Numerical methods and error estimates for a singular boundary‐value problem

Abstract

In this paper we analyze a class of equations of the form y? (x) = —g(x) x^p (y(x)) ^q where p and q are real parameters satisfying p > ^_1 , g < ^_1 and g is a positive and continuous function on [0,1]. We search for positive solutions which satisfy the boundary conditions y'(0)=y(l) = 0.

Numerical approximations of the solution are obtained by means of a finite difference scheme and the asymptotic expansion of the discretization error is deduced. Some numerical examples are analyzed.

Nagrinejama viena klase antrosios eiles netiesiniu diferencialiniu lygčiu su kraštine salyga. Uždavinys yra singuliarusis viename arba abiejuose intervalo galuose. Siūlomas skaitinis metodas taikytinas atskiriems uždaviniu klases atvejams. Darbas tesia ankstesnius autoriu darbu tyrimus. Pateikti skaitinio eksperimento rezultatai, patvirtinantys teorinius iverčius.

Key words:

• singular problems
• finite differences scheme
• asymptotic expansion