Symmetric Regularized Long Wave Research Articles

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity u(x,t) and density ρ(x,t) in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with O(τC2+τF+h2) in the discrete L∞-norm for u(x,t) and in the discrete L2-norm for ρ(x,t) are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.

Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended Tanh-Function Method

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in understanding the long-term dynamics of these events. This article primarily focuses on using the improved modified extended tanh-function algorithm to determine certain traveling wave solutions to the space-time fractional symmetric regularized long wave (SRLW) equation, which is used to discuss space-charge waves, shallow water waves, etc. The Jumarie’s modified Riemann-Liouville derivative is successfully used to deal with the fractional derivatives, which appear in the SRLW problem. We find many traveling wave solutions on the form of trigonometric, hyperbolic, complex, and rational functions. Furthermore, the performance of the employed technique is investigated in comparison to other techniques such as the Oncoming exp(−Θ(q))-expansion method and the extended Jacobi elliptic function expansion strategy. Some obtained results are graphically displayed to show their physical features. The findings of this article demonstrate that the used approach enables us to handle more NLFPDEs that emerge in mathematical physics.

CONSTRUCTION OF CLOSED FORM SOLITON SOLUTIONS TO THE SPACE-TIME FRACTIONAL SYMMETRIC REGULARIZED LONG WAVE EQUATION USING TWO RELIABLE METHODS

The employment of nonlinear fractional partial differential equations (NLFPDEs) is not limited to branches of mathematics entirely but also applicable in other science fields such as biology, physics and engineering. This paper derives some solitary wave solutions for the space-time fractional symmetric regularized long wave (SRLW) equation by means of the improved [Formula: see text]-expansion approach and the [Formula: see text]-expansion method. We use the definition of the Jumarie’s modified Riemann–Liouville derivative to handle the fractional derivatives appearing in this equation. Diverse types of soliton solutions are successfully expressed on the form of rational, hyperbolic, trigonometric, and complex functions. We extract kink wave, internal solitary wave, and solitary wave solutions. The performances of the proposed methods are compared with each other. Moreover, we compare the constructed results with some published solutions. The long behaviors of the obtained solutions are plotted in 2D and 3D figures. The resulting outcomes point out that the used techniques promise to empower us to deal with more NLFPDEs arising in mathematical physics.

Multiple exp-function method to solve the nonlinear space–time fractional partial differential symmetric regularized long wave (SRLW) equation and the (1+1)-dimensional Benjamin–Ono equation

In this study, we apply relatively analytical techniques, the multiple [Formula: see text]-function method, [Formula: see text]-function method and [Formula: see text]-expansion method to get approximate and analytic solutions of some nonlinear partial differential equations (PDEs), i.e., the nonlinear space–time fractional partial differential symmetric regularized long wave equation, an impressive model to characterize ion-acoustic and space change waves, the nonlinear [Formula: see text]-dimensional Fokas PDE, a meaningful multi-dimensional extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations, [Formula: see text]-dimensional Bateman–Burgers equation, a simplification of a more complex and sophisticated model, and the [Formula: see text]-dimensional Benjamin–Ono equation, a model for the propagation of unidirectional internal waves in stratified fluids. Finally, we propose the numerical results in tables and discuss advantages and disadvantages of the mentioned methods.

The behavior of plasma and space-charge waves represented by nonlinear mathematical models

In this study, traveling wave solutions are obtained by applying the modified exponential function method (MEFM) to the conformable time-fractional (3+1)-dimensional Zakharov-Kuznetsov (FZK) and the conformable time-fractional Symmetric Regularized Long Wave (SRLW) equations. In order to better understand the physical interpretations of the mathematical models represented by these two equations, 3D, contour, density and 2D graphics were plotted with the help of appropriate parameters. Thus, it has been shown that the modified exponential function method is effective in investigating the solutions of the conformable time-fractional (3+1)-dimensional FZK and the conformable time-fractional SRLW equations.

Exact Solutions of the Nonlinear Space-Time Fractional Partial Differential Symmetric Regularized Long Wave (SRLW) Equation by Employing Two Methods

In this article, with the aid of Maple software, the exact solutions to the space-time fractional symmetric regularized long wave (SRLW) equation are successfully examined by G ′ / G 2 -expansion and extended complex methods. Consequently, three types of traveling wave solutions are found such as Weierstrass double periodic elliptic functions, simply periodic functions, and the rational function solutions. The obtained results will play an important role in understanding and studying SRLW equation. It is easy to see that the extended complex and G ′ / G 2 -expansion methods are reliable and will be used extensively to seek exact solutions of any other fractional nonlinear partial differential equations (FNPDE).

Transcendental surface wave to the symmetric regularized long-wave equation

The symmetric regularized long-wave (SRLW) equation describes the attribute of the nonlinear ion acoustic waves, space charge waves, undular bore in meteorology etc. In this article, we investigate the SRLW equation to rummage transcendental shape of surface wave solution using the auxiliary equation approach and the improved Bernoulli sub-equation function technique. The solutions are ascertained in the form of trigonometric, hyperbolic, rational and exponential functions containing several ascendant parameters to the SRLW equation through the stated approaches which are compatible, dependable and simple to use. We extract a variety of rich structured surface solitons from the generic solution obtained including parabolic, kink, bell-shaped, lump soliton, etc. The implications of the various parameter values associated with the resulting solution are explicitly discussed to demonstrate their importance in explaining wave profiles behavior which may be convenient for analyzing nonlinear ion acoustic wave and space charge waves.

Numerical analysis of a high-order accurate compact finite difference scheme for the SRLW equation

In this paper, we develop a fourth-order accurate compact difference scheme for the symmetric regularized long wave (SRLW) equation for a single nonlinear velocity form. The discrete conservation, priori estimate, solvability, convergence and stability of the present scheme are proved by the discrete energy method. Numerical examples are given to support the theoretical analysis.

Distinct solutions of nonlinear space–time fractional evolution equations appearing in mathematical physics via a new technique

Abstract In this paper, we furnished novel and different traveling wave solutions to the nonlinear space–time fractional Klein–Gordon (KG) equation and the nonlinear space–time fractional symmetric regularized long wave (SRLW) equation. The considered equations are turned into ordinary differential equations with the aid of fractional composite transformation in the sense of conformable fractional derivative and then the proposed improved tanh method along with Riccati equation is employed. Consequently, fresh and further general solutions in the shape of kink, bell, cuspon, compacton, periodic etc. are successfully obtained, which are compared with those in the literature. Further, it is important to point out that some of our solutions is in good harmony for a special case with already published results which in turn validates our other solutions. The comparison ensures the novelty of the proposed technique which might be taken into account for further research in future.

Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, by using a direct method based on the Jacobi elliptic functions, the exact solutions of the space-time fractional symmetric regularized long wave (SRLW) equation have been obtained. The elliptic function solutions of a nonlinear ordinary differential (auxiliary) equation $\left({dF}/{d \xi}\right) ^{2} = PF^{4} (\xi)+QF^{2} (\xi) + R$ have also been examined. Besides, the solutions have been found in general form including rational, trigonometric and hyperbolic functions. Moreover, the complex valued solutions, periodic solutions, and soliton solutions, have also been gained. Some solutions have been illustrated by the graphics.

New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

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.

Exact Solutions For some nonlinear Partial Differential Equations by A Variation of (𝑮′𝑮⁄)-Expansion Method:

In this article , we study exact solutions of nonlinear of Burger’s Equation, Symmetric Regularized Long Wave (SRLW) equation, and Whitham-Broer-Kaup equations by using the variation of (G'⁄G)-expansion method, respectively. With the aid of mathematical software Maple, we can obtain the exact solutions for the above equations. Here we use the variation of (G'⁄G)-expansion method by applying it to solve the above mentioned equations, some new exact traveling wave solutions are obtained which include solitary wave solutions.When the arbitrary constants are taken some special values, the periodic and soliton solutions are obtained from the travelling wave solutions. The obtained solutions are new and not found elsewhere. It is shown that the methods are effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. This methods is effectual, uncomplicated and can also be used to tackle a number of other differential equations related to mathematical physics

Dispersive wave solutions of the nonlinear fractional Zakhorov-Kuznetsov-Benjamin-Bona-Mahony equation and fractional symmetric regularized long wave equation

Abstract In this paper, we obtain several novelty solutions by applying the improved F-expansion method to solve the space–time fractional Zakhorov Kuznetsov Benjamin Bona Mahony (ZKBBM) equation and the space–time fractional symmetric regularized long wave (SRLW) equation. These two fractional equations have been extensively used to explain numerous forms of physical mechanism in mathematical physics and engineering. The suggested fractional complex transform and the revised Riemann–Liouville derivative is used to change the two equations to its ordinary differential equations. The exact solutions of these equations include hyperbolic and trigonometric solutions has been effectively obtained.

Application of the ITEM for solving three nonlinear evolution equations arising in fluid mechanics

In this paper, by introducing new approach, the improved $$\tan \left( \phi (\xi )/2\right) $$ -expansion method (ITEM) is further extended into the Vakhnenko–Parkes (VP) equation, the generalized regularized-long-wave (GRLW) equation and the symmetric regularized-long-wave (SRLW) equation in fluid mechanic. We extended the ITEM proposed by Manafian et al. (Int J Appl Comput Math 2:243–268, 2016) to construct new types of soliton wave solutions of nonlinear partial differential equations (NPDEs). The merit of the presented method is finding the further solutions of the considering problems including soliton, periodic, kink and kink-singular wave solutions. Comparing our new results with other results shows that our results give the further solutions. The results of applying this procedure (Figs. 1, 2, 3, 4, 5, 6) to the studied cases show the high efficiency of the new technique. Finally, these solutions might play an important role in engineering, physics and applied mathematics fields.

Numerical study of a conservative weighted compact difference scheme for the symmetric regularized long wave equations

In this article, a new weighted and compact conservative difference scheme for the symmetric regularized long wave (SRLW) equations is considered. The new scheme is decoupled and linearized in practical computation, that is, at each time step only two tridiagonal systems of linear algebraic equations need to be solved. It is proved by the discrete energy method that the compact scheme is uniquely solvable, the convergence and stability of the difference scheme are obtained, and its numerical convergence order is in the ‐norm. Numerical experiment results show that the scheme is efficient and reliable.

Traveling wave solutions to some nonlinear fractional partial differential equations through the rational (G′/G)-expansion method

In this article, the analytical solutions to the space-time fractional foam drainage equation and the space-time fractional symmetric regularized long wave (SRLW) equation are successfully examined by the recently established rational (G′/G)-expansion method. The suggested equations are reduced into the nonlinear ordinary differential equations with the aid of the fractional complex transform. Consequently, the theories of the ordinary differential equations are implemented effectively. Three types closed form traveling wave solutions, such as hyperbolic function, trigonometric function and rational, are constructed by using the suggested method in the sense of conformable fractional derivative. The obtained solutions might be significant to analyze the depth and spacing of parallel subsurface drain and small-amplitude long wave on the surface of the water in a channel. It is observed that the performance of the rational (G′/G)-expansion method is reliable and will be used to establish new general closed form solutions for any other NPDEs of fractional order.

New structures for the space-time fractional simplified MCH and SRLW equations

In this paper, we constructed new solitary structures for the space-time fractional simplified modified Camassa-Holm (MCH) equation and space-time fractional symmetric regularized long wave (SRLW) equation using the modified extended tanh method. The space-time fractional derivatives are defined in the sense of the new conformable fractional derivative. Further, with the help of Mathematica software, the set of over-determined algebraic equations obtained after reducing the equations to ordinary differentials equations are treated. We finally provide graphical illustrations for some structures.

New exact traveling wave solutions for space-time fractional (1 + 1)-dimensional SRLW equation

This paper is concerned with extension of some analytical methods to obtain exact solutions of nonlinear fractional (1+1)-dimensional symmetric regularized long wave (SRLW) equation. These methods are straightforward, concise, effective and easy to use. They are powerful mathematical tools to obtain exact solutions of nonlinear fractional differential equations and can be used to solve another nonlinear fractional equations.

An Exponential Wave Integrator Pseudospectral Method for the Symmetric Regularized-Long-Wave Equation

An efficient and accurate exponential wave integrator Fourier pseudospectral (EWI-FP) method is proposed and analyzed for solving the symmetric regularized-long-wave (SRLW) equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fastFourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.