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.

It is well known that many physical phenomena in one or higher-dimensional space can be described by a soliton model. Many of these models are based on simple integrable models such as Korteweg-de Vries equation and the nonlinear Schr

A numerical study for sine-Gordon equation has been proposed including the finite difference schemes [

The meshless method is a new and interesting numerical technique. Important meshless methods have been developed and proposed, such as smooth particle hydrodynamics methods (SPH) [

The moving least-square (MLS) technique was originally used for data fitting. Nowadays, the MLS technique has been employed as the shape functions of the meshless method or element-free Galerkin (EFG) method [

The Ritz [

A new numerical method which is named the IMLS-Ritz method for the sine-Gordon equation is presented in this paper. In this paper, the unknown function is approximated by these IMLS approximations; a system of nonlinear discrete equations is obtained by the Ritz minimization procedure, and the boundary conditions are enforced by the penalty method.

Consider the following one-dimensional sine-Gordon equation:

The weighted integral form of (

The weak form of (

The energy functional

In the improved moving least-square approximation [

This defines the quadratic form

Equation (

To find the coefficients

If the functions

From (

Taking derivatives of (

Imposing boundary conditions by penalty method, the total energy functional for this problem will be obtained:

By (

Making time discretization of (

The numerical solution of the one-dimensional sine-Gordon equation will be obtained by solving the above iteration equation.

To verify the efficiency and accuracy of the proposed IMLS-Ritz method for the sine-Gordon equation, two examples are studied and the numerical results are presented. The weight function is chosen to be cubic spline and the bases are chosen to be linear in all examples.

Consider the sine-Gordon Equation (

The exact solution is

The IMLS-Ritz method is applied to solve the above equation with penalty factor

The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 41 nodes at

Node number | Exact solution | IMLS-Ritz method | EFG method |
---|---|---|---|

21 | 0.1488 | 0.1479 | 0.1473 |

22 | 0.2939 | 0.2918 | 0.2909 |

23 | 0.4318 | 0.4297 | 0.4275 |

24 | 0.5590 | 0.5583 | 0.5536 |

25 | 0.6725 | 0.6732 | 0.6661 |

26 | 0.7694 | 0.7687 | 0.7653 |

27 | 0.8474 | 0.8471 | 0.8399 |

28 | 0.9045 | 0.9032 | 0.8989 |

29 | 0.9393 | 0.9389 | 0.9380 |

30 | 0.8474 | 0.8478 | 0.8487 |

31 | 0.7694 | 0.7685 | 0.7625 |

32 | 0.6725 | 0.6732 | 0.6753 |

Numerical solution and exact solution of

Error function

Error function

Error function

The surface of exact solution (Example

The surface of numerical solution with IMLS-Ritz method (Example

Consider the case

The IMLS-Ritz method is applied to solve the above equation with penalty factor

The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 40 nodes at

Node number | Exact solution | IMLS-Ritz method | EFG method |
---|---|---|---|

10 | 0.0001 | 0.0001 | 0.0001 |

11 | 0.0004 | 0.0004 | 0.0004 |

12 | 0.0010 | 0.0010 | 0.0010 |

13 | 0.0027 | 0.0027 | 0.0026 |

14 | 0.0073 | 0.0072 | 0.0074 |

15 | 0.0198 | 0.0197 | 0.0196 |

16 | 0.0539 | 0.0536 | 0.0535 |

17 | 0.1464 | 0.1463 | 0.1462 |

18 | 0.3960 | 0.3958 | 0.3956 |

19 | 1.0392 | 1.0396 | 1.0442 |

20 | 2.3000 | 2.2993 | 2.2989 |

21 | 3.1416 | 3.1408 | 3.1406 |

Numerical solution and exact solution of

Error function

Error function

Error function

The surface of exact solution (Example

The surface of numerical solution with IMLS-Ritz method (Example

Consider the case in (

The IMLS-Ritz method is applied to solve the above equation with penalty factor

The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 81 nodes at

Node number | Exact solution | IMLS-Ritz method | EFG method |
---|---|---|---|

30 | 0.0039 | 0.0037 | 0.0036 |

31 | 0.0070 | 0.0068 | 0.0059 |

32 | 0.0124 | 0.0125 | 0.0131 |

33 | 0.0222 | 0.0231 | 0.0229 |

34 | 0.0395 | 0.0402 | 0.0423 |

35 | 0.0703 | 0.0710 | 0.0722 |

36 | 0.1252 | 0.1251 | 0.1253 |

37 | 0.2228 | 0.2225 | 0.2231 |

38 | 0.3960 | 0.3948 | 0.3952 |

39 | 0.7004 | 0.6978 | 0.6988 |

40 | 1.2212 | 1.2209 | 1.2301 |

41 | 2.0462 | 2.0471 | 2.0483 |

The surface of exact solution (Example

The surface of numerical solution with IMLS-Ritz method (Example

Error function

From these figures, it is shown that numerical results obtained by the IMLS-Ritz method are in good agreement with the exact solutions.

This paper presents a numerical method, named the IMLS-Ritz method, for the one-dimensional sine-Gordon equation. The IMLS approximation is employed to approximate the 1D displacement field. A system of discrete equations is obtained through application of the Ritz minimization. In the IMLS approximation, the basis function is chosen as the orthogonal function system with a weight function. The IMLS approximation has greater computational efficiency and precision than the MLS approximation, and it does not lead to an ill-conditioned system of equations. The numerical results show that the technique is accurate and efficient.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of Ningbo City (Grant nos. 2013A610067, 2102A610023, and 2013A610103), the Natural Science Foundation of Zhejiang Province of China (Grant no. Y6110007), and the National Natural Science of China (Grant no. 41305016).