Polynomial particular solutions for solving elliptic partial differential equations

• Published in: Computers & mathematics with applications (1987) Vol. 73; no. 1; pp. 60 - 70
• Main Authors: Dangal, Thir, Chen, C.S., Lin, Ji
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.01.2017; Elsevier BV
• Subjects: Algorithms; Basis functions; Chebyshev approximation; Constants; Convection; Functions (mathematics); Ill-conditioned problems (mathematics); Mathematical analysis; Mathematical models; Method of particular solutions; Multiple scale technique; Numerical analysis; Operators (mathematics); Partial differential equations; Particular solution; Polynomial basis function; Polynomials; Radial basis functions; Studies; Multiple scale technique; Polynomial basis function; Radial basis functions; Particular solution; Method of particular solutions
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2016.10.024

In the past, polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this paper, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solution is further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. The polynomial basis functions are well-known for yielding ill-conditioned systems when their order becomes large. The multiple scale technique is applied to circumvent the difficulty of ill-conditioning problem. Five numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.