Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance.(English)Zbl 1123.65078

The authors study numerical solutions of the singularly perturbed differential equation $\varepsilon^{2} y''(x)+a(x)y(x-\delta)+w(x)y(x)+b(x)y(x+\eta)=f(x) \tag{1}$ on $$(0,1)$$ satisfying the boundary conditions $y(x)=\phi(x), \quad -\delta\leq x\leq 0; \qquad y(x)=\gamma (x), \quad 1\leq x\leq 1+\eta.\tag{2}$ Here the coefficients $$a(x), w(x), b(x), f(x)$$ and $$\phi(x), \gamma (x)$$ are sufficiently smooth functions, $$0<\varepsilon\ll 1$$, and the delay and advance parameters $$\delta, \eta$$ are positive and hold $$\delta,\eta=o(\varepsilon).$$
In the second section of the paper, the authors take Taylor expansions of $$y(x-\delta)$$ and $$y(x+\eta)$$, and, under appropriate hypotheses, obtain that the solution $$u$$ of the new problem $-C_{\varepsilon}(x)u''(x)+A_{\varepsilon}(x)u'(x)+B(x)u(x)=F(x); \quad u(0)=\phi(0), \quad u(1)=\gamma(1),$ differing from the original one by $$O(\delta^{3}u''',\eta^{3}u''')$$ terms, is a good approximation of the solution of (1), (2).
Then, according to the introduction of the paper: “Section 3 is devoted to the construction of some exact and non-standard finite difference schemes for problems resulting from (1) whose solution possess layer behaviour. In Section 4, we analyse these non-standard schemes for convergence. We briefly give some exact and non-standard schemes in Section 5 for problems having oscillatory solutions. Several numerical examples are given in Section 6. Discussion on results as well as further research plans are indicated in Section 7.”

MSC:

 65L12 Finite difference and finite volume methods for ordinary differential equations 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 34K26 Singular perturbations of functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K10 Boundary value problems for functional-differential equations
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