# zbMATH — the first resource for mathematics

The method of approximate particular solutions for solving certain partial differential equations. (English) Zbl 1242.65267
Summary: A standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the method of approximate particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This leads to a much simpler numerical scheme with similar accuracy to the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples are given with excellent results.

##### MSC:
 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs 35G05 Linear higher-order PDEs
Full Text:
##### References:
 [1] Chen, Symmetric boundary knot method, Eng Anal Bound Elem 26 pp 489– (2002) · Zbl 1006.65500 [2] Chen, New insights into boundary-only and domain-type RBF methods, Int J Nonlinear Sci Numer Simul 1 pp 145– (2000) · Zbl 0954.65084 [3] Li, Trefftz and Collocation Methods (2008) · Zbl 1140.65005 [4] Fairweather, The method of fundamental solutions for elliptic boundary value problems, Adv Comput Math 9 pp 69– (1998) · Zbl 0922.65074 [5] Golberg, Boundary Integral Methods: Numerical and Mathematical Aspects pp 103– (1998) [6] Golberg, A mesh free method for solving nonlinear reaction-diffusion equations, Comput Model Eng Sci 2 pp 87– (2001) [7] Ingber, A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations, Int J Numer Methods Engrg 60 pp 2183– (2004) · Zbl 1178.76276 [8] Reutskiy, A boundary meshless method using Chebyshev interpolation and trigonometric basis function for solving heat conduction problems, Int J Numer Methods Eng 74 pp 1621– (2008) · Zbl 1191.80044 [9] Atkinson, The numerical evaluation of particular solutions for Poisson’s equation, IMA J Numer Anal 5 pp 319– (1985) · Zbl 0576.65114 [10] Cheng, Particular solutions of Laplacian, Helmholtz-type, and polyharmonic operators involving higher order radial basis functions, Eng Analy Bound Elem 24 pp 531– (2000) · Zbl 0966.65088 [11] Cho, Trefftz methods for time dependent partial differential equations, Comput, Mater, Continua 1 pp 1– (2004) [12] Cho, Some comments on mitigating the ill-conditioning of the method of fundamental solutions, Eng Anal Bound Elem 30 pp 405– (2006) · Zbl 1187.65136 [13] Duchon, Lecture Notes in Mathematics 571 pp 85– (1976) [14] Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics - II, Comput Math Appl 19 pp 147– (1990) · Zbl 0850.76048 [15] Alves, Advances in Meshfree Techniques pp 241– (2007) · Zbl 1323.65122 [16] Valtchev, A time-marching MFS scheme for heat conduction problems, Eng Analysis Bound Elem 32 pp 480– (2008) · Zbl 1244.80025 [17] Karageorghis, Efficient Kansa-type MFS algorithm for elliptic problems, Numer Algorithm 54 pp 261– (2010) · Zbl 1190.65184 [18] Muleshkov, Particular solutions of Helmholtz-type operators using higher order polyharmonic splines, Comp Mech 23 pp 411– (1999) · Zbl 0938.65139 [19] Chen, Derivation of particular solution using Chebyshev polynomial based functions, Int J Comput Methods 4 pp 15– (2007) · Zbl 1198.65245 [20] Tsai, Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators, Eng Analysis Bound Elem 33 pp 514– (2009) · Zbl 1244.65209 [21] Yao, A revisit on the derivation of the particular solution for the differential operator {$$\Delta$$}2 {$$\pm$$} {$$\lambda$$}2, Adv Appl Math Mech 1 pp 750– (2009) · Zbl 1262.35086 [22] Cheng, Dual reciprocity BEM based on global interpolation functions, Eng Anal Bound Elem 13 pp 303– (1994) [23] Yao, The comparison of three meshless methods using radial basis functions for solving fourth-order partial differential equations, Eng Anal Bound Elem 34 pp 625– (2010) · Zbl 1267.65198 [24] Muleshkov, Adv Comput Eng Sci pp 27– (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.