From the fitting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem

Numerical Methods for Partial Differential Equations

Volumn 22, Issue 4, 2006, Pages 761-775

  Mounim, A Serghini ^a   De Dermale, B M ^a  

^a UNIVERSITÉ DE MONCTON   (Canada)

Author keywords

Fitted method; Liouville Bratu Gelfand problem; Nonstandardfinite difference

Indexed keywords

EID: 33745801140     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20116     Document Type: Article

References (23)

1. 1 B. Gidas, W. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm Math Phys 68 ( 1979), 209–243.
2. 2 J. Liouville, Sur l'équation aux différences partielles (d 2 log λ/ dudv)±λ2 a 2 =0, J Math Pures Appl 18 ( 1853), 71–72.
3. 3 G. Bratu, Sur les équations intégrales non linéaires, Bull Soc Math France 42 ( 1914), 113–142.
4. 4 R. Buckmire, Application of a Mickens Finite‐difference Scheme to the Cylindrical Bratu‐Gelfand problem, Numer Methods Partial Differential Eq 20 ( 2004), 327–337.
5. 5 D. N. Allen, R. V. Southwell, Relaxation methods applied to determine the motion, in 2D of aviscous fluid past a fixed cylinder, Quart J Mech Appl Math 8 ( 1955), 129–145.
6. 6 A. M. Il'In, A difference scheme for a differential equation with a small parameter multiplying the second derivative, Matematičeskije zametki 6 ( 1969), 237–248.
7. 7 D. Scharfetter, H. K. Gummel, Large‐signal analysis of a silicon read diode oscillator, IEEE Trans Elec Devices ED‐16 ( 1969), 64–77.
8. 8 S. C. R. Dennis, Finite differences associated with second‐order differential equations, Quart Journ Mech Appl Math 13 ( 1960), 487–507.
9. 9 S. C. R. Dennis, J. D. Hudson, Accurate representations of partial differential equations by finite‐difference schemes, J Inst Math Appl 23 ( 1979), 43–51.
10. 10 D. B. Splading, A novel finite difference formulation for differential expressions involving both first and second derivatives, Int J Numer Methods Engng 4 ( 1972), 551–559.
11. 11 D. F. Roscoe, New methods for the derivation of stable difference representations for differential equations, J Inst Maths Appl 16 ( 1975), 291–301.
12. 12 J. I. Ramos, On diffusive methods and exponentially fitted techniques, Appl Math Comput 103 ( 1999), 69–96.
13. 13 J. I. Ramos, On exponentially‐fitted method for singularly perturbed, one‐dimensional, parabolic problems, Appl Math Comput 16 ( 2) ( 2005), 513–523.
14. 14 S. V. Patankar, Numerical heat transfer and fluid flow, Series Computat Methods Mechanics Thermal Sci ( 1980).
15. 15 M. R. Osborne, Minimizing truncation error in finite difference approximations to ordinary differential equations, Math Comp 21 ( 1967), 133–145.
16. 16 E. J. Doedel, The construction of finite‐difference approximations to ordinary differential equations, SIAM J Numer Anal 15 ( 1978), 450–465.
17. 17 A. E. Berger, J. M. Solomon, M. Ciment, S. H. Leventhal, B. C. Weinberg, Generalized operator compact implicit schemes for boundary layer problems, Math Comp v 35 ( 1980), 695–731.
18. 18 R. E. Lynch, J. R. Rice, A high order difference method for differential equations, Math Comp 34 ( 1980), 333–372.
19. 19 E. C. Gartland Jr., Uniform high‐order difference schemes for a singularly perturbed two‐point boundary value problem, Math Comput 48 ( 1987), 551–567.
20. 20 F. Brezzi, L. Marini, P. Pietra, Two‐dimensional exponential fitting and applications to drift‐diffusion models, SIAM J Num Anal 26 ( 1989), 1342–1355.
21. 21 R. E. Mickens, A nonstandard finite differene scheme for the diffusionless Burgers equation with logistic reaction, Math Comput Simul 62 ( 2003), 117–124.
22. 22 A. Serghini Mounim, A new mixed‐hybrid finite element method for convection‐diffusion problems, Int J Pure Appl Math 14 ( 2004), 119–138.
23. 23 B. M. de Dormale, A. Serghini Mounim, A Versatile Finite‐difference Scheme for the Liouville‐Bratu‐Gelfand Problem, submitted.