The method of normal splines for linear DAEs on the number semi-axis

The method of normal spline-collocation (NSC), applicable to a wide class of ordinary linear singular differential and integral equations, is specified for the boundary value problems for differential-algebraic equations of second order on the number semi-axis. The method consists in minimization of a norm of the collocation systemsʹ solutions in an appropriate Hilbert–Sobolev space. The NSC method does not use the notion of differentiation index and it is applicable to DAEs of any index as well as to equations not reducible to the normal form. The problems on the infinite interval can be solved in two ways. The first way is based on the use of the original space of functions defined on the semi-axis, and the second way is based on a singular transformation of the semi-axis into the unit segment. A new reproducing kernel, that provides the first way, is presented. An algorithm to create a non-uniform collocation grid is described.