Benjamin Bona Mahony Burgers Research Articles

Unconditional superconvergence analysis of a new energy stable nonconforming BDF2 mixed finite element method for BBM–Burgers equation

The focus of this paper is to establish a new energy stable 2-step backward differentiation formula (BDF2) fully-discrete mixed finite element method (MFEM) for the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation with the nonconforming rectangular EQ1rot element and zero-order Raviart–Thomas (R–T) element (EQ1rot/Q10×Q01). Based on the energy stable property, the existence and uniqueness of the numerical solution are proved by the Brouwer fixed point theorem. Subsequently, with the assistance of some typical properties of this element pair and the interpolation post-processing approach, the unconditional superclose and superconvergence results with O(h2+τ2) are obtained directly without any constraints between the spatial partition parameter h and the time step τ. It is worthy to mention that the method and analysis presented herein are very different from the time–space splitting approach utilized in the previous studies and simplify the implement. Finally, two numerical experiments are executed to validate the theoretical analysis.

Higher-order predictor–corrector methods for fractional Benjamin–Bona–Mahony–Burgers’ equations

Higher-order predictor–corrector methods for fractional Benjamin–Bona–Mahony–Burgers’ equations

Solitonic solutions and stability analysis of Benjamin Bona Mahony Burger equation using two versatile techniques

This study aims to explore the precise resolution of the nonlinear Benjamin Bona Mahony Burgers (BBMB) equation, which finds application in a variety of nonlinear scientific disciplines including fluid dynamics, shock generation, wave transmission, and soliton theory. Within this paper, we employ two versatile methodologies, specifically the extended exp(-Ψ(χ))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\Psi (\\chi ))$$\\end{document} expansion technique and the novel Kudryashov method, to identify the exact soliton solutions of the nonlinear BBMB equation. The solutions we discovered involve trigonometric functions, hyperbolic functions, and rational functions. The uniqueness of this research lies in uncovering the bright soliton, kink wave solution, and periodic wave solution, and conducting stability analysis. Furthermore, the solutions’ graphical characteristics were explored through the utilization of the mathematical software Maple 2022 (https://maplesoft.com/downloads/selectplatform.aspx?hash=61ab59890f2313b2241fde3423fd975e). The system’s physical interpretation is defined through various types of graphs, including contour graphs, 3D-surface graphs, and line graphs, which use appropriate parameter values. These recommended techniques hold significant importance and are applicable in diverse nonlinear evolutionary equations found in the field of nonlinear sciences for illustrating nonlinear physical models.

Investigating invariance and conservation laws for the classes of nonlinear parabolic and wave systems

We study various classes of the nonlinear dynamics of some high order parabolic equations like the Oskolkov–Benjamin–Bona–Mahony–Burgers and Benjamin–Bona–Mahony–Peregrine–Burger equations that arise in the study of some wave phenomena. Also, a broader class of partial differential equations are used in modelling ocean waves that originate from the Ostrovsky equation. We study the invariance properties via the Lie invariance method for the nonlinear systems and further establish classes of conservation laws for models arises in this study. We show how the relationship leads to double reductions of the systems. This relationship is determined by a recent result involving multipliers that lead to a total divergence or closed form of the differential equation under investigation.

Global classical solution of the Cauchy problem to the 3D Benjamin–Bona–Mahony–Burgers-type equation with nonlocal control constraints

Abstract This article focuses on the Cauchy problem of the Benjamin–Bona–Mahony–Burgers (BBMB) equation with nonlocal control constraints in ℝ 3 {\mathbb{R}^{3}} . We employ the Phragmén–Lindelöf decomposition method and Green’s function to investigate global classical solutions and their long-term behaviors, including optimal estimates for finite density initial perturbation. Additionally, optimal solutions near the minimal mass ground state are analyzed.

A high-order combined finite element/interpolation approach for multidimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers equation

A high-order combined finite element/interpolation approach is developed for solving a multidimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers equation subjected to suitable initial and boundary conditions. In the proposed high-order scheme we approximate the time derivative with piecewise polynomial interpolation of second-order and use the finite element discretization of piecewise polynomials of degree q and q+1, where q≥2 is an integer, to approximating the space derivatives. The stability together with the error estimates of the constructed technique are established in W21(Ω)-norm. The analysis suggests that the developed numerical scheme is unconditionally stable, temporal second-order accurate and convergence in space with order q. Furthermore, the new procedure is faster and more efficient than a broad range of numerical methods discussed in the literature for the given initial–boundary value problem. A wide set of examples are performed to confirming the theoretical studies.

Viscous Effect on Solitary Kelvin Wave in Open Cylindrical Channel under Precession

Viscous effect is introduced into the system of Navier–Stokes equations, that were derived to study the solitary Kelvin mode in an open cylindrical channel that precesses. Accordingly, three new weakly nonlinear models were derived: Korteweg–de Vries-Burgers, and two new Benjamin–Bona–Mahony-Burgers. The first was solved analytically by discussing the phase solution and numerically using an implicite finite difference method to track the solution with time under diffusion effect. The second two models were solved numerically only using the Quartic B-Spline collocation method. By manipulating the scaling the first model included only the gravity force effect, and the second included both gravity and Coriolis forces. The numerical method is tested experimentally by comparing the velocity solutions with ADV signal extracted from the ADV measurements under bore conditions, and the amplitude solution with the solitary kelvin mode.

A Modified Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers Equation

This paper presents a modified Runge–Kutta (MRK) scheme for the generalized Benjamin–Bona–Mahony–Burgers equation. The MRK method separate this equation into two parts which apply different finite difference scheme in space, and finally use fourth-order Runge–Kutta method in time.The scheme is conditionally stable and convergent with order of O (тау4+h2), the Comparisons of the existing numerical methods demonstrate that this scheme is capable of high accuracy and stability.

A Modified Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers Equation

A Modified Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers Equation

The Painleve Analysis of an Ordinary Differential Equation Which Describes a Traveling-Wave Solution of the Oskolkov–Benjamin–Bona–Mahony–Burgers Equation

The Painleve Analysis of an Ordinary Differential Equation Which Describes a Traveling-Wave Solution of the Oskolkov–Benjamin–Bona–Mahony–Burgers Equation

A New Highly Accurate Numerical Scheme for Benjamin–Bona–Mahony–Burgers Equation Describing Small Amplitude Long Wave Propagation

A New Highly Accurate Numerical Scheme for Benjamin–Bona–Mahony–Burgers Equation Describing Small Amplitude Long Wave Propagation

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

In this paper, we propose a high-order nonlinear algorithm based on a finite difference method modification to the regularized long wave equation and the Benjamin–Bona–Mahony–Burgers equation subject to the homogeneous boundary. The consequence system of nonlinear equations typically trades with high computation burden. This dilemma can be overcome by establishing a fast numerical algorithm procedure without a reduction of numerical accuracy. The proposed algorithm forms a linear system with constant coefficient matrix at each time step and produces numerical solutions, which remarkably gains many computational advantages. In terms of analysis, a priori estimation for the numerical solution is derived to obtain the convergence and stability analysis. Additionally, the algorithm is globally mass preserving to avoid nonphysical behavior. Two benchmarks, including a single solitary wave to both equations, are given to validate the applicability and accuracy of the proposed method. Numerical results are obtained and compared to other approaches available in the literature. From the comparisons it is clear that the proposed approach produces accurate and precise results at low computational cost. Besides, the proposed scheme is applied to study the effect of the viscous term on a single solitary wave. It is shown that the viscous term results in the amplitude and width of the initial condition but not in its velocities in the case of a single solitary wave. As a consequence, theoretical and numerical findings provide a new area to investigate and expand the high-order algorithm for the family of wave equations.

Numerical study of generalized 2-D nonlinear Benjamin–Bona–Mahony–Burgers equation using modified cubic B-spline differential quadrature method

The objective of this work is to present the modified cubic B-spline differential quadrature (MCBSDQ) method for the numerical study of the generalized 2-D nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation. A detailed description of the proposed method is provided. Formulation of the generalized BBMB equation is done using the proposed technique. The method’s stability analysis is discussed. The accuracy and efficiency are shown through numerical examples. The numerical results of this study have been compared with those of other existing methods in the literature. The proposed method has excellent agreement with other methods which are used to solve the generalized 2-D nonlinear BBMB equation.

Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the generalized Benjamin–Bona–Mahony–Burgers equation with dissipative term

In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condititon for the generalized Benjamin–Bona–Mahony–Burgers equation with a fourth-order dissipative term. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity. We can further obtain the same global asymptotic stability of the rarefaction wave to the generalized Korteweg–de Vries–Benjamin–Bona–Mahony–Burgers equation with a fourth-order dissipative term as the former one.

Unconditional superconvergence analysis of an energy-stable finite element scheme for nonlinear Benjamin–Bona–Mahony–Burgers equation

In this paper, an energy-stable Crank–Nicolson fully discrete finite element scheme is proposed for the Benjamin–Bona–Mahony–Burgers equation. Firstly, the stability of energy is proved, which leads to the boundedness of the finite element solution in H^{1}-norm. Secondly, combining with the above boundedness and the special property of bilinear element, the unconditional superclose and superconvergence results are derived. Finally, numerical examples are provided to illustrate the validity and efficiency of our theoretical analysis and method.

Error analysis and numerical solution of generalized Benjamin–Bona–Mahony–Burgers equation using 3-scale Haar wavelets

In this paper, we propose an extended numerical algorithm for the numerical solution of the Benjamin–Bona–Mahony–Burgers equation. This algorithm involves the application of wavelet theory. First, we use the Quasilinearization technique of linearization and apply the 3-scale Haar wavelet approach for truncation error. This algorithm is constructed from two wavelet functions that make it robust and highly accurate. A multi-resolution is used to generate the Haar basis function. We consider three cases of a mathematical problem for the accuracy of the presented algorithm. The obtained results show good agreement with analytical solutions and have better accuracy.

Painlevé analysis, dark and singular structures for pseudo-parabolic type equations

In this paper, two important pseudo-parabolic type equations are studied for extracting soliton solutions via the generalized projective riccati equations method. These equations are Oskolkov equation and Oskolkov–Benjamin–Bona–Mahony–Burgers equation. The proposed method extracts dark soliton and singular soliton. Furthermore, the Painlevé test (P-test) has also been employed on pseudo-parabolic type equations for investigating integrability. The proposed equations are proved to be integrable by P-test. The numerical simulations have also been carried out by 3D and 2D graphs of some of the obtained solutions.

Numerical and analytical investigation for solutions of fractional Oskolkov–Benjamin–Bona–Mahony–Burgers equation describing propagation of long surface waves

In this paper, a novel meshless numerical scheme to solve the time-fractional Oskolkov–Benjamin–Bona–Mahony–Burgers-type equation has been proposed. The proposed numerical scheme is based on finite difference and Kansa-radial basis function collocation approach. First, the finite difference scheme has been employed to discretize the time-fractional derivative and subsequently, the Kansa method is utilized to discretize the spatial derivatives. The stability and convergence analysis of the time-discretized numerical scheme are also elucidated in this paper. Moreover, the Kudryashov method has been utilized to acquire the soliton solutions for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme.

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.