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

Abstract 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.

[1]  Hung-Chang Lee,et al.  A modified group-preserving scheme for solving the initial value problems of stiff ordinary differential equations , 2002, Appl. Math. Comput..

[2]  Khaled Omrani,et al.  The convergence of fully discrete Galerkin approximations for the Benjamin-Bona-Mahony (BBM) equation , 2006, Appl. Math. Comput..

[3]  Cesar Augusto Gómez Sierra,et al.  New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations , 2010, Appl. Math. Comput..

[4]  Saeid Abbasbandy,et al.  The first integral method for modified Benjamin–Bona–Mahony equation , 2010 .

[5]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[6]  Davood Domiri Ganji,et al.  Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method , 2009 .

[7]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[8]  Saeid Abbasbandy,et al.  Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function , 2012 .

[9]  Hui Yin,et al.  Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations , 2010 .

[10]  Ernst Hairer,et al.  Geometric Integration of Ordinary Differential Equations on Manifolds , 2001 .

[11]  Dumitru Baleanu,et al.  Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line , 2016, J. Comput. Phys..

[12]  Chein-Shan Liu.  Group preserving scheme for backward heat conduction problems , 2004 .

[14]  E. Kansa,et al.  Numerical simulation of two-dimensional combustion using mesh-free methods , 2009 .

[15]  Saeid Abbasbandy,et al.  The Lie-group shooting method for solving the Bratu equation , 2011 .

[16]  Chein-Shan Liu,et al.  The backward group preserving scheme for multi-dimensional nonhomogeneous and nonlinear backward wave problems , 2014 .

[17]  Arieh Iserles,et al.  Geometric integration: numerical solution of differential equations on manifolds , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  Mehdi Dehghan,et al.  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 , 2015, J. Comput. Appl. Math..

[19]  Huijiang Zhao,et al.  Nonlinear stability of generalized Benjamin–Bona–Mahony–Burgers shock profiles in several dimensions , 2013 .

[20]  Mekki Ayadi,et al.  Numerical simulation of Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equations using finite difference method , 2013, Appl. Math. Comput..

[21]  Saeid Abbasbandy,et al.  Comparison of meshless local weak and strong forms based on particular solutions for a non-classical 2-D diffusion model , 2014 .

[22]  Chein-Shan Liu,et al.  The Fourth-Order Group Preserving Methods for the Integrations of Ordinary Differential Equations , 2009 .

[23]  Chen Zhengzheng,et al.  DEGENERATE BOUNDARY LAYER SOLUTIONS TO THE GENERALIZED BENJAMIN-BONAMAHONY-BURGERS EQUATION , 2012 .

[24]  Shaomei Fang,et al.  Optimal decay rates of solutions for a multidimensional generalized Benjamin–Bona–Mahony equation , 2012 .

[25]  Chein-Shan Liu,et al.  A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains , 2012 .

[26]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[27]  S. Abbasbandy,et al.  Meshless simulations of the two-dimensional fractional-time convection–diffusion–reaction equations , 2012 .

[28]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[29]  Mehdi Dehghan,et al.  The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions , 2014, Comput. Math. Appl..

[30]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[31]  Saeid Abbasbandy,et al.  Group preserving scheme for the Cauchy problem of the Laplace equation , 2011 .

[32]  Saeid Abbasbandy,et al.  Existence and numerical simulation of periodic traveling wave solutions to the Casimir equation for the Ito system , 2015, Commun. Nonlinear Sci. Numer. Simul..

[33]  Muhammad Aslam Noor,et al.  Some new solitonary solutions of the modified Benjamin-Bona-Mahony equation , 2011, Comput. Math. Appl..

[34]  A. Aminataei,et al.  ON THE NUMERICAL SOLUTION OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS USING MULTIQUADRIC APPROXIMATION SCHEME , 2008 .

[35]  Su-Ying Zhang,et al.  Group preserving schemes for nonlinear dynamic system based on RKMK methods , 2006, Appl. Math. Comput..

[36]  T. Achouri,et al.  On the convergence of difference schemes for the Benjamin-Bona-Mahony (BBM) equation , 2006, Appl. Math. Comput..

[37]  Chih-Wen Chang,et al.  An implicit Lie-group iterative scheme for solving the nonlinear Klein–Gordon and sine-Gordon equations , 2016 .

[38]  Saeid Abbasbandy,et al.  A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation , 2013 .

[39]  Chein-Shan Liu.  Cone of non-linear dynamical system and group preserving schemes , 2001 .