Solution of Cauchy Problems by the Multiple Scale Method of Particular Solutions Using Polynomial Basis Functions (Communications in Computational Physics, 2018)
We have recently proposed a new meshless method for solving second orderpartial differential equations where the polynomial particular solutions are obtainedanalytically [1]. In this paper, we further extend this new method for the solution ofgeneral two- and three-dimensional Cauchy problems. The resulting system of linearequations is ill-conditioned, and therefore, the solution will be regularized by usinga multiple scale technique in conjunction with the Tikhonov regularization method,while the L-curve approach is used for the determination of a suitable regularizationparameter. Numerical examples including 2D and 3D problems in both smooth andpiecewise smooth geometries are given to demonstrate the validity and applicabilityof the new approach.