Solving high order ordinary differential equations with radial basis function networks

International Journal for Numerical Methods in Engineering

Volumn 62, Issue 6, 2005, Pages 824-852

  Mai Duy, Nam ^a  

^a UNIVERSITY OF SYDNEY   (Australia)

Author keywords

Approximation; Derivatives; High order; Ordinary differential equations; RBFNs

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIATION (CALCULUS); INTEGRATION; PROBLEM SOLVING; RADIAL BASIS FUNCTION NETWORKS;

DIFFERENTIATION PROCESS; HIGH RATE OF CONVERGENCE; NUMERICAL SOLUTION;

ORDINARY DIFFERENTIAL EQUATIONS;

DIFFERENTIAL EQUATION; MATHEMATICAL METHOD;

EID: 13244255578     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.1220     Document Type: Article

References (33)

1. Dahlquist G, Bjorck A, Anderson N. Numerical Methods. Prentice-Hall: Englewood Cliffs, NJ, 1974.
2. Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press: Cambridge, 1988.
3. Viecelli JA. Exponential difference operator approximation for the sixth order Onsager equation. Journal of Computational Physics 1983; 50:162-170.
4. Sallam S, El-Hawary HM. A deficient spline function approximation of second-order differential equations. Applied Mathematical Modelling 1984; 8(6):408-412.
5. Esmail MN, Fawzy T, Ahmed M, Elmoselhi HO. A deficient spline function approximation to fourth-order differential equations. Applied Mathematical Modelling 1994; 18(12):658-664.
6. Gutierrez RH, Laura PAA. Vibrations of non-uniform rings studied by means of the differential quadrature method. Journal of Sound and Vibration 1995; 185(3):507-513.
7. Wu TY, Liu GR. Application of the generalized differential quadrature rule to sixth-order differential equations. Communications in Numerical Methods in Engineering 2000; 16:777-784.
8. Wu TY, Liu GR. The generalized differential quadrature rule for fourth-order differential equations. International Journal for Numerical Methods in Engineering 2001; 50:1907-1929.
9. Wu TY, Liu GR. Application of the generalized differential quadrature rule to eighth-order differential equations. Communications in Numerical Methods in Engineering 2001; 17:355-364.
10. Bellman R, Casti J. Differential quadraturre and long term integration. Journal of Mathematical Analysis and Application 1971; 34 235-238.
11. Wood HG, Morton JB. Onsager's pancake approximation for the fluid dynamics of a gas centrifuge. Journal of Fluid Mechanics 1980; 101(1):1-31.
12. Kansa EJ. Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers and Mathematics with Applications 1990; 19(8,9):147-161.
13. Sharan M, Kansa EJ, Gupta S. Application of the multiquadric method for numerical solution of elliptic partial differential equations. Journal of Applied Science and Computation 1997; 84:275-302.
14. Zerroukat M, Power H, Chen CS. A numerical method for heat transfer problems using collocation and radial basis functions. International Journal for Numerical Methods in Engineering 1998; 42:1263-1278.
15. Mai-Duy N, Tran-Cong T. Numerical solution of differential equations using multiquadric radial basis function networks. Neural Networks 2001; 14(2):185-199.
16. Mai-Duy N, Tran-Cong T. Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks. International Journal for Numerical Methods in Fluids 2001; 37:65-86.
17. Fedoseyev AI, Friedman MJ, Kansa EJ. Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers and Mathematics with Applications 2002; 43 3-5): 439-455.
18. Mai-Duy N, Tran-Cong T. Approximation of function and its derivatives using radial basis function network methods. Applied Mathematical Modelling 2003; 27:197-220.
19. Jin X, Li G, Aluru NR. On the equivalence between least-squares and kernel approximations in meshless methods. Computer Modelling in Engineering and Sciences 2001; 2(4):447-462.
20. Atluri SN, Shen S. The Meshless Local Petrov-Galerkin Method. Tech Science Press: Encino, 2002.
21. Li S, Liu WK. Meshfree and particle methods and their applications. Applied Mechanics Reviews 2002; 55(1):1-34.
22. Liu GR. Mesh Free Methods-Moving Beyond the Finite Element Method. CRC Press: Boca Raton, 2003.
23. Liu GR, Gu YT. A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. Journal of Sound and Vibration 2001; 246(1):29-46.
24. Wang JG, Liu GR. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering 2002; 54:1623-1648.
25. Wang JG, Liu GR. On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Computer Methods in Applied Mechanics and Engineering 2002; 191:2611-2630.
26. Wang JG, Liu GR, Lin P. Numerical analysis of Biot's consolidation process by radial point interpolation method. International Journal of Solids and Structures 2002; 39(6):1557-1573.
27. Gu YT, Liu GR. A boundary radial point interpolation method (BRPIM) for 2-D structural analyses. Structural Engineering and Mechanics 2003; 15(5):535-550.
28. Wu YL, Liu GR. A meshfree formulation of local radial point interpolation method (LRPIM) for incompressible flow simulation. Computational Mechanics 2003; 30(5,6):355-365.
29. Haykin S. Neural Nenvorks: A Comprehensive Foundation. Prentice-Hall: Englewood Cliffs, NJ, 1999.
30. Abramowitz M, Stegun IA. Handbook of Mathematical Functions. Dover Publications: New York, 1972.
31. Madych WR, Nelson SA. Multivariate interpolation and conditionally positive definite functions. Approximation Theory and its Applications 1989; 4:77-89.
32. Madych WR, Nelson SA. Multivariate interpolation and conditionally positive definite functions, II. Mathematics of Computation 1990; 54(189):211-230.
33. Kelkar VS, Sewell RT. Fundamentals of the Analysis and Design of Shell Structures. Prent-Hall Inc.: Englewood Cliffs, NJ, 1987.