Symmetric method of approximate particular solutions for solving certain partial differential equations

Abstract

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.