On numerical solution of nonlinear singularly-perturbed problems. (English. Russian original) Zbl 0658.65074

Lisejkin, V. D.; Petrenko, V. E.

Sov. Math., Dokl. 36, No. 3, 535-538 (1988); translation from Dokl. Akad. Nauk SSSR 297, 791-794 (1987).

This paper is concerned with the numerical solution of singularly- perturbed boundary value problems of type: \[ (1)\quad (\epsilon +px)^ qu''+a(x)u'-f(x,u)=0,\quad 0<x<1, \] u(0)\(=u_ 0\), \(u(1)=u_ 1\) where \(\epsilon\) is a small parameter, \(p=0,1\), \(a(x)\in C^ 2[0,1]\), \(f(x,u)\in C^ 2([0,1]\times {\mathbb{R}})\), with \(f_ u>0\) and q is a positive integer. First, using some techniques developed by the first author and N. N. Yanenko [Proc. Third Int. Conf. Boundary and Interior Layers - Computational and Asymptotic Methods, Dublin 1984, 68- 80 (1984)], estimates of the derivatives of u(x) for different values of p and q are stated. In particular a function \(\phi\) (x), depending on the coefficients of (1), is defined so that \(| u'(x)| \leq \phi (x)\). Then, with such an estimate a nonuniform mesh \(x_ i=x(ih),\) \(i=0,...,N\) is constructed by means of a mapping x(q) defined by \(q=M^{- 1}\int^{x}_{0}\phi (x)dx\) with \(M=\int^{1}_{0}\phi (x)dx.\) Finally, a standard finite difference scheme (with order \(\geq 1)\) is proposed to solve (1) on this nonuniform mesh and some numerical examples are presented.

Reviewer: M.Calvo

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
86A05	Hydrology, hydrography, oceanography
86A10	Meteorology and atmospheric physics
34B15	Nonlinear boundary value problems for ordinary differential equations
34E15	Singular perturbations, general theory for ordinary differential equations

Keywords:

singularly-perturbed boundary value problems; small parameter; numerical examples