TY - JOUR
T1 - Solution of Cauchy Problems by the Multiple Scale Method of Particular Solutions Using Polynomial Basis Functions
JO - Communications in Computational Physics
VL - 5
SP - 1409
EP - 1434
PY - 2018
DA - 2018/06
SN - 24
DO - http://doi.org/10.4208/cicp.OA-2017-0187
UR - https://global-sci.org/intro/article_detail/cicp/12483.html
KW - Method of particular solution, polynomial basis function, multiple scale technique, regularization technique, Cauchy problem.
AB - <p style="text-align: justify;">We have recently proposed a new meshless method for solving second order
partial differential equations where the polynomial particular solutions are obtained
analytically [1]. In this paper, we further extend this new method for the solution of
general two- and three-dimensional Cauchy problems. The resulting system of linear
equations is ill-conditioned, and therefore, the solution will be regularized by using
a multiple scale technique in conjunction with the Tikhonov regularization method,
while the L-curve approach is used for the determination of a suitable regularization
parameter. Numerical examples including 2D and 3D problems in both smooth and
piecewise smooth geometries are given to demonstrate the validity and applicability
of the new approach.</p>