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Zbl 1069.65086
Kadalbajoo, M.K.; Sharma, K.K.
Numerical analysis of singularly perturbed delay differential equations with layer behavior. (English)
[J] Appl. Math. Comput. 157, No. 1, 11-28 (2004). ISSN 0096-3003

The problem under consideration is the singularly perturbed boundary value problem (BVP) for the delay differential equation $$ \varepsilon y''(x)+a(x)y'(x-\delta)+b(x)y(x)=f(x), \quad 0 < x < 1, $$ under the boundary conditions $$ {y(x)}=\phi(x),\quad -\delta\leq x\leq 0, \quad y(1)=\gamma, $$ where $\varepsilon$ and $\delta$ are small positive parameters. The stated BVP for the delay differential equation is approximated by one for the ordinary differential equation (ODE), created by replacing the retarded term $y'(x-\delta)$ by its first order Taylor approximation $y'(x)-\delta y''(x)$. The approximate BVP for the ODE is approximated by a standard three points difference scheme. The stability and convergence of the method is discussed for two cases corresponding to the location the boundary layer, on the left side (when $a(x)>0$) and on the right (when $a(x)< 0$). Numerical examples are presented.
[Vladimir Gorbunov (Ul'yanovsk)]

MSC 2000:: *65L10 Boundary value problems for ODE (numerical methods)
34K28 Numerical approximation of solutions of FDE
65L20 Stability of numerical methods for ODE
34K26 Singular perturbations of functional-differential equations
34K10 Boundary value problems for functional-differential equations

Keywords: delay differential equation; negative shift; boundary layer; singular perturbation; differential-difference equation; difference scheme; stability; convergence; numerical examples

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Zentralblatt MATH Berlin [Germany]