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The method of normal splines for linear DAEs on the number semi-axis. (English) Zbl 1160.65045

Summary: 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 (DAEs) of second order on the number semi-axis. The method consists in minimization of a norm of the collocation systems’ 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.

MSC:

65L80 Numerical methods for differential-algebraic equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations

Software:

MATSLISE; COLNEW; RODAS
PDFBibTeX XMLCite
Full Text: DOI

References:

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