The lie-group shooting method for singularly perturbed two-point boundary value problems

CMES - Computer Modeling in Engineering and Sciences

Volumn 15, Issue 3, 2006, Pages 179-196

  Liu, Chein Shan ^a  

^a NATIONAL TAIWAN OCEAN UNIVERSITY   (Taiwan)

Author keywords

Boundary layer; Ill posed equation; Lie group shooting method; One step group preserving scheme; Singularly perturbed boundary value problem; Stiff equation

Indexed keywords

COMPUTATIONAL METHODS; FINITE DIFFERENCE METHOD; MATHEMATICAL TRANSFORMATIONS; NUMERICAL METHODS; PERTURBATION TECHNIQUES; STIFFNESS MATRIX;

ILL POSED EQUATION; LIE GROUP SHOOTING METHOD; ONE STEP GROUP PRESERVING SCHEME; SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM; STIFF EQUATION;

BOUNDARY VALUE PROBLEMS;

EID: 33845427623     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (32)

1. Ascher, U. M.; Mattheij, R. M. M.; Russell, R. D. (1995): Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia.
2. Bender, C. M.; Orszag, S. A. (1978): Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York.
3. Carrier, G.; Pearson, C. (1991): Ordinary Differential Equations. SIAM, Philadelphia.
4. De Jager, E. M.; Jiang, F. R. (1996): The Theory of Singular Perturbation. Noth-Holland, Amsterdam.
5. Gasparo, M. G.; Macconi, M (1989): New initial-value method for singularly perturbed boundary value problems. J. Optim. Theory Appl., vol. 63, pp. 213-224.
6. Gasparo, M. G.; Macconi, M. (1990): Initial-value methods for second-order singularly perturbed boundary value problems. J. Optim. Theory Appl., vol. 66, pp. 197-210.
7. Ilicasu, F. O.; Schultz, D. H. (2004): High-order finite-difference techniques for linear singular perturbation boundary value problems. Int. J. Comput. Math. Appl., vol. 47, pp. 391-417.
8. Kadalbajoo, M. K.; Patidar, K. C. (2002): A survey of numerical techniques for solving singularly perturbed ordinary differential equations. Appl. Math. Comp., vol. 130, pp. 457-510.
9. Kadalbajoo, M. K.; Reddy, Y. N. (1986): Initial-value technique for a class of nonlinear singular perturbation problems. J. Optim. Theory Appl., vol. 53, pp. 395-406.
10. Keller, H. B. (1992): Numerical Methods for Two-point Boundary Value Problems. Dover, New York.
11. Kevorkian, J.; Cole, J. D. (1981): Perturbation Methods in Applied Mathematics. Springer-Verlag, New York.
12. Kevorkian, J.; Cole, J. D. (1996): Multiple Scale and Singular Perturbation Methods. Springer-Verlag, New York.
13. Kubicek, M.; Hlavacek, V. (1983): Numerical Solution of Nonlinear Boundary Value Problems with Applications. Prentice-Hall, New York.
14. Li, Z.; Wang, W. (2005): Mechanization for solving SPP by reducing order method. Appl. Math. Comput., vol. 169, pp. 1028-1037.
15. Liu, C.-S. (2001): Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Mech., vol. 36, pp. 1047-1068.
16. Liu, C.-S. (2005): Nonstandard group-preserving schemes for very stiff ordinary differential equations. CMES: Computer Modeling in Engineering & Sciences, vol. 9, pp. 255-272.
17. Liu, C.-S. (2006a): Preserving constraints of differential equations by numerical methods based on integrating factors. CMES: Computer Modeling in Engineering & Sciences, vol. 12, pp. 83-108.
18. Liu, C.-S. (2006b): The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions. CMES: Computer Modeling in Engineering & Sciences, vol. 13, pp. 149-163.
19. Liu, C.-S. (2006c): One-step GPS for the estimation of temperature-dependent thermal conductivity. Int. J. Heat Mass Transf., vol. 49, pp. 3084-3093.
20. Liu, C.-S. (2006d): An efficient simultaneous estimation of temperature-dependent thermophysical properties. CMES: Computer Modeling in Engineering & Sciences, vol. 14, pp. 77-90.
21. Liu, C.-S. (2006e): An efficient backward group preserving scheme for the backward in time Burgers equation. CMES: Computer Modeling in Engineering & Sciences, vol. 12, pp. 55-65.
22. Liu, C.-S. (2006f): A group preserving scheme for Burgers equation with very large Reynolds number. CMES: Computer Modeling in Engineering & Sciences, vol. 12, pp. 197-211.
23. Liu, C.-S.; Chang, C.-W.; Chang, J.-R. (2006): Past cone dynamics and backward group preserving schemes for backward heat conduction problems. CMES: Computer Modeling in Engineering & Sciences, vol. 12, pp. 67-81.
24. Natesan, S.; Ramanujam, M. (1998): Initial-value technique for singularly-perturbed turning-point problems exhibiting twin boundary layers. J. Optim. Theory Appl., vol. 99, pp. 37-52.
25. Nayfeh, A.M. (1981): Introduction to Perturbation Techniques. Wiley, New York.
26. O'Malley, R. E. (1991): Singular Perturbation Methods for Ordinary Differential Equations. Spring-Verlag, New York.
27. Reddy, Y. N.; Chakravarthy, P. P. (2004): An initial-value approach for solving singularly perturbed two-point boundary value problems. Appl. Math. Comput., vol. 155, pp. 95-110.
28. Roberts, S. M. (1982): A boundary-value technique for singular perturbation problems. J. Math. Anal. Appl., vol. 87, pp. 489-503.
29. Roos, H. G.; Stynes, M.; Tobiska, L. (1996): Numerical Methods for Singularly Perturbed Differential Equations. Spring-Verlag, Berlin.
30. Shampine, L. F.; Gear, C. W. (1979): A user's view of solving stiff ordinary differential equations. SIAM Rev., vol. 21, pp. 1-17.
31. Varner, T. N.; Choudhury, S. R. (1998): Non-standard difference schemes for singular perturbation problems revisited. Appl. Math. Comput., vol. 92, pp. 101-123.
32. Wang, L. (2004): A novel method for a class of nonlinear singular perturbation problems. Appl. Math. Comput., vol. 156, pp. 847-856.