Riccati Bernoulli sub-ODE Method Research Articles

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps.

Dynamics of the Traveling Wave Solutions of Fractional Date–Jimbo–Kashiwara–Miwa Equation via Riccati–Bernoulli Sub-ODE Method through Bäcklund Transformation

Dynamics of the Traveling Wave Solutions of Fractional Date–Jimbo–Kashiwara–Miwa Equation via Riccati–Bernoulli Sub-ODE Method through Bäcklund Transformation

Novel exact traveling wave solutions of the space-time fractional Sharma Tasso-Olver equation via three reliable methods

The dominant intention of this article is to extract the new exact traveling waves solutions of the nonlinear space-time fractional Sharma-Tasso-Olver equation in the sense of beta-derivative by using three integration schemes namely, Riccati-Bernoulli (RB) sub-ODE method, Generalized Bernoulli (GB) sub-ODE method and Generalized tanh (GT) method. By the virtue of employed techniques, different types of solutions are obtained in the form of trigonometric, hyperbolic, and exponential functions respectively. The obtained solutions are also verified for the aforesaid equation through symbolic soft computations. To promote the vital propagated features; some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the specific values to the parameters under the confined conditions. Further, based on the bifurcation theory, we examine the phase portrait of the proposed nonlinear equation. Furthermore, we ensure that all the solutions are innovative and have remarkable impacts on the prevailing solitary wave theory literature.

Wave solutions to the Landau-Ginzburg-Higgs equation and the modified KdV-Zakharov-Kuznetsov equation by the Riccati-Bernoulli sub-ODE method

Wave solutions to the Landau-Ginzburg-Higgs equation and the modified KdV-Zakharov-Kuznetsov equation by the Riccati-Bernoulli sub-ODE method

Riccati–Bernoulli sub-ode method-based exact solution of new coupled Konno–Oono equation

Nonlinear partial differential equations (NLPDEs) have emerged as a major focus of study in a variety of nonlinear science disciplines. This is because in general, mathematical, physics and engineering problems may be stated by NLPDEs. It may be noted that a specific kind of exact/analytical solution called traveling wave solutions for NLPDEs has great significance. In this regard, this paper addresses the traveling wave solution to the new coupled Konno–Oono equation (NCKOE), a set of NLPDE. A traveling wave transformation and the Riccati–Bernoulli sub-ode method (RBSOM) are used here to retrieve the exact/analytical traveling wave solution of the NCKOE. By using the above-mentioned approach, NLPDEs can be updated toward a collection of algebraic equations. Further, different types of solitary wave solitons, Trigonometric function solutions, Hyperbolic soliton solutions and dark bright solitons to the NCKOE have been successfully retrieved. Importantly, the newly derived solutions adhere to the main equation when they are inserted into the governing equations. Furthermore, through an appropriate selection of parameters, three-dimensional and two-dimensional figures are presented for physical illustration of the obtained solutions. These visual representations serve to showcase the effectiveness, conciseness and efficiency of the applied techniques.

Analytical solutions to time-space fractional Kuramoto-Sivashinsky Model using the integrated Bäcklund transformation and Riccati-Bernoulli sub-ODE method

<abstract><p>This paper solves an example of a time-space fractional Kuramoto-Sivashinsky (KS) equation using the integrated Bäcklund transformation and the Riccati-Bernoulli sub-ODE method. A specific version of the KS equation with power nonlinearity of a given degree is examined. Using symbolic computation, we find new analytical solutions to the current problem for modeling many nonlinear phenomena that are described by this equation, like how the flame front moves back and forth, how fluids move down a vertical wall, or how chemical reactions happen in a uniform medium while they oscillate uniformly across space. In the field of mathematical physics, the Riccati-Bernoulli sub-ODE approach is shown to be a valuable tool for producing a variety of single solutions.</p></abstract>

Exact solutions to the fractional nonlinear phenomena in fluid dynamics via the Riccati-Bernoulli sub-ODE method

<p>The Riccati-Bernoulli sub-ODE method has been used in recent research to efficiently investigate the analytical solutions of a non-linear equation widely used in fluid dynamics research. By utilizing this method, exact solutions are obtained for the space-time fractional symmetric regularized long-wave equation. These results comprehensively understand the long wave equation widely used in numerous fluid dynamics and wave propagation scenarios. The approach to studying these phenomena and using conceptual representation to understand their essential characteristics opens the door to valuable insights that may help improve both the theoretical and applied aspects of fluid dynamics and similar fields. Thus, as these complex equations demonstrate, the suggested approach is a valuable tool for conducting further research into non-linear phenomena across several disciplines.</p>

Lump and kink soliton phenomena of Vakhnenko equation

Understanding natural processes often involves intricate nonlinear dynamics. Nonlinear evolution equations are crucial for examining the behavior and possible solutions of specific nonlinear systems. The Vakhnenko equation is a typical example, considering that this equation demonstrates kink and lump soliton solutions. These solitons are possible waves with several intriguing features and have been characterized in other naturalistic nonlinear systems. The solution of nonlinear equations demands advanced analytical techniques. This work ultimately sought to find and study the kink and lump soliton solutions using the Riccati–Bernoulli sub-ode method for the Vakhnenko equation (VE). The results obtained in this work are lump and kink soliton solutions presented in hyperbolic trigonometric and rational functions. This work reveals the effectiveness and future of our method for solving complex solitary wave problems.

Kink phenomena of the time-space fractional Oskolkov equation

<p>In this study, we applied the Riccati-Bernoulli sub-ODE method and Bäcklund transformation to analyze the time-space fractional Oskolkov equation for kink solutions by matching the coefficients and optimal series parameters. The time-space fractional Oskolkov equation is used to analyze the behavior of solitons for different applications such as fluid dynamics and viscoelastic flow. The kink solutions derived have important consequences for stability analysis and interaction dynamic in these systems, and these are useful in controlling the physical behaviour of systems described by this equation. Such effects are illustrated by 2D and 3D plots, showing that the proposed model can handle both fractional and integer-order solitons with different but equally efficient outcomes. This research contributes to a valuable analytical method that can determine and manage processes in diversified systems based on fractional differential equations. This work provides a basis for subsequent analysis in other branches of science and technology in which the fractional Oskolkov model is used.</p>

Perturbed Gerdjikov–Ivanov equation: Soliton solutions via Backlund transformation

The perturbed Gerdjikov–Ivanov (pGI) equation, a mathematical model that depicts the behaviour of optical pulses during propagation while accounting for perturbation influences, has been thoroughly studied by us. This equation has significant applications in the field of optical fibres, notably for photonic crystal fibres. In this paper, we combined the Riccati-Bernoulli sub-ODE method with the Backlund transformation to construct travelling wave solutions for the perturbed Gerdjikov–Ivanov (pGI) equation. This technique offered us families of solutions as well as a fresh way of dealing with nonlinear models. The results that emerge could greatly advance our knowledge of the physical consequences that are present in the nonlinear model that we are studying. The resulting solutions broaden the scope of earlier findings made using various approaches by including solitons, trigonometric functions, and rational expressions. The Backlund transformation used in this study is distinguished by its simplicity and brevity, and it produces results that are noticeably more thorough than those often obtained by alternative methods. Additionally, we create graphical representations of our results using the computational capabilities of the Maple software. In order to do this, the proper parameter values must be chosen, allowing for a visual study of the results. Through the display of two-dimensional graphical depictions, these effects have been clearly explained, permitting a thorough understanding of their physical implications.

The dynamical study of fractional complex coupled maccari system in nonlinear optics via two analytical approaches

In this work, the modified auxiliary equation method (MAEM) and the Riccati–Bernoulli sub-ODE method (RBM) are used to investigate the soliton solutions of the fractional complex coupled maccari system (FCCMS). Nonlinear partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilizing a travelling wave transformation, the MAEM, and the RBM. As a result, solutions to hyperbolic, trigonometric and rational functions with unconstrained parameters are obtained. The travelling wave solutions can also be used to generate the solitary wave solutions when the parameters are given particular values. There are several solutions that are modelled for different parameter combinations. We have developed a number of novel solutions, such as the kink, periodic, M-waved, W-shaped, bright soliton, dark soliton, and singular soliton solution. We simulate our figures in Mathematica and provide many 2D and 3D graphs to show how the beta derivative, M-truncated derivative and conformable derivative impacts the analytical solutions of the FCCMS.The results show how effectively the MAEM and RBM work together to extract solitons for fractional-order nonlinear evolution equations in science, technology, and engineering.

Solution Structures of an Electrical Transmission Line Model with Bifurcation and Chaos in Hamiltonian Dynamics

Employing the Riccati–Bernoulli sub-ODE method (RBSM) and improved Weierstrass elliptic function method, we handle the proposed [Formula: see text]-dimensional nonlinear fractional electrical transmission line equation (NFETLE) system in this paper. An infinite sequence of solutions and Weierstrass elliptic function solutions may be obtained through solving the NFETLE. These new exact and solitary wave solutions are derived in the forms of trigonometric function, Weierstrass elliptic function and hyperbolic function. The graphs of soliton solutions of the NFETLE describe the dynamics of the solutions in the figures. We also discuss elaborately the effects of fraction and arbitrary parameters on a part of obtained soliton solutions which are presented graphically. Moreover, we also draw meaningful conclusions via a comparison of our partially explored areas with other different fractional derivatives. From our perspectives, by rewriting the equation as Hamiltonian system, we study the phase portrait and bifurcation of the system about NFETLE and we also for the first time discuss sensitivity of the system and chaotic behaviors. To our best knowledge, we discover a variety of new solutions that have not been reported in existing articles [Formula: see text], [Formula: see text]. The most important thing is that there are iterative ideas in finding the solution process, which have not been seen before from relevant articles such as [Tala-Tebue et al., 2014; Fendzi-Donfack et al., 2018; Ashraf et al., 2022; Ndzana et al., 2022; Halidou et al., 2022] in seeking for exact solutions about NFETLE.

New diverse soliton solutions for the coupled Konno-Oono equations

The main aim of this article is to establish new impressive diverse soliton solutions to the nonlinear Coupled Konno-Oono Model (NCKOM) that represents current-field string interact with an external magnetic field. The achieved soliton solutions will give stretch study for this model and all related phenomena’s. Three different schemes have been called for this purpose. The first one is the extended direct algebraic method (EDAM), while the second is the Paul-Painlevé approach method (PPAM) and the third one is the Riccati-Bernoulli Sub-ODE method (RBSODM). Brief comparisons between our results and that achieved previously have been listed.

Diverse Precise Traveling Wave Solutions Possessing Beta Derivative of the Fractional Differential Equations Arising in Mathematical Physics

In this paper, we obtain the novel exact traveling wave solutions in the form of trigonometric, hyperbolic and exponential functions for the nonlinear time fractional generalized reaction Duffing model and density dependent fractional diffusion-reaction equation in the sense of beta-derivative by using three fertile methods, namely, Generalized tanh (GT) method, Generalized Bernoulli (GB) sub-ODE method, and Riccati-Bernoulli (RB) sub-ODE method. The derived solutions to the aforementioned equations are validated through symbolic soft computations. To promote the vital propagated features; some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the specific values to the parameters under the confine conditions. The accomplished solutions show that the presented methods are not only powerful mathematical tools for generating more solutions of nonlinear time fractional partial differential equations but also can be applied to nonlinear space-time fractional partial differential equations.

Effects of the Wiener Process on the Solutions of the Stochastic Fractional Zakharov System

We consider in this article the stochastic fractional Zakharov system derived by the multiplicative Wiener process in the Stratonovich sense. We utilize two distinct methods, the Riccati–Bernoulli sub-ODE method and Jacobi elliptic function method, to obtain new rational, trigonometric, hyperbolic, and elliptic stochastic solutions. The acquired solutions are helpful in explaining certain fascinating physical phenomena due to the importance of the Zakharov system in the theory of turbulence for plasma waves. In order to show the influence of the multiplicative Wiener process on the exact solutions of the Zakharov system, we employ the MATLAB tools to plot our figures to introduce a number of 2D and 3D graphs. We establish that the multiplicative Wiener process stabilizes the solutions of the Zakharov system around zero.

Dark-soliton behaviors arising from a coupled nonlinear Schrödinger system

The principal objective of this work focuses on achieving a class of advanced and impressive N-dark–dark solitons estimations to the coupled nonlinear Schrödinger system (CNLSS) that represents the propagation of optical pluses in the model-locked fiber lasers. The achieved results will perfectly describe different nonlinear coherent structures for the behavior of the resultant solitons to this model as well as ottosecond pulses in optical fibers that play fundamental rule for all recent telecommunications. The N-dark-dark solitons arise when nonlinearities are all removed, or when there is mixture between both focusing and defocusing nonlinearities. If various focusing and defocusing solitons components are collide with each other, the energies will emerge. The N–dark-dark solitons for this model will be established by using five significant schemas that were checked before to extracting the bright, dark and other types of solitons for many nonlinear problems that arise in many branches of science. The proposed methods that are bided for this goal are the extended direct algebraic method (EDAM), the (G′G) -expansion method, the extended simple equation method (ESEM), the Paul-Painleve approach method (PPAM) and the Riccati-Bernoulli Sub-ODE method (RBSODM). The suggested methods have been checked before for many other NLPDE and continuously achieve distinct results. The achieved results will add future studies for all recent applications of the internet communications.

Brownian motion effects on analytical solutions of a fractional-space long–short-wave interaction with conformable derivative

In this article, we consider the stochastic fractional-space long–short-wave interaction system (SFS-LSWIs) forced by multiplicative Brownian motion. To obtain a new exact stochastic fractional-space solutions, we apply two different methods such as sin–cos method and the Riccati–Bernoulli sub-ODE method. These solutions are essential for explaining some difficult and complicated physical phenomena. Because this system has never been investigated using a combination of multiplicative noise and fractional space, we will generalize certain previously obtained results as special cases. Furthermore, we use Matlab to plot 3D surfaces of analytical solutions produced in this study to show the impact of Brownian motion on the analytical solutions of the SFS-LSWIs.

New diverse variety for the exact solutions to Keller-Segel-Fisher system

In our current paper, we will extract new unexpected diverse variety of the exact solutions to the Keller-Segel -Fisher system (KSFS) which is a famous mathematical biological model that governs the mechanism of bacteria to discovery food and gets rid of venoms. The suggested model plays a vital rule for health of humans, animals as well as all other organisms. There are three famous different techniques are introduced for extract and document these new unexpected behaviors of the exact solutions are the (G′G) -expansion method, the extended simple equation method (ESEM) and the Riccati-Bernoulli Sub-ODE method (RBSODM). We will implement these three proposed techniques in the same time and in arranged sequence. Furthermore, we will document and list a comparison between our new achieved exact solutions of this model each with each other as well as the previously documented solutions by other authors who studied this model before.

Impressive and accurate solutions to the generalized Fokas-Lenells model

In this article, we utilize the generalized full nonlinearity perturbed complex Fokas-Lenells model (GFLM) which is a general dynamics representation of modern electronic communications "Internet blogs, Facebook communication and Twitter comments". The modified simple equation method (MSEM) has been applied effectively to generate closed form solution. On the other hand, the Riccati-Bernoulli Sub-ODE method (RPSOM) which reduces the steps of calculation has been applied perfectly to achieve accurate solution to this equation. We established the solutions achieved by these distinct manners in same vein and parallel.

Traveling wave structures of some fourth-order nonlinear partial differential equations

This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method. In this method, utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation, the fourth-order equation can be transformed into a set of algebraic equations. Solving the set of algebraic equations, we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper. The physical interpretation of the nonlinear models are also detailed through the exact solutions, which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations. Finally, 3D graphs of some derived solutions in this paper are depicted through suitable parameter values.