Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 613082, 11 pages
http://dx.doi.org/10.1155/2013/613082
Research Article

Analytical Particular Solutions of Multiquadrics Associated with Polyharmonic Operators

Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan

Received 9 January 2013; Accepted 29 March 2013

Academic Editor: Kue-Hong Chen

Copyright © 2013 Chia-Cheng Tsai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces,” Journal of Geopphysical Research, vol. 76, pp. 1905–1915, 1971. View at Google Scholar
  2. J. Duchon, “Splines minimizing rotation-invariant semi-norms in Sobolev spaces,” in Constructive Theory of Functions of Several Variables, vol. 571 of Lecture Notes in Mathematics, pp. 85–100, Springer, Berlin, Germany, 1977. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree,” Advances in Computational Mathematics, vol. 4, no. 4, pp. 389–396, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. E. J. Kansa, “Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates,” Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 127–145, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. E. J. Kansa, “Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations,” Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 147–161, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. D. Nardini and C. A. Brebbia, “A new approach to free vibration analysis using boundary elements,” Applied Mathematical Modelling, vol. 7, no. 3, pp. 157–162, 1983. View at Google Scholar · View at Scopus
  7. A. Bogomolny, “Fundamental solutions method for elliptic boundary value problems,” SIAM Journal on Numerical Analysis, vol. 22, no. 4, pp. 644–669, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. G. Fairweather and A. Karageorghis, “The method of fundamental solutions for elliptic boundary value problems,” Advances in Computational Mathematics, vol. 9, no. 1-2, pp. 69–95, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. A. Golberg, “The method of fundamental solutions for Poisson's equation,” Engineering Analysis with Boundary Elements, vol. 16, no. 3, pp. 205–213, 1995. View at Google Scholar · View at Scopus
  10. V. D. Kupradze and M. A. Aleksidze, “The method of functional equations for the approximate solution of certain boundary-value problems,” USSR Computational Mathematics and Mathematical Physics, vol. 4, no. 4, pp. 82–126, 1964. View at Google Scholar · View at MathSciNet
  11. R. Mathon and R. L. Johnston, “The approximate solution of elliptic boundary-value problems by fundamental solutions,” SIAM Journal on Numerical Analysis, vol. 14, no. 4, pp. 638–650, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. C.-S. Liu, “A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain,” Computer Modeling in Engineering & Sciences, vol. 20, no. 2, pp. 111–122, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. C. S. Liu, “A highly accurate MCTM for direct and inverse problems of biharmonic equation in arbitrary plane domains,” Computer Modeling in Engineering & Sciences, vol. 30, no. 2, pp. 65–75, 2008. View at Google Scholar · View at Scopus
  14. M. A. Golberg and C. S. Chen, “The theory of radial basis functions applied to the bem for inhomogeneous partial differential equations,” Boundary Elements Communications, vol. 5, pp. 57–61, 1994. View at Google Scholar
  15. S. R. Karur and P. A. Ramachandran, “Radial basis function approximation in the dual reciprocity method,” Mathematical and Computer Modelling, vol. 20, no. 7, pp. 59–70, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. A. Golberg, “Numerical evaluation of particular solutions in the BEM—a review,” Boundary elements communications, vol. 6, no. 3, pp. 99–106, 1995. View at Google Scholar · View at Scopus
  17. C. S. Chen, “The method of fundamental solutions for non-linear thermal explosions,” Communications in Numerical Methods in Engineering, vol. 11, no. 8, pp. 675–681, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. A. Golberg, C. S. Chen, and S. R. Karur, “Improved multiquadric approximation for partial differential equations,” Engineering Analysis with Boundary Elements, vol. 18, no. 1, pp. 9–17, 1996. View at Publisher · View at Google Scholar · View at Scopus
  19. M. A. Golberg and C. S. Chen, Eds., The Method of Fundamental Solutions for Potential, Helmholtz and Diffusion Problems, Computational Mechanics Publications, Southampton, UK, 1999.
  20. M. F. Samaan and Y. F. Rashed, “BEM for transient 2D elastodynamics using multiquadric functions,” International Journal of Solids and Structures, vol. 44, no. 25-26, pp. 8517–8531, 2007. View at Publisher · View at Google Scholar · View at Scopus
  21. M. F. Samaan and Y. F. Rashed, “Free vibration multiquadric boundary elements applied to plane elasticity,” Applied Mathematical Modelling, vol. 33, no. 5, pp. 2421–2432, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. A. H. D. Cheng, C. S. Chen, M. A. Golberg, and Y. F. Rashed, “BEM for theomoelasticity and elasticity with body force—a revisit,” Engineering Analysis with Boundary Elements, vol. 25, no. 4-5, pp. 377–387, 2001. View at Publisher · View at Google Scholar · View at Scopus
  23. G. Yao, The method of approximate particular solutions for solving partial differential equations [Ph.D. thesis], Department of Mathematics, The University of Southern Mississippi, 2010.
  24. C.-C. Tsai, “Automatic particular solutions of arbitrary high-order splines associated with polyharmonic and poly-Helmholtz equations,” Engineering Analysis with Boundary Elements, vol. 35, no. 7, pp. 925–934, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  25. S. Wolfram, The Mathematica Book, Cambridge University Press, Cambridge, UK, 1996. View at MathSciNet
  26. H. Wendland, “On the smoothness of positive definite and radial functions,” Journal of Computational and Applied Mathematics, vol. 101, no. 1-2, pp. 177–188, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet