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

Volumn 20, Issue 3, 2004, Pages 167-182

  Kadalbajoo, M K ^a   Sharma, K K ^a  

^a INDIAN INSTITUTE OF TECHNOLOGY KANPUR   (India)

Author keywords

Action potential; Convergence; Differential difference equations; Mixed type; Negative shift; Positive shift; Rapid oscillations; Singular perturbation; Stability

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; NEUROLOGY; OSCILLATIONS; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

OSCILLATORY SOLUTIONS;

BOUNDARY VALUE PROBLEMS;

MATHEMATICAL METHOD; OSCILLATION;

EID: 1542347723     PISSN: 10698299     EISSN: None     Source Type: Journal    
DOI: 10.1002/cnm.657     Document Type: Article

References (17)

1. Tuckwell HC, Richter W. On the first-exit time problem for temporally homogeneous Markov processes. Journal of Applied Probability 1976; 13:39-48.
2. Stein RB, A theoretical analysis of neuronal variability. Biophysical Journal 1965; 5:173-194.
3. Holden AV, Models of the Stochastic Activity of Neurones. Springer: New York, Heidelberg, Berlin, 1976.
4. Nichols NK, On the numerical integration of a class of singular perturbation problems. Journal of Optimization Theory and Applications 1989; 60:439-452.
5. Gene H. Golub, Ortega JM. Scientific Computing and Differential Equations: an Introduction to Numerical Methods. Academic Press: London, 1991.
6. Tuckwell HC, Cope DK. Accuracy of neuronal inter spike times calculated from a diffusion approximation. Journal of Theoretical Biology 1980; 83:377-387.
7. Stein RB, Some models of neuronal variability. Biophysical Journal 1967; 7:37-68.
8. Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential difference equations. V. Small shifts with layer behavior. SIAM Journal on Applied Mathematics 1994; 54:249-272.
9. Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential difference equations. SIAM Journal on Applied Mathematics 1982; 42:502-531.
10. Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential difference equations II. Rapid oscillations and resonances. SIAM Journal on Applied Mathematics 1985; 45:687-707.
11. Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential difference equations III.Turning point problems. SIAM Journal on Applied Mathematics 1985; 45:708-734.
12. 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.
13. Tian H. Numerical Treatment of Singularly Perturbed Delay Differential Equations. Ph.D. Thesis, Department of Mathematics, University of Manchester, 2000.
14. Kadalbajoo MK, Sharma KK. Numerical analysis of boundary-value problems for singularly-perturbed differential-difference equations with small shifts of mixed type. Journal of Optimization Theory and Applications 2002; 115:145-163.
15. Hutchinson GE. Circular casual systems in ecology. Annals of the New York Academy of Science 1948; 50: 221-246.
16. Kuang Y, Delay Differential Equations with Application in Population Dynamics, vol. 191. Academic Press: London, 1993.
17. Wilbur WJ, Rinzel J. An analysis of Stein's model for stochastic neuronal excitation. Biological Cybernetics 1982; 45:107-114.