Polynomial particular solutions for solving elliptic partial differential equations

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• 作者：Thir Dangal^a ; C.S. Chen^a ; Ji Lin^b ; ^{linji861103@126.com}
• 关键词：Method of particular solutions ; Polynomial basis function ; Multiple scale technique ; Particular solution ; Radial basis functions
• 刊名：Computers & Mathematics with Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：73
• 期：1
• 页码：60-70
• 全文大小：613 K
• 卷排序：73

文摘

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.