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A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations. (English) Zbl 1141.65062
A numerical method is proposed for linear second-order singularly perturbed delay differential equations of the form $$\varepsilon y''(x) + a(x)y'(x-\delta) + b(x)y(x) = f(x),\,\text{on}\,\, (0,1),$$ subject to Dirichlet boundary conditions. Here, $0 < \varepsilon \ll 1$ is the perturbation parameter and $\delta$ is the small shift parameter. The authors are mainly focused on the case $\delta = O(\varepsilon)$. In order to solve this problem classical finite difference schemes are used with the mesh parameter $h = \delta/m$, where $m = pq$, $p$ is a positive integer and $q$ is the mantissa of $\delta$. The truncation error contains the higher-order derivatives of the solution of the continuous problem which involve negative powers of the small (perturbation and delay) parameters. Therefore, the convergence result provided here may not be independent of the parameters, that is, they are not uniformly-convergent. Some numerical examples are presented.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 65L12 Finite difference methods for ODE (numerical methods) 34K28 Numerical approximation of solutions of functional-differential equations 34K26 Singular perturbations of functional-differential equations 65L70 Error bounds (numerical methods for ODE) 65L20 Stability and convergence of numerical methods for ODE
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##### References:
 [1] Derstine, M. W.; Gibbs, F. A. H.H.M.; Kaplan, D. L.: Bifurcation gap in a hybrid optical system. Phys. rev. A 26, 3720-3722 (1982) [2] Doolan, E. P.; Miller, J. J. H.; Schilders, W. H. A.: Uniform numerical methods for problems with initial and boundary layers. (1980) · Zbl 0459.65058 [3] Glizer, V. Y.: Asymptotic analysis and solution of a finite-horizon H$\infty$ control problem for singularly-perturbed linear systems with small state delay. J. optim. Theory appl. 117, 295-325 (2003) · Zbl 1036.93013 [4] Kadalbajoo, M. K.; Sharma, K. K.: Numerical analysis of singularly perturbed delay differential equations with layer behavior. Appl. math. Comput. 157, 11-28 (2004) · Zbl 1069.65086 [5] Lange, C. G.; Miura, R. M.: Singular perturbation analysis of boundary-value problems for differential-difference equations. V. small shifts with layer behavior. SIAM J. Appl. math. 54, 249-272 (1994) · Zbl 0796.34049 [6] Lange, C. G.; Miura, R. M.: Singular perturbation analysis of boundary-value problems for differential-difference equations. Vi. small shifts with rapid oscillations. SIAM J. Appl. math. 54, 273-283 (1994) · Zbl 0796.34050 [7] Longtin, A.; Milton, J.: Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Math. biosci. 90, 183-199 (1988) [8] M.C. Mackey, G.L. Oscillations and chaos in physiological control systems, Science, 197 (1977), pp. 287 -- 289. [9] H. Tian, Numerical treatment of singularly perturbed delay differential equations, PhD thesis, University of Manchester, 2000. [10] Wazewska-Czyzewska, M.; Lasota, A.: Mathematical models of the red cell system. Mat. stos. 6, 25-40 (1976)