modelinin yeni salınımlı dalga çözümleri üzerinde

The aim of this study is to present some new travelling wave solutions of conformable time-fractional Fitzhugh–Nagumo equation that model the transmission of nerve impulses. For this purpose, the improved Bernoulli sub-equation function method has been used. The obtained results are shown by way of the the 3D-2D graphs and contour surfaces for the suitable values.

New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model

The Konopelchenko-Dubrovsky (KD) system is presented by the application of the improved Bernoulli subequation function method (IBSEFM). First, The KD system being Nonlinear partial differential equations system is transformed into nonlinear ordinary differential equation by using a wave transformation. Last, the resulting equation is successfully explored for new explicit exact solutions including singular soliton, kink, and periodic wave solutions. All the obtained solutions in this study satisfy the Konopelchenko-Dubrovsky model. Under suitable choice of the parameter values, interesting two- and three-dimensional graphs of all the obtained solutions are plotted.

Regarding some novel exponential travelling wave solutions to the Wu–Zhang system arising in nonlinear water wave model

In this paper, our main aim is to find the new travelling wave solutions to the Wu–Zhang system which describes dispersive long waves. Some new and important travelling wave solutions with rational and exponential function structures are successfully constructed by using the improved Bernoulli subequation function method. All the obtained solutions in this study satisfy the Wu–Zhang system. The interesting three- and two-dimensional surfaces to all the obtained solutions are plotted. We carried out all the computations and the graphics plots in this paper with aid of the Wolfram Mathematica 9.

The new complex rational function prototype structures for the nonlinear Schrödinger‐inviscid burgers system

In this paper, a system representing the coupling between the nonlinear Schrödinger equation and the inviscid burgers equation in modeling the interactions between short and long waves in fluids has been investigated. The new algorithms, like improved tan(Φ(ξ)/2)‐expansion method and improved Bernoulli subequation function method, have been proposed. The proposed methods have been discussed comprehensively in this article. Using these methods, some new prototype for exact solutions such as complex exponential, complex hyperbolic, and complex trigonometric function solutions have been obtained for nonlinear Schrödinger‐inviscid burgers system. Through the present analysis, it has been established that, with the help of symbolic computation, these methods provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations.