Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations: small shifts of mixed type with rapid oscillations
✍ Scribed by Kadalbajoo, M. K. ;Sharma, K. K.
- Publisher
- John Wiley and Sons
- Year
- 2004
- Tongue
- English
- Weight
- 917 KB
- Volume
- 20
- Category
- Article
- ISSN
- 1069-8299
- DOI
- 10.1002/cnm.657
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✦ Synopsis
Abstract
We study the boundary‐value problems for singularly perturbed differential‐difference equations with small shifts. Similar boundary‐value problems are associated with expected first‐exit time problems of the membrane potential in models for activity of neurons (SIAM J. Appl. Math. 1994; 54: 249–283; 1982; 42: 502–531; 1985; 45: 687–734) and in variational problems in control theory. In this paper, we present a numerical method to solve boundary‐value problems for a singularly perturbed differential‐difference equation of mixed type, i.e. which contains both type of terms having negative shifts as well as positive shifts, and consider the case in which the solution of the problem exhibits rapid oscillations. The stability and convergence analysis of the method is given. The effect of small shift on the oscillatory solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method. Copyright © 2004 John Wiley & Sons, Ltd.