Development of the Normal Spline Method for Linear Integro-Differential Equations

Abstract

The normal spline method is developed for the initial and boundary-value problems for linear integro-differential equations, probably being unresolved with respect to the derivatives, in Sobolev spaces of the arbitrary smoothness. It allows to solve a high-order systems without the reduction to first-order ones. The solving system can be arbitrary degenerate (with high differentiation index or irreducible to normal form). The method of nonuniform collocation grid creation for stiff problems is offered. Results of numerical solution to test problems are demonstrated. Supported by Russian Foundation for Basic Research, project N o 01-01-00731

The Parameterization Method in Singular Differential-Algebraic Equations

“…The problem (5), (6) with the parameterized control (8) is reduced to the finite-dimensional minimization of the function (10) under time-parameter restrictions (7).…”

“…It allows ones to apply the Newton method on the base of the penalty function one for the parameterized functional (10) minimization under conditions (7). Note, the variational problem (5), (6), corresponding to the initial problem for DAE (1), can be solved sequentially on sufficiently small intervals [t k−1 , t k ].…”

“…The considered DE problem is linear and the initial problem was solved by the normal spline collocation method in the [7] on the [0; 1]. To solve this problem on the [0; 1] by the parameterization method it was reduced to VC problem:…”