In this study, a symmetric method of approximate particular solutions (MAPS) is proposed for solving certain partial differential equations (PDEs). Inspired by the unsymmetric MAPS and symmetric radial basis function collocation method (RBFCM), the symmetric MAPS is developed by using the bi-particular solutions of the multiquadrics (MQ). Similar to the unsymmetric MAPS, the right-hand-side of the governing equation is mainly approximated by the MQ in the proposed method. In addition, the system matrix of the prescribed method is symmetric. Numerical examples are solved by the unsymmetric & symmetric RBFCM and MAPS for different problems with different types of governing equations and boundary conditions. Numerical results with different shape parameters are analyzed to show that the symmetric methods are more stable. In addition, the accuracy improvement of the symmetric MAPS is studied. Finally, the stability performance of the symmetric MAPS is further studied for convection-diffusion problems at high Péclet numbers.

在这项研究中，提出了一种近似的特殊解（MAPS）的对称方法来求解某些偏微分方程（PDE）。受非对称MAPS和对称径向基函数配置方法（RBFCM）的启发，通过使用多二次方程（MQ）的双向解来开发对称MAPS。与非对称MAPS相似，在所提出的方法中，控制方程的右侧主要由MQ近似。另外，规定方法的系统矩阵是对称的。通过不对称和对称的RBFCM和MAPS解决了具有不同类型的控制方程和边界条件的不同问题的数值示例。分析了不同形状参数的数值结果，表明对称方法更加稳定。此外，研究了对称MAPS的精度提高。最后，针对高Péclet数下的对流扩散问题，进一步研究了对称MAPS的稳定性能。