Volume 10, Issue 5
Polynomial Particular Solutions for the Solutions of PDEs with Variables Coefficients

Jun Lu, Hao Yu, Ji Lin & Thir Dangal

Adv. Appl. Math. Mech., 10 (2018), pp. 1247-1260.

Published online: 2018-07

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  • Abstract

The closed-form particular solutions with polynomial basis functions for general partial differential equations (PDEs) with constant coefficients have been derived and applied for solving various kinds of problems in the context of the method of approximate particular solutions (MAPS). In this paper, we propose to extend the above-mentioned method to PDEs with variable coefficients by the substituting and adding-back technique. Since the linear system derived from the polynomial particular solutions is notoriously ill-conditioned, the multiple scale method is applied to alleviate this difficulty. To validate our proposed method, four numerical examples are considered and compared with those obtained by the MAPS using the radial basis functions.

  • Keywords

Polynomial particular solutions variable coefficients multiple scale methods collocation approach method of approximate particular solutions

  • AMS Subject Headings

65N80 65N35

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COPYRIGHT: © Global Science Press

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@Article{AAMM-10-1247, author = {Jun Lu, Hao Yu, Ji Lin and Thir Dangal}, title = {Polynomial Particular Solutions for the Solutions of PDEs with Variables Coefficients}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {5}, pages = {1247--1260}, abstract = {

The closed-form particular solutions with polynomial basis functions for general partial differential equations (PDEs) with constant coefficients have been derived and applied for solving various kinds of problems in the context of the method of approximate particular solutions (MAPS). In this paper, we propose to extend the above-mentioned method to PDEs with variable coefficients by the substituting and adding-back technique. Since the linear system derived from the polynomial particular solutions is notoriously ill-conditioned, the multiple scale method is applied to alleviate this difficulty. To validate our proposed method, four numerical examples are considered and compared with those obtained by the MAPS using the radial basis functions.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0016}, url = {http://global-sci.org/intro/article_detail/aamm/12597.html} }
TY - JOUR T1 - Polynomial Particular Solutions for the Solutions of PDEs with Variables Coefficients AU - Jun Lu, Hao Yu, Ji Lin & Thir Dangal JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1247 EP - 1260 PY - 2018 DA - 2018/07 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2018-0016 UR - https://global-sci.org/intro/article_detail/aamm/12597.html KW - Polynomial particular solutions KW - variable coefficients KW - multiple scale methods KW - collocation approach KW - method of approximate particular solutions AB -

The closed-form particular solutions with polynomial basis functions for general partial differential equations (PDEs) with constant coefficients have been derived and applied for solving various kinds of problems in the context of the method of approximate particular solutions (MAPS). In this paper, we propose to extend the above-mentioned method to PDEs with variable coefficients by the substituting and adding-back technique. Since the linear system derived from the polynomial particular solutions is notoriously ill-conditioned, the multiple scale method is applied to alleviate this difficulty. To validate our proposed method, four numerical examples are considered and compared with those obtained by the MAPS using the radial basis functions.

Jun Lu, Hao Yu, Ji Lin & Thir Dangal. (1970). Polynomial Particular Solutions for the Solutions of PDEs with Variables Coefficients. Advances in Applied Mathematics and Mechanics. 10 (5). 1247-1260. doi:10.4208/aamm.OA-2018-0016
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