Generalized Regularized Long Wave Research Articles

Local radial basis functions and scale-3 Haar wavelets operational matrices based numerical algorithms for generalized regularized long wave model

In this article, two numerical algorithms are designed for the simulation of generalized regularized long wave (GRLW) model via local radial basis functions (LRBFs) and Scale-3 Haar wavelets (S3HWs). First of all, the well-posedness of the model is discuss for the initial data u0x∈H02Ω. After that in the design of two numerical algorithms, the first step is semi-discretization in time with a finite difference, quasilinearization technique (QT) for linearization and then the obtained semi-discrete model is analyzed for truncation errors and stability. In the end, the semi-discrete model is fully discretized via LRBFs and S3HWs. Finally, the fully discretized system is simulated by developing MATLAB routines. In the last section, some numerical problems are considered to inspect the chastity of the developed algorithms.

Two efficient methods for solving the generalized regularized long wave equation

ABSTRACT In this paper, an exact method named Riccati–Bernoulli sub-ODE method and a numerical method named Subdomain finite element method are proposed for solving the nonlinear generalized regularized long wave (GRLW) equation. For this purpose, Bäcklund transformation of the Riccati–Bernoulli equation and sextic B-spline functions are used for the exact and numerical solutions, respectively. The single soliton wave motion is used to confirm the methods which are found to be correct and effective. The three invariants (, and ) of motion have been assessed to indicate the conservation properties of the numerical algorithm. For the motion of single solitary and error norms are handled to evaluate differences between the analytical and numerical solutions. Unconditional stability is demonstrated using von-Neumann method. The procured outcomes show that our new schemes guarantee an evident and a functional mathematical equipment for solving nonlinear evolution equations.

Conservative Difference Scheme of Solitary Wave Solutions of the Generalized Regularized Long-Wave Equation

Conservative difference scheme for the nonlinear dispersive generalized regularized long-wave (GRLW) equation is proposed. Existence of its difference solutions has been shown. It is proved by the discrete energy method that the difference scheme is uniquely solvable, unconditionally stable and the convergence is of second-order in the maximum norm. The particular case known as the modified regularized long-wave (MRLW) equation is also discussed numerically in details. Furthemore, three invariants of motion are evaluated to determine the conservation properties of the problem. Interaction of two and three solitary waves are shown. Some numerical examples are given in order to validate the theoretical results.

Numerical investigation of nonlinear generalized regularized long wave equation via delta-shaped basis functions

In this study we will investigate generalized regularized long wave (GRLW)equation numerically. The GRLW equation is a highly nonlinear partialdifferential equation. We use finite difference approach for timederivatives and linearize the nonlinear equation. Then for space discretizationwe use delta-shaped basis functions which are relatively few studiedbasis functions. By doing so we obtain a linear system of equationswhose solution is used for constructing numerical solution of theGRLW equation. To see efficiency of the proposed method four classictest problems namely the motion of a single solitary wave, interactionof two solitary waves, interaction of three solitary waves and Maxwellianinitial condition are solved. Further, invariants are calculated.The results of numerical simulations are compared with exact solutionsif available and with finite difference, finite element and some collocationmethods. The comparison indicates that the proposed method is favorableand gives accurate results.

New Application for Generalized Regularized Long Wave (GRLW) Equation, Modified Dispersive Water Wave (MDWW) Equation and Kawahara Equation by Homogeneous Balance Method

New Application for Generalized Regularized Long Wave (GRLW) Equation, Modified Dispersive Water Wave (MDWW) Equation and Kawahara Equation by Homogeneous Balance Method

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

AbstractThe generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L2, L∞ and three invariants I1, I2, and I3 are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.

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.

A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation

In this article, a collocation algorithm based on quintic B-splines is proposed for the numerical solution of the non-linear generalized regularized long wave (GRLW) equation. Also, to analyse the linear stability of the numerical scheme, the von-Neumann technique is used. The numerical approach is discussed on three test examples consisting of a single solitary wave, the collision of two solitary waves and the growth of an undular bore. The accuracy of the method is demonstrated by calculating the error in the L2 and L¥ norms and the conservative quantities I1, I2 and I3. The findings are compared with those of previously reported in the literature. Finally, the motion of solitary waves is graphically plotted according to different parameters.

Solitary-wave solutions of the GRLW equation using septic B-spline collocation method

In this work, solitary-wave solutions of the generalized regularized long wave (GRLW) equation are obtained by using septic B-spline collocation method with two different linearization techniques. To demonstrate the accuracy and efficiency of the numerical scheme, three test problems are studied by calculating the error norms L2 and L∞ and the invariants I1, I2 and I3. A linear stability analysis based on the von Neumann method of the numerical scheme is also investigated. Consequently, our findings indicate that our numerical scheme is preferable to some recent numerical schemes.

On the accuracy of some explicit and implicit methods for the inviscid GRLW equation subject to initial Gaussian conditions

Purpose – The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions. Design/methodology/approach – Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained. Findings – It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation. Originality/value – This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

Chebyshev–Chebyshev spectral collocation method for solving the generalized regularized long wave (GRLW) equation

In this research, a Chebyshev–Chebyshev spectral collocation method based on Kronecker and Hadamard products is proposed for solving the generalized regularized long wave (GRLW) equation. Chebyshev–Gauss–Lobatto collocation points are used in both time and space directions. Three invariants of motion: mass, momentum and energy are evaluated to determine the conservation properties of the GRLW equation. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

Solitary waves generated by bell-shaped initial conditions in the inviscid and viscous GRLW equations

The objective of this paper is three-fold. First, the accuracy of three methods of lines that employ either piecewise analytical integration and yield exponential methods or three-point, fourth-order accurate finite difference approximations for the spatial derivatives, is assessed by comparing the numerical solutions with the exact one of the generalized regularized long wave (GRLW) equation. Second, the three-point, fourth-order accurate, compact method is used to determine the disintegration or breakup of bell-shaped initial conditions for the GRLW and viscous GRLW equations. Third, the dynamics of the GRLW and viscous GRLW equations is studied as a function of the amplitude and width of Gaussian and triangular initial conditions. It is shown that the compact operator method yields slightly more accurate results than an exponential method that treats the source terms with fourth-order accuracy. It is also shown that the disintegration of the bell-shaped initial conditions is characterized by an initial transient, steepening of the leading front and the formation of solitary waves in the absence of viscosity. The number, speed and distance between successive solitary waves are shown to increase as the dispersion parameter is decreased in accord with a linear stability analysis. The number and speed of solitary waves are also functions of the amplitude and width of the bell-shaped initial conditions, whereas the effect of the linear advective terms is to push the solitary waves towards the downstream boundary. In the presence of viscosity, it is shown that the amplitude of the leading solitary wave that results from the disintegration of the initial conditions, decreases as a function of time and the waves exhibit a curvature which is a function of the linear and nonlinear advective and dispersion terms, and the viscosity coefficient. For large viscosities, it is shown that the initial flow adjustment results in a steep leading front whose slope decreases as time increases and waves analogous to the ones observed in the viscous Burgers’ equation are formed. It is also shown that, for the GRLW equation, the number and speed of solitary waves that are formed after the break-up of the initial profile increase as the amplitude and width of the initial conditions is increased, and that Gaussian initial conditions result in faster and larger amplitude solitary waves than triangular ones. The long-term dynamics of the viscous GRLW equation is found to be nearly independent of the initial conditions.

A 2N order compact finite difference method for solving the generalized regularized long wave (GRLW) equation

The generalized regularized long wave (GRLW) equation is solved by fully different numerical scheme. The equation is discretized in space by 2N order compact finite difference method and in time by a backward finite difference method. At the inner and the boundary nodes, the first and the second order derivatives with 2N order of accuracy are obtained. To determine the conservation properties of the GRLW equation three invariants of motion are evaluated. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

A Fourth-Order Conservative Compact Finite Difference Scheme for the Generalized RLW Equation

The generalized regularized long-wave (GRLW) equation is studied by finite difference method. A new fourth-order energy conservative compact finite difference scheme was proposed. It is proved by the discrete energy method that the compact scheme is solvable, the convergence and stability of the difference schemes are obtained, and its numerical convergence order isO(τ2+h4)in theL∞-norm. Further, the compact schemes are conservative. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method

The generalized regularized long wave (GRLW) equation is solved numerically by using a distributed approximating functional (DAF) method realized by the regularized Hermite local spectral kernel. Test problems including propagation of single solitons, interaction of two and three solitons, and conservation properties of mass, energy, and momentum of the GRLW equation are discussed to test the efficiency and accuracy of the method. Furthermore, using the Maxwellian initial condition, we show that the number of solitons which are generated can be approximately determined. Comparisons are made between the results of the proposed method, analytical solutions, and numerical methods. It is found that the method under consideration is a viable alternative to existing numerical methods.

Variational Homotopy Perturbation Method for the Nonlinear Generalized Regularized Long Wave Equation

This paper presents Variational Homotopy Perturbation method for the nonlinear Generalized Regularized Long Wave (GRLW) equation. The solution of nonlinear GRLW equation is obtained and is solved using the iteration method which is combination of Variational Iteration method and Homotopy Perturbation Method. An example of the propagation of single soliton is given to show the precision of this method.

Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method

A numerical analysis of the generalized regularized long wave (GRLW) equation by application of the element-free kp-Ritz method is performed. To approximate the displacement field, an element-free kernel particle estimation is employed and a system of nonlinear discrete equations is achieved by using the Ritz minimization procedure to the energy expressions. Validation of the present numerical solutions with the exact results has been conducted by numerical examples and evidenced a good agreement.

Element-Free Approximation of Generalized Regularized Long Wave Equation

The generalized regularized long wave (GRLW) equation is an important nonlinear equation for describing a large number of physical phenomena, for examples, the shallow water waves and plasma waves. In this study, numerical approximation of the GRLW using the element-free improved moving least-squares Ritz (IMLS-Ritz) method is performed. In the solution procedure, the IMLS approximation is employed to reduce the number of unknown coefficients in the trial functions. The Ritz minimization procedure is then used to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear GRLW equation. Time difference technique and Newton-Raphson method are adopted to solve the nonlinear equation system. Numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element-free IMLS-Ritz method by comparing the computed IMLS-Ritz results with the existing available analytical solutions.

A New Compact Finite Difference Method for Solving the Generalized Long Wave Equation

The aim of this article is to analyze a new compact finite difference method (CFDM) for solving the generalized regularized long wave (GRLW) equation. This method leads to a system of linear equations involving tridiagonal matrices and the rate of convergence of the method is of order O(k 2 + h 4) where k and h are mesh sizes of time and space variables, respectively. Stability analysis of the method is investigated by the energy method and an error estimate is given. The propagation of single solitons and interaction of two solitary waves are applied to validate the method which is found to be accurate and efficient. Three invariants of the motion are evaluated to determine conservation properties of the method.

Optimal Homotopy Asymptotic Method to Nonlinear Damped Generalized Regularized Long-Wave Equation

A new semianalytical technique optimal homology asymptotic method (OHAM) is introduced for deriving approximate solution of the homogeneous and nonhomogeneous nonlinear Damped Generalized Regularized Long-Wave (DGRLW) equation. We tested numerical examples designed to confine the features of the proposed scheme. We drew 3D and 2D images of the DGRLW equations and the results are compared with that of variational iteration method (VIM). Results reveal that OHAM is operative and very easy to use.