Generalized Benjamin Bona Mahony Burgers 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

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.

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.

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.

VARIATIONAL PRINCIPLE AND APPROXIMATE SOLUTION FOR THE FRACTAL GENERALIZED BENJAMIN–BONA–MAHONY–BURGERS EQUATION IN FLUID MECHANICS

The well-known generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation is widely used in the fluid mechanics, but it becomes invalid under the non-smooth boundary. So this paper, for the first time ever, extends the gBBMB equation into the fractal form that still works under the non-smooth boundary. By using the semi-inverse method, we develop the fractal variational formulations for the problem, which can provide the conservation laws in an energy form, and reveal the possible solution structures of the equation. Furthermore, the two-scale transform method combined with the variational iteration method is used to solve the fractal gBBMB equation. The obtained results show a good agreement with the existed results.

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.

A new class of polynomial functions for approximate solution of generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equations

In this paper, a new class of polynomial functions equipped with a parameter is used, which is applied to approximate the gBBMB. In this method, the first, time discretization of the gBBMB equations is made by using finite difference approaches. Then, at any spatial point, a nonlinear equations system is generated. Using the collocation points and solving a nonlinear system in the least squares method, the coefficients of functions are obtained. In some examples, a linearization is performed first, and then the collocation method is applied.

Meshless formulation to two‐dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence

AbstractIn this study, the singular boundary method (SBM) is employed for the simulation of nonlinear generalized Benjamin–Bona–Mahony–Burgers problem with initial and Dirichlet‐type boundary conditions. The θ‐weighted finite difference method is used to discretize the time derivatives. Then the original equations are split into a system of partial differential equations. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. To solve this system, the method of particular solution (MPS) in combination with the SBM is used where the SBM is used for homogeneous solution and MPS is used for particular solution. Furthermore, the stability and convergence of the proposed method is conducted. Finally, several numerical examples with different domains are provided and compared with the exact analytical solutions to show the accuracy and efficiency in comparison with existing other methods.

The numerical solution of nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations via the meshless method of integrated radial basis functions

In this paper, a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. The used numerical method is based on the integrated radial basis functions (IRBFs). First, the time derivative has been approximated using a finite difference scheme. Then, the IRBF method is developed to approximate the spatial derivatives. The two-dimensional version of these equations is solved using the presented method on different computational geometries such as the rectangular, triangular, circular and butterfly domains and also other irregular regions. The aim of this paper is to show that the integrated radial basis function method is also suitable for solving nonlinear partial differential equations. Numerical examples confirm the efficiency of the proposed scheme.

Applications of two numerical methods for solving inverse Benjamin–Bona–Mahony–Burgers equation

In this paper, two numerical techniques are presented to solve the nonlinear inverse generalized Benjamin–Bona–Mahony–Burgers equation using noisy data. These two methods are the quartic B-spline and Haar wavelet methods combined with the Tikhonov regularization method. We show that the convergence rate of the proposed methods is $$\textit{O}(k^2+h^3)$$ and $$\textit{O}\left( \frac{1}{M}\right) $$ for the quartic B-spline and Haar wavelet method, respectively. In fact, this work considers a comparative study between quartic B-spline and Haar wavelet methods to solve some nonlinear inverse problems. Results show that an excellent estimation of the unknown functions of the nonlinear inverse problem has been obtained.

On the convergence of difference schemes for the generalized BBM–Burgers equation

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order k - 1 {k-1} when the exact solution belongs to the Sobolev space W 2 k ⁢ ( Q ) {W_{2}^{k}(Q)} , 1 < k ≤ 3 {1<k\leq 3} .

Numerical solutions of two dimensional Sobolev and generalized Benjamin–Bona–Mahony–Burgers equations via Haar wavelets

This work addresses, numerical method based on Haar wavelets and finite differences to solve two dimensional linear, nonlinear Sobolev and non-linear generalized Benjamin–Bona–Mahony–Burgers (NGBBMB) equations. The temporal part is discretized using finite differences while spatial part is approximated by two dimensional Haar wavelets. With this strategy, computing solution of two dimensional PDEs reduces to computing solution of linear system of algebraic equations. Collocation approach is then applied to determine the wavelet coefficients from linear system. This paper shows that two dimensional Haar wavelets are suitable and effective for two dimensional linear and non-linear PDEs. For validation of the proposed scheme different problems have been solved and error norms L∞,L2 are computed. Computation verifies that current scheme has good outcome.

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LG–RBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in Dehghan et al. (2014) to confirm the accuracy and efficiency of the presented method.

A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equation

In this paper, a new method based on hybridization of Lucas and Fibonacci polynomials is developed for approximate solutions of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equations. Firstly time discretization is made by using finite difference approaches. After that unknown function and its derivatives are expanded to Lucas series. Based on these series expansion, differentiation matrices are derived by utilizing Fibonacci polynomials. By doing so, the solution of the mentioned equations is reduced to the solution of an algebraic system of equations. By solving this system of equations the Lucas series coefficients are obtained. Then substituting these coefficients into Lucas series expansion approximate solutions can be constructed successively. The main goal of this paper is to indicate that Lucas polynomial based method is appropriate for 1D and 2D nonlinear problems. Efficiency and performance of the proposed method are judged on six test problems which consists of the 1D and 2D version of mentioned equation by calculating L2 and L∞ error norms. Feasibility of the method is verified by obtained accurate results.

Conservation laws and exact solutions of a Generalized Benjamin–Bona–Mahony–Burgers equation

The concept of nonlinear self-adjointness given by Ibragimov is applied to a Generalized Benjamin–Bona–Mahony–Burgers equation. Then, a nonlinear self-adjoint classification has been achieved. Moreover, some nontrivial conservation laws are constructed by using the multipliers method which does not require the use of a variational principle. Finally, by applying the modified simplest equation method we derive new travelling wave solutions.

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite element method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(τ). Since the shape functions of moving least squares approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed. An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme.

The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers equation. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we use Kansa’s approach to approximate the spatial derivatives. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. Also, we show that convergence order of the time discrete scheme is O(τ). We solve the two-dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular, triangular and circular domains and also the three-dimensional case is solved on the cubical and spherical domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations. Also, several test problems including the three-dimensional case are given. Numerical examples confirm the efficiency of the proposed scheme.