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A higher order approximation to a singular perturbation problem. (English) Zbl 1029.65505

The authors give an analysis of a method of higher order for numerical solution of the model differential equation \[ \begin{gathered} Ly\equiv\varepsilon y''(x)+p(x)y'(x)-d(x)y(x)=f(x),\quad x\in (0,1)\\ By\equiv (y(0),y(1))=(\alpha_0,\alpha_1) \end{gathered} \] where \(\varepsilon\in(0,\varepsilon_0]\) is a small parameter, and \(p,d,f\) are sufficiently smooth. The proposed scheme is a linear combination of the known schemes (of the same authors) which belong to the class of spline difference methods derived by collocation with piecewise exponential splines from \(C^1[0,1]\) on a regular mesh. The errors at the grid points of a new scheme are bounded by \(Ch^4/(\varepsilon^2+h^2)\), where \(C\) is a constant independent of \(\varepsilon\) and mesh size \(h\). Numerical examples illustrate the behavior of the proposed scheme and other schemes.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
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