The Improved Moving Least-Square Ritz Method for the One-Dimensional Sine-Gordon Equation

Analysis of the one-dimensional sine-Gordon equation is performed using the improved moving least-square Ritz method (IMLS-Ritz method). The improved moving least-square approximation is employed to approximate the 1D displacement field. A system of discrete equations is obtained by application of the Ritz minimization procedure. The effectiveness and accuracy of the IMLS-Ritz method for the sine-Gordon equation are investigated by numerical examples in this paper.

[1]  Guo-Wei Wei,et al.  Discrete singular convolution for the sine-Gordon equation , 2000 .

[2]  Mehdi Dehghan,et al.  The boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation , 2008 .

[3]  Baodong Dai,et al.  A meshless local moving Kriging method for two-dimensional solids , 2011, Appl. Math. Comput..

[4]  Yumin Cheng,et al.  The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems , 2011 .

[5]  Boundary element model for electrochemical dissolution under externally applied low level stress , 2013 .

[6]  W. Ritz.  Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. , 1909 .

[7]  Mehdi Dehghan,et al.  A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions , 2008, Math. Comput. Simul..

[8]  Wang Jian-fei,et al.  A new complex variable meshless method for transient heat conduction problems , 2012 .

[9]  John Argyris,et al.  Finite element approximation to two-dimensional sine-Gordon solitons , 1991 .

[10]  Mehdi Dehghan,et al.  A numerical method for one‐dimensional nonlinear Sine‐Gordon equation using collocation and radial basis functions , 2008 .

[11]  K. M. Liew,et al.  A meshless analysis of three-dimensional transient heat conduction problems , 2012 .

[12]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[13]  Mingrong Cui.  Fourth‐order compact scheme for the one‐dimensional sine‐Gordon equation , 2009 .

[14]  Wang Jian-fei,et al.  New complex variable meshless method for advection—diffusion problems , 2013 .

[15]  Mehdi Dehghan,et al.  The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation , 2008 .

[16]  Michael Griebel,et al.  Meshfree Methods for Partial Differential Equations , 2002 .

[17]  Joseph J Monaghan,et al.  An introduction to SPH , 1987 .

[18]  K. M. Liew,et al.  The reproducing kernel particle method for two-dimensional unsteady heat conduction problems , 2009 .

[19]  L. W. Zhang,et al.  Modeling of biological population problems using the element-free kp-Ritz method , 2014, Appl. Math. Comput..

[20]  K. M. Liew,et al.  Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method , 2004 .

[21]  Yumin Cheng,et al.  The complex variable reproducing kernel particle method for elasto-plasticity problems , 2010 .

[22]  K. Liew,et al.  Analyzing the 2D fracture problems via the enriched boundary element-free method , 2007 .

[23]  Yumin Cheng,et al.  A boundary element-free method (BEFM) for two-dimensional potential problems , 2009 .

[24]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[25]  Dogan Kaya.  A numerical solution of the sine-Gordon equation using the modified decomposition method , 2003, Appl. Math. Comput..

[26]  Yumin Cheng,et al.  The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems , 2011 .

[27]  K. M. Liew,et al.  THE ELEMENT-FREE kp-RITZ METHOD FOR VIBRATION OF LAMINATED ROTATING CYLINDRICAL PANELS , 2002 .

[28]  K. Liew,et al.  Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems , 2008 .

[29]  Wang Ling-hui,et al.  A moving Kriging interpolation-based boundary node method for two-dimensional potential problems , 2010 .

[30]  F. Sun,et al.  An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems , 2012 .

[31]  K. M. Liew,et al.  Analyzing 2D fracture problems with the improved element-free Galerkin method , 2008 .

[32]  K. M. Liew,et al.  Analyzing two-dimensional sine–Gordon equation with the mesh-free reproducing kernel particle Ritz method , 2012 .

[33]  Chen Li,et al.  Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems , 2013 .

[34]  Mehdi Dehghan,et al.  Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation , 2010, J. Comput. Appl. Math..

[35]  K. Liew,et al.  The improved element-free Galerkin method for two-dimensional elastodynamics problems , 2013 .

[36]  Cheng Yumin,et al.  Boundary element-free method for elastodynamics , 2005 .

[37]  程玉民,et al.  Complex variable element-free Galerkin method for viscoelasticity problems , 2012 .

[38]  Pei Liu,et al.  THE COMPLEX VARIABLE ELEMENT-FREE GALERKIN (CVEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS , 2009 .

[39]  Mehdi Dehghan,et al.  Implementation of meshless LBIE method to the 2D non‐linear SG problem , 2009 .

[40]  Zheng Bao-Jing,et al.  Improved meshless local Petrov-Galerkin method for two-dimensional potential problems , 2010 .

[41]  H. Ge,et al.  The element-free Galerkin method of numerically solving a regularized long-wave equation , 2012 .

[42]  Wei Qi,et al.  Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method , 2013 .

[43]  Juan I. Ramos,et al.  The sine-Gordon equation in the finite line , 2001, Appl. Math. Comput..

[44]  K. Liew,et al.  The improved element-free Galerkin method for three-dimensional wave equation , 2012 .

[45]  Jianfeng Wang,et al.  The Complex Variable Element-Free Galerkin (cvefg) Method for Two-Dimensional Elastodynamics Problems , 2012 .

[46]  K. M. Liew,et al.  Nonlinear analysis of laminated composite plates using the mesh-free kp-Ritz method based on FSDT , 2004 .

[47]  F. Sun,et al.  AN INTERPOLATING BOUNDARY ELEMENT-FREE METHOD WITH NONSINGULAR WEIGHT FUNCTION FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS , 2013 .

[48]  K. M. Liew,et al.  Numerical study of the three-dimensional wave equation using the mesh-free kp-Ritz method , 2013 .

[49]  Qin Sheng,et al.  A predictor‐‐corrector scheme for the sine‐Gordon equation , 2000 .

[50]  YuMin Cheng,et al.  The improved element-free Galerkin method for three-dimensional transient heat conduction problems , 2013 .

[51]  Baodong Dai,et al.  AN IMPROVED LOCAL BOUNDARY INTEGRAL EQUATION METHOD FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS , 2010 .

[52]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[53]  Linghui Wang,et al.  Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method , 2013, Appl. Math. Comput..

[54]  F. Esposito,et al.  Theory and applications of the sine-gordon equation , 1971 .

[55]  John Argyris,et al.  An engineer's guide to soliton phenomena: Application of the finite element method , 1987 .