(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we combine the Local Method of Approximate Particular Solutions (LMAPS) with polynomial basis functions to solve near-singular problems. Due to the unique feature of the local approach, the LMAPS is capable of capturing the rapid variation of the solution. Polynomial basis functions can become very unstable when the order of the polynomial becomes large. However, since the LMAPS is a local method, the order does not need to be very high; an order of 5 can achieve sufficient accuracy. In order to show the effectiveness of the LMAPS using polynomial basis functions for solving near singular problems, we compare the results with the LMAPS using radial basis functions (RBFs). The advantage of using polynomials as a basis rather than RBFs is that finding an appropriate shape parameter is not necessary.