Benjamin Bona Mahony Burgers Equation Research Articles

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.

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.

Septic Hermite collocation method for the numerical solution of Benjamin–Bona–Mahony–Burgers equation

In this paper, the nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation is solved using the septic Hermite collocation method (SHCM) for the space discretization and Crank–Nicholson scheme for the time discretization. The Von-Neumann stability analysis is used to show the unconditional stability of the proposed method. The error bounds for the septic Hermite interpolation polynomials are estimated using the Peano's theorem. The method is applied to one, two and three interaction solitary wave problems. The and error norms are calculated to demonstrate the efficiency and the robustness of the proposed technique.

Numerical investigation of variable‐order fractional Benjamin–Bona–Mahony–Burgers equation using a pseudo‐spectral method

This article introduces a new local variable‐order (VO) fractional differentiation called the VO conformable fractional derivative. This derivative is employed to define the VO time fractional Benjamin–Bona–Mahony–Burgers (BBMB) equation. A pseudo‐spectral algorithm using the Chebyshev cardinal functions (CCFs) is adopted to find a numerical solution for this equation. The presented method with the help of the CCFs derivative matrices (which are obtained in this research) turns problem solving into solving an algebraic system of equations. The proposed approach is applied on some test problems, and its accuracy is examined in terms ofL∞andL2error norms. A numerical comparison is performed between the achieved results of the method with the results obtained from the cubic B‐spline method and the hybrid method generated using the quintic Hermite approach and the finite difference technique.

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

AbstractThis article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction isO(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

AbstractIn the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two‐level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ2 + h2). For the three‐level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on ψ function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(τ2 + h2) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.

Computational method for generalized fractional Benjamin–Bona–Mahony–Burgers equations arising from the propagation of water waves

In this research, by utilizing the concept of the mixed Caputo fractional derivative and left-sided mixed Riemann–Liouville fractional integral, we approximate the solution of generalized fractional Benjamin–Bona–Mahony–Burgers equations (GF-BBMBEs). In addition, using Genocchi polynomial properties, we obtain a new formula to approximate the functions by Genocchi polynomials. In the process of computation, we discuss a method of obtaining the operational matrix of integration and pseudo-operational matrices of the fractional order of derivative. Also, an algorithm of obtaining the mixed fractional integral operational matrix is presented. Using the collocation method and matrices introduced, the proposed equations are converted to a system of nonlinear algebraic equations with unknown Genocchi coefficients. In addition, we discuss the upper bound of the error for the proposed method. Finally, we examine several problems to demonstrate the validity and applicability of the proposed method.

An efficient numerical method for solving Benjamin–Bona–Mahony–Burgers equation using difference scheme

ABSTRACTIn this paper, we propose a numerical solution of the Benjamin–Bona–Mahony–Burgers equation (BBMB) based on finite difference scheme of sixth order. It is proved that the scheme is convergent. The existence and stability analysis of numerical solutions are discussed. The method is applied to several examples and accuracy of the method is tested in terms of and error norms. Furthermore, the numerical results have been compared with some other methods.