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} }
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://dor.org/10.4208/cicp.OA-2017-0187 UR - https://global-sci.org/intro/cicp/12483.html KW - Method of particular solution, polynomial basis function, multiple scale technique, regularization technique, Cauchy problem. AB -

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.

Ji Lin, Yuhui Zhang, Thir Dangal & C. S. Chen. (2020). Solution of Cauchy Problems by the Multiple Scale Method of Particular Solutions Using Polynomial Basis Functions. Communications in Computational Physics. 24 (5). 1409-1434. doi:10.4208/cicp.OA-2017-0187
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