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Volume 1, Issue 6
A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆2 ± λ2

Guangming Yao, C. S. Chen & Chia Cheng Tsai

DOI: 10.4208/aamm.09-m09S01

Adv. Appl. Math. Mech., 1 (2009), pp. 750-768.

Published online: 2009-01

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  • Abstract

In this paper, we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆2 ± λ2. Similar to the derivation of fundamental solutions, it is non-trivial to derive particular solutions for higher order differential operators. In this paper, we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D. The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration. Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.

  • Keywords

The method of fundamental solutions, radial basis functions, meshless methods, polyharmonic splines, the method of particular solutions.

  • AMS Subject Headings

35J05, 35J25, 65D05, 65D15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-1-750, author = {Yao , GuangmingChen , C. S. and Tsai , Chia Cheng}, title = {A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆2 ± λ2}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {6}, pages = {750--768}, abstract = {

In this paper, we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆2 ± λ2. Similar to the derivation of fundamental solutions, it is non-trivial to derive particular solutions for higher order differential operators. In this paper, we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D. The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration. Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m09S01}, url = {http://global-sci.org/intro/article_detail/aamm/8395.html} }
TY - JOUR T1 - A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆2 ± λ2 AU - Yao , Guangming AU - Chen , C. S. AU - Tsai , Chia Cheng JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 750 EP - 768 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/10.4208/aamm.09-m09S01 UR - https://global-sci.org/intro/article_detail/aamm/8395.html KW - The method of fundamental solutions, radial basis functions, meshless methods, polyharmonic splines, the method of particular solutions. AB -

In this paper, we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆2 ± λ2. Similar to the derivation of fundamental solutions, it is non-trivial to derive particular solutions for higher order differential operators. In this paper, we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D. The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration. Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.

Yao , GuangmingChen , C. S. and Tsai , Chia Cheng. (2009). A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆2 ± λ2. Advances in Applied Mathematics and Mechanics. 1 (6). 750-768. doi:10.4208/aamm.09-m09S01
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