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Volume 14, Issue 3
A Comparative Study on Polynomial Expansion Method and Polynomial Method of Particular Solutions

Jen-Yi Chang, Ru-Yun Chen & Chia-Cheng Tsai

Adv. Appl. Math. Mech., 14 (2022), pp. 577-595.

Published online: 2022-02

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  • Abstract

In this study, the polynomial expansion method (PEM) and the polynomial method of particular solutions (PMPS) are applied to solve a class of linear elliptic partial differential equations (PDEs) in two dimensions with constant coefficients. In the solution procedure, the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS, respectively. The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS. Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space. Then, some numerical experiments were conducted to validate the implementation of the PEM and PMPS. Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.

  • Keywords

Pascal polynomial, polynomial expansion method, polynomial method of particular solutions, collocation method, multiple-scale technique.

  • AMS Subject Headings

35C11, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-14-577, author = {Chang , Jen-YiChen , Ru-Yun and Tsai , Chia-Cheng}, title = {A Comparative Study on Polynomial Expansion Method and Polynomial Method of Particular Solutions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {3}, pages = {577--595}, abstract = {

In this study, the polynomial expansion method (PEM) and the polynomial method of particular solutions (PMPS) are applied to solve a class of linear elliptic partial differential equations (PDEs) in two dimensions with constant coefficients. In the solution procedure, the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS, respectively. The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS. Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space. Then, some numerical experiments were conducted to validate the implementation of the PEM and PMPS. Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0385}, url = {http://global-sci.org/intro/article_detail/aamm/20276.html} }
TY - JOUR T1 - A Comparative Study on Polynomial Expansion Method and Polynomial Method of Particular Solutions AU - Chang , Jen-Yi AU - Chen , Ru-Yun AU - Tsai , Chia-Cheng JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 577 EP - 595 PY - 2022 DA - 2022/02 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2020-0385 UR - https://global-sci.org/intro/article_detail/aamm/20276.html KW - Pascal polynomial, polynomial expansion method, polynomial method of particular solutions, collocation method, multiple-scale technique. AB -

In this study, the polynomial expansion method (PEM) and the polynomial method of particular solutions (PMPS) are applied to solve a class of linear elliptic partial differential equations (PDEs) in two dimensions with constant coefficients. In the solution procedure, the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS, respectively. The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS. Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space. Then, some numerical experiments were conducted to validate the implementation of the PEM and PMPS. Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.

Jen-Yi Chang, Ru-Yun Chen & Chia-Cheng Tsai. (2022). A Comparative Study on Polynomial Expansion Method and Polynomial Method of Particular Solutions. Advances in Applied Mathematics and Mechanics. 14 (3). 577-595. doi:10.4208/aamm.OA-2020-0385
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