TY - JOUR
T1 - A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆<sup>2</sup> ± λ<sup>2</sup>
AU - Yao , Guangming
AU - Chen , C. S.
AU - Tsai , Chia Cheng
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 750
EP - 768
PY - 2009
DA - 2009/01
SN - 1
DO - http://doi.org/10.4208/aamm.09-m09S01
UR - https://global-sci.org/intro/article_detail/aamm/8395.html
KW - The method of fundamental solutions, radial basis functions, meshless methods, polyharmonic splines, the method of particular solutions.
AB - <p style="text-align: justify;">In this paper, we applied the polyharmonic splines as the basis functions
to derive particular solutions for the differential operator ∆<sup>2</sup> ± λ<sup>2</sup>. Similar to the
derivation of fundamental solutions, it is non-trivial to derive particular solutions
for higher order differential operators. In this paper, we provide a simple algebraic
factorization approach to derive particular solutions for these types of differential
operators in 2D and 3D. The main focus of this paper is its simplicity in the sense
that minimal mathematical background is required for numerically solving higher
order partial differential equations such as thin plate vibration. Three numerical
examples in both 2D and 3D are given to validate particular solutions we derived.</p>