Commun. Comput. Phys., 5 (2009), pp. 942-958.

The Recursive Formulation of Particular Solutions for Some Elliptic PDEs with Polynomial Source Functions

J. Ding 1, H. Y. Tian 1, C. S. Chen 1*

1 Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA.

Received 2 April 2008; Accepted (in revised version) 3 September 2008
Available online 14 October 2008


In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations. We first approximate the source function by Chebyshev polynomials. We then focus on how to find a polynomial particular solution when the source function is a polynomial. Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined, the coefficients of the particular solution satisfy a triangular system of linear algebraic equations. Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs. The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs. Numerical results show that our approach is efficient and accurate.

AMS subject classifications: 35J05, 35J25, 65D05, 65D15

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Key words: The method of fundamental solutions, particular solution, Helmholtz equation, Chebyshev polynomial, Laplace-Helmholtz equation, convection-reaction equation.

*Corresponding author.
Email: (J. Ding), (H. Y. Tian), (C. S. Chen)

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