The method of fundamental solutions with eigenfunctions expansion method for 3D nonhomogeneous diffusion equations

Numerical Methods for Partial Differential Equations

Volumn 25, Issue 1, 2009, Pages 195-211

  Young, D L ^a   Chen, C H ^a   Fan, C M ^a   Shen, L H ^a  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

3D nonhomogeneous diffusion equations; Eigenfunctions expansion method; Method of fundamental solutions; Method of particular solutions; Singular value decomposition

Indexed keywords

EID: 58449093808     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20336     Document Type: Article

References (25)

1. 1 S. P. Zhu, Solving transient diffusion problems: time‐dependent fundamental solution approaches versus LTDRM approaches, Eng Anal Bound Elem 21 (1998), 87–90.
2. 2 M. Zerroukat, A boundary element scheme for diffusion problems using compactly supported radial basis functions, Eng Anal Bound Elem 23 (1999), 201–209.
3. 3 A. Sutradhar, G. H. Paulino, and L. J. Gray, Transient heat conduction in homogeneous and nonhomogeneous materials by the Laplace transform Galerkin boundary element method, Eng Anal Bound Elem 26 (2002), 119–132.
4. 4 D. Nardini and C. A. Brebbia, A new approach to free vibrations analysis using boundary elements. Boundary element in engineering, C. A. Brebbia, editor, CMP, Southamptom and Springer, Berlin 1982, pp. 312–326.
5. 5 C. S. Chen, M. A. Golberg, and Y. C. Hon, The method of fundamental solutions and quasi‐Monte‐Carlo method for diffusion equations, Int J Numer Methods Eng 43 (1998), 1421–1435.
6. 6 C. S. Chen, M. A. Golberg, and Y. F. Rashed, A mesh free method for linear diffusion equations, Numer Heat Transf B 33 (1998), 469–486.
7. 7 M. A. Golberg and C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, Boundary integral methods: Numerical and mathematical aspects, Computational Mechanics Publications, Boston, 1999.
8. 8 K. Balakrishnan and P. A. Ramachandran, The method of fundamental solutions for linear diffusion‐reaction equations, Math Comput Model 31 (2002), 221–237.
9. 9 D. L. Young, C. C. Tsai, K. Murugesan, C. M. Fan, and C. W. Chen, Time‐dependent fundamental solutions for homogeneous diffusion problems, Eng Anal Bound Elem 28 (2004), 1463–1473.
10. 10 D. L. Young, C. C. Tsai, and C. M. Fan, Direct approach to solve nonhomogeneous diffusion problems using fundamental solutions and dual reciprocity methods, J Chin Inst Eng 27 (2004), 597–609.
11. 11 D. L. Young, C. M. Fan, C. C. Tsai, C. W. Chen, and K. Murugesan, Eulerian‐Lagrangian method of fundamental solutions for multi‐dimensional advection‐diffusion equation, Int Math Forum 1 (2006), 687–706.
12. 12 D. L. Young, S. P. Hu, C. M. Fan, and C. W. Chen, Method of fundamental solutions and Cole‐Hopf transformation for Burgers' equations, Comput Fluid Dyn J 14 (2006), 454–467.
13. 13 D. L. Young, C. M. Fan, S. P. Hu, and S. N. Atluri, The Eulerian‐Lagrangian method of fundamental solutions for two‐dimensional unsteady Burgers' equations, Eng Anal Bound Elem (doi: 10.1016/ j.enganabound.2007.08.011).
14. 14 Z. Yao and G. F. Margrave, Wavefield extrapolation in laterally inhomogeneous media by generalized eigenfunction transform method, CREWES Res Rep 11, 1999.
15. 15 D. L. Young, C. W. Chen, C. M. Fan, and C. C. Tsai, The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation, Numer Meth Part Differ Equat 22 (2006), 1173–1196.
16. 16 J. T. Chen, H.‐K. Hong, C. S. Yeh, and S. W. Chyuan, Integral representations and regularizations for a divergent series solution of a beam subjected to support motions, Earthquake Eng Struct Dyn 25 (1996), 909–925.
17. 17 M. A. Golberg, The method of fundamental solutions for Poisson's equation, Eng Anal Bound Elem 16 (1995), 205–213.
18. 18 C. S. Chen and R. A. Schaback, Recent developments of the dual reciprocity method using compactly supported radial basis functions, Transformation of the domain effects to the boundary, editor, Y. F. Rashed, WIT Press, 2003, pp. 183–225.
19. 19 W. Madych, Miscellaneous error bounds for multiquadric and related interpolants, Comput Math Appl 24 (1992), 121–138.
20. 20 C. S. Chen, C. A. Brebbia, and H. Power, Dual reciprocity method using compactly supported radial basis functions, Commun Numer Methods Eng 15 (1999), 137–150.
21. 21 A. H.‐D. Cheng, D. L. Young, and C. C. Tsai, Solution of Poisson's equation by iterative DRBEM using compactly‐supported, positive‐definite radial basis function, Eng Anal Bound Elem 24 (2000), 549–557.
22. 22 D. L. Young, C. C. Tsai, and A. H.‐D. Cheng, Solution of Stokes flow using an iterative DRBEM based on compactly‐supported, positive‐definite radial basis function, Comput Math Appl 43 (2002), 607–619.
23. 23 J. T. Chen, I. L. Chen, and Y. T. Lee, Eigensolutions of multiply connected membranes using the method of fundamental solutions, Eng Anal Bound Elem 29 (2005), 166–174.
24. 24 C. C. Tsai, D. L. Young, C. W. Chen, and C. M. Fan, The method of fundamental solutions for eigenproblems in domains with and without interior holes, Proc R Soc A 462 (2006), 1443–1466.
25. 25 D. L. Young, S. P. Hu, C. W. Chen, C. M. Fan, and K. Murugesan, Analysis of elliptical waveguides by the method of fundamental solutions, Microw Opt Technol Lett 44 (6) (2005), 552–558.