Regularized Long Wave Research Articles

A space-time Petrov-Galerkin method for the two-dimensional regularized long-wave equation

In this work, a space-time Petrov-Galerkin (STPG) method is used to numerically analyze the two-dimensional regularized long-wave (RLW) equation. The STPG method is a nonstandard finite element method, that is, both of the spatial and temporal variables of this method are discretized by finite element method. Therefore, it can display the superiority of the finite element method in both spatial and temporal directions. In particular, it is easy to obtain space-time high accuracy and is unconditionally stable, thus this method is very appropriate for solving the nonlinear convection-diffusion equations. We prove the existence and uniqueness of the approximate solution and derive an optimal order error estimate in the maximum norm without requiring any compatibility between the space and time mesh size. In addition, some numerical experiments are presented to validate the accuracy and efficiency of the proposed method. Also, we provide several numerical experiments to confirm the conservation properties of discrete mass, energy and momentum for both the single and double-soliton waves.

New soliton solutions and modulation instability analysis of the regularized long-wave equation in the conformable sense

In this manuscript, we explore the solitary soliton of the regularized long-wave (RLW) equation which involving with weedy nonlinearity effects and dispersion relations which arises in shallow water, phonon packets in crystals, plasma wave and ion auditory waves. To execute the soliton solutions, we applied the advance Exp( − φ(ξ)) expansion method and the new form of modified Kudryashov's technique with conformable derivative from the fractional RLW model. For the explanation of the nature of RLW model, we obtained some behavior such as on the obtained solitons as bell-wave, rogue wave, periodic and double-periodic waves. This article offered the effects of conformable derivative to check the stability of the obtained phenomena. To check the stability of this model, we use the modulation instability analysis also. This work has a decent sense to endorse the extensive proposal of the model. Some 3-D and 2-D plots with their graphical explanation provides this research to characterized the obtained waves of RLW model.

Subdivision collocation method for numerical treatment of regularized long wave (RLW) equation

This research project introduces a novel computational approach for solving the regularized long wave equation. The proposed method utilizes a subdivision scheme with appropriate basis functions to transform the equation into a system of linear algebraic equations. A suitable numerical technique is employed to compute the solution of the transformed equations. Theoretical analysis of stability and error for the proposed method is also conducted. Furthermore, the invariants of three physical properties, waves, mass (M), momentum (P), and energy (ɛ), are calculated. Additionally, numerical evidence is presented to demonstrate the effectiveness and accuracy of the method. The results of the numerical experiments confirm the efficiency and high accuracy of the proposed method. Moreover, the numerical results of the invariants validate the conservation laws and align with the theoretical results.

Machine Learning Study through Physics-Informed Neural Networks: Analysis of the Stable Vortices in Quasi-Integrable Systems

Vortices in the nonlinear equations, including Zakharov-Kuznetsov (ZK) equation and the regularized long-wave (RLW) equation are studied. The Physics-Informed Neural Networks solve these equations in the forward process and obtain the solutions. In the inverse process, the proper equations can successfully be derived from a given training data. However, between the ZK equation and the RLW equation, sometimes serious misidentification occurs. In order to improve the resolution of the identification, we introduce two methods: a friction method and deformations of the initial profile which offers a nice discrimination of the equations.

An efficient wavelet‐based approximation method for solving nonlinear fractional‐time long wave equations: An operational matrix approach

This research paper develops an efficient Gegenbauer wavelet‐based approximation method to solve nonlinear fractional‐time regularized long wave (RLW) equations arising in ocean engineering. The operational matrices of derivatives, as well as the fractional order integration, are engaged in the proposed method. Using the operational matrix method and collocation point, the nonlinear RLW equations are converted into a system of algebraic equations, and these equations are further solved. Some results regarding the error‐bound estimation of this method have been developed. The method's accuracy and efficiency are confirmed by error estimation. The obtained results are compared with other available results and exact solutions. Moreover, the use of Gegenbauer wavelets is found to be a simple, straightforward forward and efficient tool for solving nonlinear PDEs arising in ocean engineering.

Abundant time-wavering solutions of a modified regularized long wave model using the EMSE technique

This research article presents the application of the enhanced modified simple equation integral technique to obtain abundant time-wavering solutions of the modified regularized long wave (RLW) model. The model is known for its significance in exploring wave phenomena in various fields, including shallow water dynamics, pressure waves in liquids and gas bubbles, nonlinear transverse waves in magneto-hydrodynamics, and ion-acoustic waves in plasma. By implementing the enhanced modified simple equation integral technique, we construct arbitrary time-varying wave solutions for the model, resulting in a diverse range of solitonic waveforms. These solutions include kink waves, anti-kink waves, bright and dark bell waves, double periodic waves, and combinations of solitons and periodic waves. The obtained solutions are visually presented through 2D and 3D density and contour plots. Our findings demonstrate the effectiveness and potential of the enhanced modified simple equation integral technique in identifying innovative time-varying soliton solutions within the modified RLW model.

On soliton solutions of the modified equal width equation

PurposeThe soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived.Design/methodology/approachThe main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal.FindingsThus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained.Originality/valueIllustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems.

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.

Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation

In this paper, high accurate numerical solutions of the regularised long-wave (RLW) equation is going to be obtained by using effective algorithm including finite difference method, differential quadrature and Rubin-Graves type linearization technique. Solitary wave solutions and Maxwellian initial condition based wave generation solutions are obtained successfully. To observe the development of the present algorithm, the present numerical results are compared with many earlier works. The present results are seen as superior among the given ones. The rates of the convergence are also given.

Dynamical Analysis of Nonlocalized Wave Solutions of 2 + 1 -Dimensional CBS and RLW Equation with the Impact of Fractionality and Free Parameters

This study retrieves some new exact solutions to the 2 + 1 -dimensional Calogero-Bogoyavlenskii-Schilf (CSB) equation and regularized long wave (RLW) equation in the context of nonlinear traveling wave phenomena. In this regard, the advanced exp − φ ξ -expansion method is imposed to the 2 + 1 -dimensional CBS and RLW equation, and consequently, rogue, kink, singular kink, periodic, singular, and multiple soliton solutions are exhibited in terms of trigonometric, hyperbolic, and rational function solutions. To enucleate the underlying nonlocalized traveling wave features, accomplished exact solutions are established by making their dynamic behavior of the exact solutions exhibited in three-dimensional (3D) and two-dimensional (2D) combined chart with the help of computational software Maple 18. All of our accomplished solutions are claimed to be new in the sense of conformable derivative, chosen a unique fractional type wave transformation, dynamical behavior of fractionally and free variable, and the imposed method on our preferred equations.

Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform

A new integral transform method for regularized long-wave (RLW) models having fractional-order is presented in this study. Although analytical approaches are challenging to apply to such models, semianalytical or numerical techniques have received much attention in the literature. We propose a new technique combining integral transformation, the Elzaki transform (ET), and apply it to regularized long-wave equations in this study. The RLW equations describe ion-acoustic waves in plasma and shallow water waves in seas. The results obtained are extremely important and necessary for describing various physical phenomena. This work considers an up-to-date approach and fractional operators in this context to obtain satisfactory approximate solutions to the proposed problems. We first define the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for solving RLW equations. We can readily obtain numerical results that provide us with improved approximations after only a few iterations. The derived solutions were found to be in close contact with the exact solutions. Furthermore, the suggested procedure has attained the best level of accuracy. In fact, when compared to other analytical techniques for solving nonlinear fractional partial differential equations, the present method might be considered one of the finest.

A Legendre spectral element method for the family of regularized long wave equations

Regularized Long Wave (RLW) equation is one of the most important nonlinear PDEs. It is related to the famous KdV equation and acts as an alternative for scrutinizing soliton phenomena. In this paper, we use the Legendre Spectral Element Method (LSEM) for the numerical study of the family of the RLW equations containing RLW, modified RLW, and generalized RLW. We perform the space discretization by the LSEM, and the Crank–Nicolson method is applied to discretize the time. To confirm the accuracy and efficiency of the proposed method, the conservation properties of the equations are checked, and various types of errors are reported.

A novel outlook to the an alternative equation for modelling shallow water wave: Regularised Long Wave (RLW) equation

A novel outlook to the an alternative equation for modelling shallow water wave: Regularised Long Wave (RLW) equation

Higher order Haar wavelet method integrated with strang splitting for solving regularized long wave equation

In this paper, we are going to utilize newly developed Higher Order Haar wavelet method (HOHWM) and classical Haar wavelet method (HWM) to numerically solve the Regularized Long Wave (RLW) equation. Spatial variable of the RLW equation is treated with HOHWM and HWM separately. On the other hand temporal variable is discretized by finite differences combined with Strang splitting approach. The presented methods applied to three different test problems and the obtained results are given in tables as well as depicted in figures. The obtained results are compared with analytical results wherever they exist. The error norms L2 and L∞ and invariants I1, I2 and I3 are used to show the accuracy of the methods when comparing the present results with those in the literature.

New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

Abstract The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

Traveling wave solutions of the generalized Rosenau–Kawahara-RLW equation via the sine–cosine method and a generalized auxiliary equation method

Abstract In this paper, we have approached a complicated nonlinear wave equation which links the Rosenau–Kawahara equation to the regularized long wave (RLW) equation. Taking advantages from the sine–cosine method as well as from the generalized auxiliary equation method, we have successfully reached to three types of traveling wave solutions: periodic, hyperbolic and exponential ones. Our results do constitute themselves as a challenge to apply the mentioned techniques in order to solve other generalized dynamical models, for example, the ones which involve phenomena such as a fully nonlinear dispersion and a fully nonlinear convection.

Bright-dark wave envelopes of nonlinear regularized-long-wave and Riemann wave models in plasma physics

The nonlinear regularized-long-wave (RLW) and the Riemann wave (RW) models are physically significant in plasma physics and in further study of nonlinear dispersive waves, namely shallow water, ion-acoustic and magneto-sound waves in plasmas, anharmonic lattice and pressure waves in liquid-gas bubble mixtures, tidal and tsunami waves in rivers and oceans, longitudinal electromagnetic wave propagation in plasma, etc. The advanced generalized (G'/G)-expansion scheme is facilitated in this article for retrieving the bright-dark exact solitary wave solutions generated in the form of hyperbolic, trigonometric, and rational structures to the above-stated models. To figure out the internal contrivance of the solutions, 3D and contour plots concerning bright parabolic wave shape, compacton wave shape, bell-shaped soliton, kink wave type soliton, dark anti-parabolic wave shape, anti-compacton wave shape, bright propagation of solitary wave shape solutions are depicted for definite free parametric values. The characteristic feature of the wave envelops fluctuates taking into consideration the changes of free parameters and they are essentially dominated by the linear and nonlinear impact.

The cubic B-spline least-squares finite-element method for the numerical solutions of regularized long-wave equation

A new least-squares method based on cubic B-spline functions is presented to solve regularized long-wave (RLW) equation numerically. An algebraic system of equations is obtained by employing the least-squares method based on cubic B-spline approximation for spatial discretization and the Crank–Nicolson technique for time discretization for integrating the RLW equation. The two test problems are implemented to show suitability of the proposed scheme. New results have been compared with those of some previous studies available in the literature.

Highly accurate numerical scheme based on polynomial scaling functions for equal width equation

In this paper we established a numerical method for Equal Width (EW) Equation using Polynomial Scaling Functions. The EW equation is a simpler alternative to well known Korteweg de Vries (KdV) and regularized long wave (RLW) equations which have many applications in nonlinear wave phenomena. According to Polynomial scaling method, algebraic polynomials are used to get the orthogonality between the wavelets and corresponding scaling functions with respect to the Chebyshev weight. First we introduce polynomial scaling functions, how are the functions are approximated according to these and Operational matrix of derivatives are given. For time discretization of the function we use finite difference method with Rubin Graves linearization and polynomial scaling functions are used for the space discretization. The method is applied to four different problem and the obtained results are compared with the results in the literature and with the exact results to give the efficiency of the method.

English

An energy conservative scheme is proposed for the regularized long wave (RLW) equation. The integral method with variational limit is used to discretize the spatial derivative and the finite difference method is used to discretize the time derivative. The energy conservation of the scheme and existence of the numerical solution are proved. The convergence of the order O(h2 + τ2) and unconditional stability are also derived. Numerical examples are carried out to verify the correctness of the theoretical analysis.