Sharma Tasso Olver Research Articles

Kink phenomena of the time-space fractional Sharma-Tasso-Olver (STO) equation

Abstract This paper aims to obtain exact solutions of solitary waves for the conformable fractional Sharma-Tasso-Olver (STO) equation which plays an important role in nuclear physics to describe the physical occurrences such as the fission and fusion processes. Solitary waves operate central parts in different areas of study such as electromagnetism, atomic quantum theory, as well as special relativity. By means of sub-ode approach with the aid of the modified fractional Riccati-Bernoulli equation, the exact forms of generalized solitary solution of the fractional (STO) equation are found and specified in hyperbolic, trigonometric, and rational functions. This makes the visualization of the fractional effects and the dynamic behaviors of these solutions in 3D and 2D help in establishing practicality for application of the results. The novel analytical results benefit general engineering and mathematical physics in demonstrating that the proposed employment of the given technique allows solving nonlinear problem analytically. These findings are significant for the progress of wave proceedings in the number of applications.

Exact Solutions for the Sharma–Tasso–Olver Equation via the Sardar Subequation Method with a Comparison between Atangana Space–Time Beta-Derivatives and Classical Derivatives

The Sharma–Tasso–Olver (STO) equation is a nonlinear, double-dispersive, partial differential equation that is physically important because it provides insights into the behavior of nonlinear waves and solitons in various physical areas, including fluid dynamics, optical fibers, and plasma physics. In this paper, the STO equation is generalized to a fractional equation by using Atangana (or Atangana–Baleanu) fractional space and time beta-derivatives since they have been found to be useful as a model for a variety of traveling-wave phenomena. Exact solutions are obtained for the integer-order and fractional-order equations by using the Sardar subequation method and an appropriate traveling-wave transformation. The exact solutions are obtained in terms of generalized trigonometric and hyperbolic functions. The exact solutions are derived for the integer-order STO and for a range of values of fractional orders. Numerical solutions are also obtained for a range of parameter values for both the fractional and integer orders to show some of the types of solutions that can occur. As examples, the solutions are obtained showing the physical behavior, such as the solitary wave solutions of the singular kink-type and periodic wave solutions. The results show that the Sardar subequation method provides a straightforward and efficient method for deriving new exact solutions for fractional nonlinear partial differential equations of the STO type.

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.

Stability analysis, lump and exact solutions to Sharma–Tasso–Olver–Burgers equation

Stability analysis, lump and exact solutions to Sharma–Tasso–Olver–Burgers equation

Theoretical examination of solitary waves for Sharma–Tasso–Olver Burger equation by stability and sensitivity analysis

Theoretical examination of solitary waves for Sharma–Tasso–Olver Burger equation by stability and sensitivity analysis

Soliton solutions of the time-fractional Sharma–Tasso–Olver equations arise in nonlinear optics

Soliton solutions of the time-fractional Sharma–Tasso–Olver equations arise in nonlinear optics

Analysis of the dynamical perspective of chaos, Lie symmetry, and soliton solution to the Sharma–Tasso–Olver system

Analysis of the dynamical perspective of chaos, Lie symmetry, and soliton solution to the Sharma–Tasso–Olver system

The dynamical structures of the Sharma–Tasso–Olver model in doubly dispersive medium

In this article, the Sharma–Tasso–Olver model is investigated which characterizes the nonlinear double-dispersive evolution dispersive waves’ dynamical propagation in heterogeneous mediums. To study the dynamical nature of moving waves with a periodic and isolated nature analytically, the governing model is converted into an ordinary differential equation via a wave transformation to employ the extended modified auxiliary equation mapping approach and the G′G2 expansion method. In order to validate the computations, the stability of the obtained results is also established. It has been found that the model supports nonlinear solitary waves, periodic waves, shock waves and stable oscillatory waves. For a set of appropriate parameters, Mathematica is used to represent the dynamics of various wave structures as 3D, 2D and contour visualizations. Additionally, we may remark that the results discussed here are novel and innovative.

Utilizing the extended tanh-function technique to scrutinize fractional order nonlinear partial differential equations

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

New analytical technique to solve fractional-order Sharma–Tasso–Olver differential equation using Caputo and Atangana–Baleanu derivative operators

Abstract The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method’s distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM’s approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems.

Numerical solutions to the Sharma–Tasso–Olver equation using Lattice Boltzmann method

AbstractA novel numerical scheme depending on Lattice Boltzmann method is proposed to solve the Sharma–Tasso–Olver equation in one, two and three dimension. The local equilibrium distribution functions and modified functions are obtained via Taylor expansion and Chapman–Enskog multiscale expansion techniques. The macro equation is recovered accurately based on the above functions, and the stability conditions of the equations are deduced. In addition, through simulating numerically some of the initial‐boundary value problems of multidimensional STO equations, the results demonstrate that the numerical solutions and the exact solutions tend to be identical. It also indicates that the model is feasible within a specific range and provides a reliable the approach for solving the multidimensional STO equations.

Variable coefficient exact solution of Sharma–Tasso–Olver model by enhanced modified simple equation method

This article presents variable coefficients solution in the form of soliton, bell shape solution, periodic solution, parabolic periodic soliton of Sharma–Tasso–Olver equation which fundamentally apply to the fission and fusion mechanism of particle in nuclear physics. Solitary wave takes the leading position in electromagnetic field, atomic quantum theory, theoretical relativistic relation etc. The enhanced modified simple equation (EMSE) method is used to generalize solitary wave solutions for Sharma–Tasso–Olver (STO) equation. The computational software maple 18 is used to derive the corresponding results in the 3D plot, density plot and counter plots for the different types of time function used in EMSE scheme.

New structures for exact solution of nonlinear fractional Sharma–Tasso–Olver equation by conformable fractional derivative

The Atangana–Baleanu conformable differential operator has been employed in this study to solve the conformable fractional Sharma–Tasso–Olever equation. The new extended direct algebraic method has then been employed in order to obtain the precise solutions. The results are obtained as hyperbolic, trigonometric, and rational solutions. To see the fractional effects and dynamical behavior, graphic visualization has been demonstrated in 3D, contour, and 2D plots. The graphical representation of these data is very helpful in identifying the equation’s true physical significance. The acquired results are brand-new and more broadly applicable, and they show the value of the advised strategy for the analytical handling of nonlinear problems in mathematical physics and engineering. They are helpful in several circumstances for a better understanding of the dynamics of waves that are propagating.

The study of nonlinear dispersive wave propagation pattern to Sharma–Tasso–Olver–Burgers equation

This paper discusses the wave propagation to the nonlinear Sharma–Tasso–Olver–Burgers (STOB) equation which is used as the governing model in different fields. Natural phenomena are typically complex and nonlinear, defying simple linear superposition. Researchers have been studying a wide range of natural phenomena in depth, and nonlinear science has gradually become a part of people’s consciousness. One of the most significant research questions in nonlinear science centers around the nonlinear evolution equation and its precise solution. We have secured different shapes of the solitary wave solutions including kink-type, shock-type and combined solitary wave solutions with the assistance of recently developed integration tool, namely the new extended direct algebraic method (NEDAM). Additionally, the solutions for the hyperbolic, exponential and trigonometric functions are retrieved. Moreover, based on a comparison of our results to those that are well known, the study indicates that our solutions are innovative. Using proper parameters in numerical simulations and physical explanations, it is possible to demonstrate the significance of the results. The results of this research can improve the nonlinear dynamic behavior of a system and indicate that the methodology employed is adequate. It is proposed that the offered method can be utilized to support nonlinear dynamical models applicable to a wide variety of physical situations. We hope that a wide spectrum of engineering model professionals will find this study to be beneficial.

On dynamical behavior for approximate solutions sustained by nonlinear fractional damped Burger and Sharma–Tasso–Olver equation

In this paper, we study a nonlinear fractional Damped Burger and Sharma–Tasso–Olver equation using a new novel technique, called homotopy perturbation transform method (FHPTM). There are three examples used to demonstrate and validate the proposed algorithm’s efficiency. This nonlinear model depicts nonlinear wave processes in fluid dynamics, ecology, solid-state physics, shallow-water wave propagation, optical fibers, fluid mechanics, plasma physics, and other applied science, engineering, and mathematical physics disciplines, as well as other phenomena. Numerous algebraic properties of the fractional derivative Caputo–Fabrizio operator are illustrated concerning the Laplace transformation to demonstrate their utility. Different graphs and tables compare the results obtained by R. Nawaz et al. [Alex. Eng. J. 60, 3205 (2021)] and M. S. Rawashdeh [Appl. Math. Inform. Sci. 9, 1239 (2015)]. The proposed scheme accelerates the convergence of the series solution and guarantees the convergence associated with the homotopy parameter. Furthermore, the physical nature of various fractional orders has been captured in plots. The obtained results demonstrate that the employed solution procedure is dependable and methodical in investigating the behaviors of nonlinear models of both integer and fractional orders.

Riemann–Hilbert approach of the complex Sharma–Tasso–Olver equation and its N-soliton solutions

We study the complex Sharma–Tasso–Olver equation using the Riemann–Hilbert approach. The associated Riemann–Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair. Subsequently, in the case that the Riemann–Hilbert problem is irregular, the N-soliton solutions of the equation can be deduced. In addition, the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.

Dynamics and Exact Traveling Wave Solutions of the Sharma–Tasso–Olver–Burgers Equation

In this paper, to study the Sharma–Tasso–Olver–Burgers equation, we focus on the geometric properties and the exact traveling wave solutions. The corresponding traveling system is a cubic oscillator with damping, and it has time-dependent and time-independent first integral. For all bounded orbits of the traveling system, we give the exact explicit kink wave solutions.

The Caputo–Fabrizio time-fractional Sharma–Tasso–Olver–Burgers equation and its valid approximations

Studying the dynamics of solitons in nonlinear time-fractional partial differential equations has received substantial attention, in the last decades. The main aim of the current investigation is to consider the time-fractional Sharma–Tasso–Olver–Burgers (STOB) equation in the Caputo–Fabrizio (CF) context and obtain its valid approximations through adopting a mixed approach composed of the homotopy analysis method (HAM) and the Laplace transform. The existence and uniqueness of the solution of the time-fractional STOB equation in the CF context are investigated by demonstrating the Lipschitz condition for as the kernel and giving some theorems. To illustrate the CF operator effect on the dynamics of the obtained solitons, several two- and three-dimensional plots are formally considered. It is shown that the mixed approach is capable of producing valid approximations to the time-fractional STOB equation in the CF context.

A FRACTAL MODIFICATION OF THE SHARMA–TASSO–OLVER EQUATION AND ITS FRACTAL GENERALIZED VARIATIONAL PRINCIPLE

The Sharma–Tasso–Olver equation can well describe the wave motion in physics, however, it becomes ineffective when the boundary is non-smooth, so a modification of the equation is urgently needed. In this study, we derive a new fractal Sharma–Tasso–Olver equation that can model the wave motion with the non-smooth boundary by applying He’s fractal derivative. By means of the semi-inverse method, we successfully establish its fractal generalized variational principle, which provides the conservation laws in an energy form in the fractal space and reveals the possible solution structures of the equation. The obtained generalized variational principle can be used for the numerical and analytical studies of the solitary wave properties in the fractal PDEs.

Traveling Wave Solution for Sharma–Tasso–Olver–Burgers（STOB）Equation by the (G’/G)-Expansion Method

In this paper, we use the (G'/G)-expansion method to construct the traveling wave solution of the Sharma–Tasso–Olver–Burgers (STOB) equation based on the idea of homogeneous equilibrium. At the same time, it is verified that the (G'/G) -expansion method has wider applicability for dealing with nonlinear evolution equations.