The role of the multiquadric shape parameters in solving elliptic partial differential equations

Computers and Mathematics with Applications

Volumn 51, Issue 8 SPEC. ISS., 2006, Pages 1335-1348

  Wertz, J ^a   Kansa, E J ^b   Ling, L ^c  

^a Aeronautical Engineering University of California ^*  (United States)

^b University of California   (United States)

^c UNIVERSITY OF TOKYO   (Japan)

Author keywords

Different shape parameters; Elliptic PDEs; Generalized multiquadrics

Indexed keywords

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; HEURISTIC METHODS; NUMERICAL METHODS; PARAMETER ESTIMATION; PROBLEM SOLVING;

DATA CENTERS; DIFFERENT SHAPE PARAMETERS; ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS; GENERALIZED MULTIQUADRICS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 33746166037     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2006.04.009     Document Type: Article

References (25)

1. Fedoseyev A.I., Friedman M.J., and Kansa E.J. Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers Math. Applic. 43 3-5 (2002) 491-500
2. Cheng A.H.D., Golberg M.A., Kansa E.J., and Zammito T. Exponential convergence and h-c multiquadric collocation method for partial differential equations. Num. Meth. PDEs 19 (2003) 571-594
3. Wu Y.L., and Shu C. Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli. Comput. Mech. 29 (2002) 477-485
4. Shu C., Ding H., and Yee K.S. Local radial basis function based differential quadrature method and its applications to solve two-dimensional incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Engng. 192 (2003) 941-954
5. Madych W.R., and Nelson S.A. Multivariate interpolation and conditionally positive definite functions, Ap-prox. Theory Appl. 4 (1988) 77-89
6. Madych W.R. Miscellaneous error bounds for multiquadric and related interpolators. Computers Math. Applic. 24 12 (1992) 121-138
7. Kansa E.J. Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics: H. Solutions to parabolic, hyperbolic, and elliptic partial differential equations. Computers Math. Applic. 19 6-8 (1990) 147-161
8. Golberg M.A., Chen C.S., and Karur S.R. Improved multiquadric approximation for partial differential equations. Eng. Anal. Bound. Elem. 18 (1996) 9-17
9. Hardy R.L. Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76 (1971) 1905-1915
10. Foley T.A. Interpolation and approximation of 3-D and 4-D scattered data. Computers Math. Applic. 13 8 (1987) 711-740
11. Carlson R.E., and Foley T.A. The parameter R2 in multiquadric interpolation. Computers Math. Applic. 21 9 (1991) 29-42
12. Kansa E.J., and Carlson R.E. Improved accuracy of multiquadric interpolation using variable shape parameters. Computers Math. Applic. 24 12 (1992) 99-120
13. Rippa S. An algorithm for selecting a good value for the parameter c in radial basis functions interpolation. Adv. Comp. Math. 11 (1999) 193-210
14. Kansa E.J., and Hon Y.C. Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations. Computers Math. Applic. 39 7/8 (2000) 123-137
15. Driscoll T.A., and Fornberg B. Interpolation in the limit of increasingly.at radial basis functions. Computers Math. Applic. 43 3-5 (2002) 413-422
16. Fornberg B., Wright G., and Larsson E. A numerical study of some radial basis function based solution methods for elliptic PDEs. Computers Math. Applic. 46 5/6 (2003) 891-902
17. Fornberg B., Wright G., and Larsson E. Some observations regarding interpolants in the limit of at radial basis functions. Computers Math. Applic. 47 1 (2004) 37-55
18. Larsson E., and Fornberg B. Theoretical and computational aspects of multivariate interpolation with increasingly at radial basis functions. Computers Math. Applic. 49 1 (2005) 103-130
19. Wang J.G., and Liu G.R. On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Comput. Meth. Appl. Mech. Eng. 191 (2002) 2611-2630
20. Xiao J.R., and McCarthy M.A. A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions. Computat. Mech. 31 (2003) 301-315
21. Xiao J.R., Gama B.A., Gillespie Jr. J.W., and Kansa E.J. Meshless solutions of 2D contact problems by subdomain variational inequality and MLPG method with radial basis functions. Eng. Anal. Bound. Elem. 29 (2005) 95-106
22. Ling L., and Kansa E.J. A least squares preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 23 (2005) 31-54
23. Ling L., and Kansa E.J. Preconditioning for radial basis functions with domain decomposition. Mathl. Comput. Modelling 40 13 (2004) 1413-1427
24. Brown D., Ling L., Kansa E.J., and Levesley J. On approximate cardinal preconditioning methods for radial functions. Eng. Anal. Bound. Elem. 29 (2005) 343-353
25. Ling L., and Hon Y.C. Improved numerical solver for Kansa's method based on affine space decomposition. Eng. Anal. Bound. Elem. 29 (2005) 1077-1085