Application of a Mickens Finite-difference Scheme to the Cylindrical Bratu-Gelfand Problem

Numerical Methods for Partial Differential Equations

Volumn 20, Issue 3, 2004, Pages 327-337

  Buckmire, Ron ^a  

^a OCCIDENTAL COLLEGE   (United States)

Author keywords

Bratu problem; Gelfand problem; Mickens discretization; Nonstandard finite differences

Indexed keywords

EID: 2342505312     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/num.10093     Document Type: Article

References (20)

1. 1 I. M. Gelfand, Some problems in the theory of quasi‐linear equations, Am Math Soc Transl Ser 2 29 (1963), 295–381.
2. 2 G. Bratu, Sur les équations intégrales non‐linéaires, Bull Math Soc France 42 (1914), 113–142.
3. 3 D. A. Frank‐Kamenetski, Diffusion and heat exchange in chemical kinetics, Princeton University Press, Princeton, NJ, 1955.
4. 4 S. Chandrasekhar, An introduction to the study of stellar structure, Dover Publications, New York, 1957.
5. 5 J. P. Abbott, An efficient algorithm for the determination of certain bifurcation points, J Comput Appl Math 4 (1) (1978), 19–27.
6. 6 U. M. Ascher, R. M. M. Matheij, and R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1995.
7. 7 J. Jacobsen and K. Schmitt, The Liouville‐Bratu‐Gelfand problem for radial operators, J Differ Eq 184 (2002), 283–298.
8. 8 R. E. Mickens, Exact solutions to difference equation models of Burgers' equation, Numer Methods Partial Differ Eq 2 (2) (1986), 123–129.
9. 9 R. E. Mickens, Nonstandard finite difference schemes for differential equations, J Differ Eq Appl 8 (9) (2002), 823–847.
10. 10 R. E. Mickens, Difference equation models of differential equations having zero local truncation errors, Differential equations (Birmingham, Ala., 1983), North‐Holland Math. Stud., Vol. 92, North‐Holland, Amsterdam‐New York, 1984, pp. 445–449.
11. 11 R. Buckmire, The design of shock‐free transonic slender bodies of revolution, Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, NY, 1994.
12. 12 R. Buckmire, A new finite‐difference scheme for singular boundary value problems in cylindrical or spherical coordinates, L. P. Cook and V. Roytburd, editors, Mathematics is for solving problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1996, pp. 3–9.
13. 13 R. Buckmire, Investigations of nonstandard, Mickens‐type, finite‐difference schemes for singular boundary value problems in cylindrical or spherical coordinates, Numer Methods Partial Differ Eq 19 (3) (2003), 380–398.
14. 14 K. Wanelik, On the iterative solutions of some nonlinear eigenvalue problems, J Math Phys 30 (8) (1989), 1707–1712.
15. 15 R. E. Mickens, editor, Applications of nonstandard finite differences, World Scientific, Singapore, 2000.
16. 16 R. E. Mickens, Nonstandard difference models of differential equations, World Scientific, Singapore, 1994.
17. 17 R. E. Mickens and A. Smith, Finite‐difference models of ordinary differential equations: Influence of denominator functions, J Franklin Instit 327 (1990), 143–145.
18. 18 R. E. Mickens, Difference equation models of differential equations, Math Comput Model 11 (1988), 528–530.
19. 19 J. Ockendon, S. Howison, A. Lacey, and A. Movchan, Applied partial differential equations, Oxford University Press, Oxford, October 1999.
20. 20 R. Buckmire, On numerical solutions of the one‐dimensional planar Bratu problem, Preprint. Available at http://faculty.oxy.edu/ron/research.