Numerical solution of a singularly perturbed problem via exponential splines (Q913461)

A spline difference scheme for the singularly perturbed problem: \(- \epsilon u''(x)+p(x)u(x)=f(x),\) p(x)\(\geq \bar p>0\), \(x\in (0,1)\), \(\epsilon\in (0,1]\), \(u(0)=\alpha_ 0\), \(u(1)=\alpha_ 1\); \(\alpha_ 0,\alpha_ 1\in {\mathbb{R}}\), derived by using exponential cubic splines at midpoints is discussed. Its second order uniform convergence in the small parameter \(\epsilon\) is proved. Numerical results from the verifying of the scheme on a special example are presented.