Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines

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

We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed $L_{\infty }$
 , $L_{2}$
 , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

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An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

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 ∞ , L 2 are computed. Computation verifies that current scheme has good outcome.

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

In the paper, a newly developed three-point fourth-order compact operator is utilized to construct an efficient compact finite difference scheme for the Benjamin–Bona–Mahony–Burgers’ (BBMB) equation. The detailed derivation is carried out based on the reduction order method together with a three-level linearized technique. The conservative invariant, boundedness and unique solvability are studied at length. The convergence is proved by the technical energy argument and induction method with the optimal convergence order $$\mathcal {O}(\tau ^2+h^4)$$
 in the sense of the maximum norm. The stability under mild conditions can be achieved based on the uniform boundedness of the numerical solution. The present scheme is very efficient in practical computation since only a linear system needs to be solved at each time. The extensive numerical examples verify our theoretical results and demonstrate the scheme’s superiority when compared with state-of-the-art those in the references.

Convergence and Stability in Maximum Norms of Linearized Fourth-Order Conservative Compact Scheme for 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 BenjaminBonaMahonyBurgers 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.

A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized BenjaminBonaMahonyBurgers equation

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A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized BenjaminBonaMahonyBurgers equation