Sokolov Wilson System Research Articles

Structure of Analytical and Numerical Wave Solutions for the Nonlinear (1 + 1)-Coupled Drinfel’d–Sokolov–Wilson System Arising in Shallow Water Waves

In this article, we successfully obtain novel solutions for the coupled Drinfel’d–Sokolov–Wilson DSW system utilizing various methods. These include soliton solutions characterized by hyperbolic, rational, and trigonometric functions. Specifically, the generalized exponential rational function method (GERFM) and a modified version of the new Kudryashov method (MVNK) are employed to derive diverse soliton solutions for the system. Additionally, we demonstrate numerical solutions for the coupled Drinfel’d–Sokolov–Wilson system using adaptive moving mesh and uniform mesh methods. Also, we study the stability and error analysis of the numerical schemes. To validate the accuracy and reliability of the exact solutions obtained through analytical methods, we compare them with the numerical solutions both analytically and graphically. The techniques presented in this article are deemed suitable and acceptable and can be effectively applied to solve other nonlinear evolution systems.

A Comparative Study of Time Fractional Nonlinear Drinfeld–Sokolov–Wilson System via Modified Auxiliary Equation Method

The time-fractional nonlinear Drinfeld–Sokolov–Wilson system, which has significance in the study of traveling waves, shallow water waves, water dispersion, and fluid mechanics, is examined in the presented work. Analytic exact solutions of the system are produced using the modified auxiliary equation method. The fractional implications on the model are examined under β-fractional derivative and a new fractional local derivative. Extracted solutions include rational, trigonometric, and hyperbolic functions with dark, periodic, and kink solitons. Additionally, by specifying values for fractional parameters, graphs are utilized to comprehend the fractional effects on the obtained solutions.

Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves

We adopted the Lie symmetry technique to obtain some new exact waveform solutions for the coupled Drinfeld Sokolov Wilson (DSW) system. This system emerges from phenomena such as water waves, theoretical physics, fluid dynamics, biology, and chemical sciences. We use the Jacobi elliptic function (JEF) method to deal with some of the reduced systems during the symmetry reductions, which reveals rational, exponential, and hyperbolic functions based waveform solutions. The properties of these solutions indicate that these are periodic waves, solitary waves, double periodic waves, and shock wave solutions. Several of the answers that have been found are entirely new and might be important for researchers to recognize. By assigning specific values to parameters, some chosen solutions are graphically validated. This method can be applied more broadly to handle several different types of nonlinear evolution systems.

Persistence of Kink and Periodic Waves to Singularly Perturbed Two-Component Drinfel’d–Sokolov–Wilson System

This paper concerns the persistence of kink and periodic waves to singularly perturbed two-component Drinfel’d-Sokolov-Wilson system. Geometric singular perturbation theory is first employed to reduce the high-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we then are able to find the sufficient conditions about the wave speed to guarantee the existence of heteroclinic orbit and periodic orbits, which indicates the existence of kink and periodic waves. Furthermore, we also show that the limit wave speed c_0(k) is increasing.

Investigating the Dynamics of Time-Fractional Drinfeld–Sokolov–Wilson System through Analytical Solutions

This study addresses a nonlinear fractional Drinfeld–Sokolov–Wilson problem in dispersive water waves, which requires appropriate numerical techniques to obtain an approximative solution. The Adomian decomposition approach, the homotopy perturbation method, and Sumudu transform are combined to tackle the problem. The Caputo manner is used to describe fractional derivative, and He’s polynomials and Adomian polynomials are employed to address nonlinearity. By following these approaches, we obtain solutions in the form of convergent series. We verify and demonstrate the effectiveness of our suggested strategies by examining the assumed model in terms of fractional order. We use plots for various fractional orders to represent the physical behavior of the suggested technique solutions, and show a numerical simulation. The results demonstrate that the suggested algorithms are systematic, simple to use, effective, and accurate in analyzing the behavior of coupled nonlinear differential equations of fractional order in related scientific and engineering fields.

Single- and Double-Peak Solitary Waves of Two-Component Drinfel’d–Sokolov–Wilson System with Kuramoto–Sivashinsky Perturbation

Solitary wave solutions of two-component Drinfel’d–Sokolov–Wilson system with Kuramoto–Sivashinsky perturbation are considered. We first employ geometric singular perturbation theory to reduce the higher-dimensional system of equations to the perturbed planar system. We then further exploit the Melnikov method to explore the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit, indicating the existence of single- and double-peak solitary waves. Of particular interest is the appearance of the double-peak solitary wave solution. Finally, we include the numerical simulations to verify the theoretical results.

Improved Finite Difference Technique via Adomian Polynomial to Solve the Coupled Drinfeld’s–Sokolov–Wilson System

This study presents a new algorithm for effectively solving the nonlinear coupled Drinfeld’s–Sokolov–Wilson (DSW) system using a hybrid explicit finite difference technique with the Adomian polynomial (EFD‐AP). The suggested approach addresses the problem of accurately solving the DSW system. Numerical results are obtained by comparing the exact solution with absolute and mean square errors using a test problem to assess the EFD‐AP accuracy against the exact solution and the conventional explicit finite difference (EFD) method. The results exhibit excellent agreement between the approximate and exact solutions at different time values, and the results showed that the proposed EFD‐AP method achieves superior accuracy and efficiency compared to the EFD method, which makes it a promising method for solving nonlinear partial differential systems of higher order.

A Robust Semi Analytical Approach to Study Fractional Coupled Sokolov Wilson System in Shallow Water Waves

A Robust Semi Analytical Approach to Study Fractional Coupled Sokolov Wilson System in Shallow Water Waves

A robust semi-analytical approach to study fractional coupled Sokolov Wilson system in shallow water waves

A robust semi-analytical approach to study fractional coupled Sokolov Wilson system in shallow water waves

Applications of Two Methods in Exact Wave Solutions in the Space-Time Fractional Drinfeld–Sokolov–Wilson System

The fractional differential equations (FDEs) are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. Therefore, FDEs have been the focus of many studies due to their frequent appearance in several applications such as physics, engineering, signal processing, systems identification, sound, heat, diffusion, electrostatics and fluid mechanics, and other sciences. The perusal of these nonlinear physical models through wave solutions analysis, corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, the exp-function method and the rational G ′ / G -expansion method are presented to establish the exact wave solutions of the space-time fractional Drinfeld–Sokolov–Wilson system in the sense of the conformable fractional derivative. The fractional Drinfeld–Sokolov–Wilson system contains fractional derivatives of the unknown function in terms of all independent variables. This system describes the shallow water wave models in fluid mechanics. These presented methods are a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences, especially in physics.

THE COMPARATIVE REPORT ON DYNAMICAL ANALYSIS ABOUT FRACTIONAL NONLINEAR DRINFELD–SOKOLOV–WILSON SYSTEM

The new extended direct algebraic technique is used to attain the new exact solitary wave solutions of the nonlinear fractional couple Drinfeld–Sokolov–Wilson system (DSWS). This DSWS has a lot of applications in water waves, fluid mechanics, quantum mechanics, chemistry, and biology. This system is developed in the perspective of fractional theory of calculus, for this purpose Riemann–Liouville fractional derivative (RLFD), [Formula: see text]-fractional differential operator and Atangana–Baleanu fractional derivative (ABFD) are utilized. The attained solutions contain some new families such as trigonometric functions, hyperbolic functions, rational functions, logarithmic functions, and exponential functions. The graphically analysis and comparison of solutions are plotted by taking appropriate values of the constant parameters to enhance the understanding of physical real-life phenomena. The approach is said to be compatible and effective as a scientific approach for the examination of the various spacetime fractional models in engineering and physics concerning real-life issues.

Investigation of Fractal Fractional nonlinear Drinfeld–Sokolov–Wilson system with Non-singular Operators

The aim of this article is to study the Drinfeld–Sokolov–Wilson equation considered in fractal-fractional sense with exponential decay and Mittag-Leffler type kernels. The Laplace transform combined with Adomian decomposition method is used to calculate the general solution of the system in series form. The convergence of the obtained series solution is also presented. For validity of our results, we consider a numerical example with suitable initial conditions. The results reveal that by decreasing fractal parameter enhances the amplitude of the soliton solution of the system. Moreover, decreasing fractional parameter, rapidly increases the amplitude of solitary wave of the system. We also observed that for small value of time t, the wave solutions of the system are very close to each other, while increasing in time t enhances the system when one of the parameters fractional or fractal is equal to one. We also studied the error analysis of the system reveals that the absolute error decreases as spatial variable x increases at small time t, while, increase in iterations, reduces the absolute error in the system.

Applied Sumudu Transform with Adomian Decomposition Method to the Coupled Drinfeld–Sokolov–Wilson System

In this paper, we studied and applied a modern numerical method, which is combining Sumudu transform with Adomian decomposition Method to obtain approximate solutions of the nonlinear the Coupled Drinfeld– Sokolov–Wilson (DSW) system. Positive and negative values of the variable x and various values of the variable t were taken with the initial conditions of the system as well as the values of the parameters . The efficiency of the method was verified, as the results obtained were compared with the accurate solution of the system. We noticed that the results are very accurate and the effectiveness of the method was confirmed.

Numerical solution of Drinfeld–Sokolov–Wilso system by using modified Adomian decomposition method

<p class="Char">In this paper, the modified Adomian decomposition method (MADM) is usedto solve different types of differential equations, one of the numerical analysis methods for solving non linear partial differential equations (Drinfeld–Sokolov–Wilson system) and short (DSWS) that occur in shallow water flows. A Genetic Algorithm was used to find the optimal value for the parameter (a). We numerically solved the system (DSWS) and compared the result to the exact solution. When the value of it is low and close to zero, the MADM provides an excellent approximation to the exact solution. As well as the lower value of leads to the numerical algorithm of (MADM) approaching the real solution. Finally, found the optimal value when a=-10 by using the Genetic Algorithm (G-MADM). All the computations were carried out with the aid of Maple 18 and Matlab to find the parameter value (a) by using the genetic algorithm as well as to figures drawing. The errors in this paper resulted from cut errors and mean square errors.</p>

He's variational method for the time–space fractional nonlinear Drinfeld–Sokolov–Wilson system

The well‐known Drinfeld–Sokolov–Wilson system is widely used to describe the flow of the shallow water. This paper considers the time–space fractional Drinfeld–Sokolov–Wilson system, where He's variational method together with the two‐scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations.

Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics

Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. The aim of the current work is threefold. First, we present a new technique which is constructed by combining the Laplace transform tool with the residual power series method. Precisely, we provide the details of implementing the proposed method to treat time-fractional nonlinear problems. Second, we test the validity and the efficiency of the method on the temporal-fractional Newell–Whitehead–Segel model. Then, we implement this new methodology to study the temporal-fractional (1+1)-dimensional Burger’s equation and the Drinfeld–Sokolov–Wilson system. Further, for accuracy and reliability purposes, we compare our findings with other methods being used in the literature. Finally, we provide 2-D and 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.

Some new and modified fractional analysis of the time-fractional Drinfeld-Sokolov-Wilson system.

In this paper, we present a presumably new approach in order to solve the time-fractional Drinfeld-Sokolov-Wilson system, which is based upon the Liouville-Caputo fractional integral (LCFI), the Caputo-Fabrizio fractional integral, and the Atangana-Baleanu fractional integral in the sense of the LCFI by using the Adomian decomposition method. We compare the approximate solutions with the exact solution (if available), and we find an excellent agreement between them. In the case of a non-integer order, we evaluate the residual error function, thereby showing that the order of the error is very small. In all of our calculations, we apply the software package, Mathematica (Version 9).

Space time fractional Drinfel'd-Sokolov-Wilson system with time-dependent variable coefficients: Symmetry analysis, power series solutions and conservation laws

In this work we aim to apply the Lie-symmetry method via the Riemann-Liouville fractional derivative based on continuous group of transformations to examine the symmetry reduction of space time fractional Drinfel'd-Sokolov-Wilson (DSW) system with time-dependent variable coefficients. The reduced Nonlinear fractional ordinary differential equations (NLFODEs) with variable coefficients are further studied for the exact solution by using the power series method. The exact solutions obtained in form of power series and their convergence show the accuracy and efficiency of the proposed method, which is simple and accurate in comparison to other methods to find the exact solution of nonlinear fractional partial differential equations (NLFPDEs). Also, the new conservation theorem and Noether's operators are used to construct conservation laws of the governing system.

The classification of single traveling wave solutions to coupled time-fractional KdV-Drinfel’d-Sokolov-Wilson system

Abstract This article aims to carry out novel research on obtaining exact solutions to a generalized time-fractional KdV-Drinfel’d-Sokolov-Wilson system. The complete discrimination system for polynomial method is applied to handle such a problem. Description of the method is given and all the possible exact traveling wave solutions including hyper-elliptic functions solutions are obtained, which are very difficult to get by other methods. In particular, to show the existence of the solutions, concrete examples are constructed for the concrete parameters of the corresponding solutions, from which we can see that each kind of the solutions in the classification can be realized except one.

Trigonometric Function Solutions of Fractional Drinfeld's Sokolov -Wilson System

In this paper, we construct exact trigonometric solutions of the space-time fractional classical Drinfeld's Sokolov-Wilson system by Modified Trial Equation Method (MTEM). These solutions may explain some physical phenomena and lead to researchers in physics and engineering.