The asymptotics of eigenvalues for schemes of a high order of accuracy in the class of piecewise continuous coefficients

Abstract

FOR THE solution by a difference method of the problem on eigenvalues Lu + λru = 0, u(0) = u ( i) = 0, Lu = ( u′ p )′ − qu , 0 < c i ⩽ p( x) ⩽ c 2, 0 ⩽ q( x) ⩽ c 3, 0 < c 4 ⩽ r( x) ⩽ c 5, on the assumption that the coefficients p( x), q( x) and r( x) are sufficiently smooth, the asymptotics of the eigenvalues, calculated by schemes of the 2nd and 4th orders of accuracy, are obtained in [1]. In [2] taking into account certain terms of the asymptotic expansion of the error of the eigenvalues, the evaluation Δλ was found for the solution, by the simplest scheme of the second order of accuracy, of the problem u′' + λru = 0, u(0) = u(1) = 0, r( x) ϵ C (4) (0,1). For the boundary value problem Lu = − f, u(0) = u 0, u(1) = u 1 in [3] an exact scheme is put forward and also schemes of any order of accuracy obtained from it, which are constructed on a non-uniform network in the class of piecewise continuous functions p, q, and r. These schemes are used in [4] in the construction of schemes of a higher order of accuracy for problem (1). In the present paper asymptotic formulae are given for the error of the eigenvalues which appear in the schemes of [4]. The peculiarity of the given formulae lies in the fact that they are obtained only on the assumption of piecewise continuity of the original data p( x), q( x) and r( x).