Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance

International Journal for Numerical Methods in Engineering

Volumn 66, Issue 2, 2006, Pages 272-296

  Patidar, Kailash C ^a   Sharma, Kapil K ^b  

^a UNIVERSITY OF PRETORIA   (South Africa)

^b PANJAB UNIVERSITY   (India)

Author keywords

Boundary value problems; Differential difference equations; Non standard finite difference methods; Singular perturbations

Indexed keywords

BOUNDARY VALUE PROBLEMS; DIFFERENCE EQUATIONS; DIFFERENTIATION (CALCULUS); FINITE DIFFERENCE METHOD; PERTURBATION TECHNIQUES;

DENOMINATOR FUNCTION; DIFFERENTIAL DIFFERENCE EQUATIONS; NON-STANDARD FINITE DIFFERENCE METHODS; SECOND-ORDER DERIVATIVE;

NUMERICAL METHODS;

BOUNDARY VALUE PROBLEMS; DIFFERENCE EQUATIONS; DIFFERENTIATION (CALCULUS); FINITE DIFFERENCE METHOD; NUMERICAL METHODS; PERTURBATION TECHNIQUES;

EID: 33645464501     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1555     Document Type: Article

References (35)

1. Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential difference equations. V. Small shifts with layer behaviour. SIAM Journal on Applied Mathematics 1994; 54:249-272.
2. Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential difference equations. VI. Small shifts with rapid oscillations. SIAM Journal on Applied Mathematics 1994; 54:273-283.
3. Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Academic Press: New York, 1993
4. Hutchinson GE. Circular casual systems in ecology. Annals of the New York Academy of Sciences 1948; 50:518-543.
5. Kadalbajoo MK, Sharma KK. Numerical analysis of singularly perturbed delay differential equations with layer behaviour. Applied Mathematics and Computation 2004; 157(1):11-28.
6. Kadalbajoo MK, Sharma KK. Numerical analysis of boundary value problems for singularly perturbed differential-difference equations: small shifts of mixed type with rapid oscillations. Communications in Numerical Methods in Engineering 2004; 20:167-182.
7. Kadalbajoo MK, Sharma KK. Numerical methods for solving a mathematical model of neuronal variability. Journal of Mathematical Analysis and Applications, in press.
8. Kadalbajoo MK, Patidar KC, Sharma KK. ε-uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs, submitted.
9. Derstine MW, Gibbs HM, Kaplan DL. Bifurcation gap in a hybrid optical system. Physical Review A 1982; 26:3720-3722.
10. Longtin A, Milton J. Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Mathematical Biosciences 1988; 90:183-199.
11. Mackey MC, Glass L. Oscillations and chaos in physiological control systems. Science 1977; 197:287-289.
12. Stein RB. A theoretical analysis of neuronal variability. Biophysical Journal 1965; 5:173-194.
13. Stein RB Some models of neuronal variability. Biophysical Journal 1967; 7:37-68.
14. Wazewska-Czyzewska M, Lasota A. Mathematical models of the red cell system. Matmatyks Stosowans 1976; 6:25-40.
15. Patidar KC, Sharma KK. ε-uniformly convergent nonstandard finite difference methods for singularly perturbed differential difference equations with small delay, Applied Mathematics and Computations, in press.
16. Buckmire R. Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem. Numerical Methods for Partial Differential Equations 2004; 20(3):327-337.
17. Dai W, Su S. A nonstandard finite difference scheme for solving one dimensional nonlinear heat transfer. Journal of Difference Equations and Applications 2004; 10(11):1025-1032.
18. Gumel AB, Moghadas SM, Micken RE. Effect of a preventive vaccine on the dynamics of HIV transmission. Communications in Nonlinear Science and Numerical Simulation 2004; 9(6):649-659.
19. Kojouharov HV, Chen BM. Nonstandard Eulerian-Lagrangian methods for multi-dimensional reactive transport problems. Applied Numerical Mathematics 2004; 49(2):225-243.
20. Letellier C, Elaydi S, Aguirrec LA, Aziz A. Difference equations versus differential equations, a possible equivalence for the Rössler system? Physica D: Nonlinear Phenomena 2004; 195(1-2):29-49.
21. Lubuma JM-S, Patidar KC. Nonstandard finite difference method for self-adjoint singular perturbation problems. In VSP Lecture Series on Computer and Computational Sciences, Simos T (ed.), vol. 1, 2004; 328-331.
22. Lubuma JM-S, Patidar KC. Nonstandard finite difference methods for dissipative singular perturbation problems. Proceedings of the 9th International Conference on Difference Equations and Applications, Los Angeles, 2-6 August 2004, to appear.
23. Mickens RE, Jordan PM. A positivity-preserving nonstandard finite difference scheme for the damped wave equation. Numerical Methods for Partial Differential Equations 2004; 20(5):639-649.
24. Sekhavat P, Sepehri N, Wu Q. Calculation of Lyapunov exponents using nonstandard finite difference discretization scheme: a case study. Journal of Difference Equations and Applications 2004; 10(4):369-378.
25. Patidar KC. On the use of nonstandard finite difference methods. Journal of Difference Equations and Applications 2005; 11(8):735-758.
26. Mickens RE. Difference Equations: Theory and Applications. Van Nostrand Reinhold: New York, 1990.
27. Hilderbrand FB. Finite Difference Equations and Simulations. Prentice-Hall: Englewood Cliffs, NJ, 1968.
28. Jain MK. Numerical Solution of Differential Equations. Wiley: New York, 1984.
29. Mitchell AR, Griffiths DF. Finite Difference Methods in Partial Differential Equations. Wiley: New York, 1980.
30. Richtmyer RD, Morton KW. Difference Methods for Initial-Value Problems. Wiley-Interscience: New York, 1967.
31. Doolan EP, Miller JJH, Schilders WHA. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press: Dublin, 1980.
32. Mickens RE. Nonstandard Finite Difference Models of Differential Equations. World Scientific: Singapore, 1994.
33. Patidar KC, Sharma KK. Some uniformly convergent nonstandard finite difference methods for nonlinear singularly perturbed differential difference equations, in preparation.
34. Patidar KC, Sharma KK. ε-uniform nonstandard finite difference methods for turning point problems for singularly perturbed differential difference equations, in preparation.
35. Moghadas SM, Alexander ME, Corbett BD. A nonstandard numerical scheme for a generalized Gause-type predatorprey model. Physica D: Nonlinear Phenomena 2004; 188(1-2):134-151.