Solution of Cauchy Problems by the Multiple Scale Method of Particular Solutions Using Polynomial Basis Functions

Abstract

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.