Abstract
The study of liquid crystals is one of the active areas of physics research. In this paper, the MLPG and direct MLPG (DMLPG) methods are used for the numerical study of the coupled nonlinear sine–Gordon equations in two dimensions arising in the modeling of some phenomena in liquid crystals and superconductors. To approximate numerical integrals in the local weak forms, the MLS and GMLS approximations are used in MLPG and DMLPG methods, respectively. The distribution of regular and scattered points on rectangular and irregular domains has been used to extract the numerical results. By comparing the numerical results, it can be seen that the DMLPG methods are faster, more accurate, and more efficient than the MLPG methods. These are because the GMLS approximation uses the basis polynomials instead of the complex shape functions of the MLS approximation.
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