Solving partial differential equation by using multiquadric quasi-interpolation

Applied Mathematics and Computation

Volumn 186, Issue 2, 2007, Pages 1502-1510

  Chen, Ronghua ^a   Wu, Zongmin ^b  

^a HUNAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^b FUDAN UNIVERSITY   (China)

Author keywords

Burgers' equation; Interaction; Multiquadric quasi interpolation; Radial basis function; Two wave travelling in opposite direction

Indexed keywords

COMPUTER SIMULATION; DIFFERENCE EQUATIONS; INTERPOLATION; NUMERICAL METHODS; PROBLEM SOLVING; RADIAL BASIS FUNCTION NETWORKS;

BURGERS' EQUATION; INTERACTIONS; MULTIQUADRIC QUASI INTERPOLATION; WAVE TRAVELING IN OPPOSITE DIRECTIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 33947667875     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2006.07.160     Document Type: Article

References (25)

1. Beatson R.K., and Dyn N. Multiquadric B-splines. J. Approx. Theory 87 (1996) 1-24
2. Beatson R.K., and Powell M.J.D. Univariate multiquadric approximation: quasi-interpolation to scattered data. Constr. Approx. 8 (1992) 275-288
3. Chen C.S., Kuhn G., Li J.C., and Mishuris G. Radial basis functions for solving near singular Poisson problems. Commun. Numer. Meth. En. 19 (2003) 333-347
4. Chen R.H., and Wu Z.M. Applied multiquadric quasi-interpolation to solve Burgers' equation. Appl. Math. Comput. 172 (2006) 472-484
5. R.H. Chen, Z.M. Wu, Solving hyperbolic conservation laws using multiquadric quasi-interpolation, Numer. Meth. PDE, in press.
6. Fasshauer G.E. Newton iteration with multiquadrics for the solution of nonlinear PDEs. Comput. Math. Appl. 43 (2002) 423-438
7. Fedoseyev A.I., Friedman M.J., and Kansa E.J. Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method. Int. J. Bifurcat. Chaos 10 (2000) 481-492
8. Friedman M.J. Improved detection of bifurcations in large nonlinear system via the continuation of invariant subspace algorithm. Int. J. Bifurcat. Chaos 11 (2000) 2277-2285
9. Furzeland R.M., Verwer J.G., and Zegeling P.A. A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of line. J. Comput. Phys. 89 (1990) 349-388
10. M.A. Golberg, C.S. Chen, Improved multiqudric approximation for partial differential equations, Eng. Anal. Bound. Elem. 18, 9-17.
11. Hon Y.C. Multiquadric collocation method with adaptive technique for problem with boundary layer. Int. J. Appl. Sci. Comput. 6 (1999) 173-184
12. Hon Y.C., Lu M.W., Xue W.M., and Zhu Y.M. Multiquadric method for the numerical solution of a biphasic model. Appl. Math. Comput. 88 (1997) 153-175
13. Hon Y.C., and Mao X.Z. A multiquadric interpolation method for solving initial value problems. J. Sci. Comput. 12 (1997) 51-55
14. Hon Y.C., and Mao X.Z. An efficient numerical scheme for Burgers' equation. Appl. Math. Comput. 95 (1998) 37-50
15. Hon Y.C., and Wu Z.M. A quasi-interpolation method for solving ordinary differential equations. Int. J. Numer. Meth. Eng. 48 (2000) 1187-1197
16. Hardy R.L. Theory and applications of the multiquadric-biharmonic method, 20 years of discovery 1968-1988. Comput. Math. Appl. 19 (1990) 163-208
17. Kansa E.J. Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics I. Comput. Math. Appl. 19 (1990) 127-145
18. Kansa E.J. Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics II. Comput. Math. Appl. 19 (1990) 147-161
19. Kansa E.J., and Hon Y.C. Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to ellipitic partial differential equations. Comput. Math. Appl. 39 (2000) 123-137
20. Li J.C., and Chen C.S. Some observations on unsymmetric radial basis function collocation methods for convection-diffusion problems. Int. J. Numer. Meth. Eng. 57 (2003) 1085-1094
21. Sharan M., Kansa E.J., and Gupta S. Application of the multiquadric method for numerical solution of elliptic partial differential equations. Appl. Math. Comput. 84 (1997) 275-302
22. Wong A.S.M., Hon Y.C., Li T.S., Chung S.L., and Kansa E.J. Multizone decomposition for simulation of time-dependent problem using the multiquadric scheme. Comput. Math. Appl. 37 (1999) 23-43
23. Wu Z.M. Solving Differential Equation with Radial Basis Function. Advances Computational Mathematics. Lecture Notes in Pure and Applied Mathematics vol. 202 (1999), Dekker pp. 537-544
24. Wu Z.M. Dynamically knots setting in meshless method for solving time dependent propagations equation. Comput. Meth. Appl. Mech. Eng. 193 (2004) 1221-1229
25. Wu Z.M., and Schaback R. Shape preserving properties and convergence of univariate multiquadric quasi-interpolation. ACTA Math. Appl. Sinica 10 (1994) 441-446