Volume 24, Issue 5
Solution of Cauchy Problems by the Multiple Scale Method of Particular Solutions Using Polynomial Basis Functions

Ji Lin, Yuhui Zhang, Thir Dangal & C. S. Chen

Commun. Comput. Phys., 24 (2018), pp. 1409-1434.

Published online: 2018-06

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  • 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.

  • Keywords

Method of particular solution, polynomial basis function, multiple scale technique, regularization technique, Cauchy problem.

  • AMS Subject Headings

65N21, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-1409, author = {}, title = {Solution of Cauchy Problems by the Multiple Scale Method of Particular Solutions Using Polynomial Basis Functions}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {5}, pages = {1409--1434}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0187}, url = {http://global-sci.org/intro/article_detail/cicp/12483.html} }
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