Collocation by singular splines

Abstract

Splines determined by the kernel of the differential operator D k (D √x D) are known to be useful to solve the singular boundary value problems of the form D √x Du = f (x, u). One of the most successful methods is the collocation method based on special Chebyshev splines. We investigate the construction of the associated B-splines based on knot-insertion algorithms for their evaluation, and their application in collocation at generalized Gaussian points. Specially, we show how to obtain these points as eigenvalues of a symmetric tridiagonal matrix of order k.

On Wavelet Decomposition of the Singular Splines

“…In papers [11], [15] the non-classical approach is based on the construction of a system of embedded spaces, and abandoning the idea of using only two functions (scaling function and parent wavelet). In this case, the multiple-scale ratio is replaced by calibration ratios.…”

“…The first of them is the extraction (additive or multiplicative) of singularities from the discussed function, and the approximation of the function rest. The second one is an introduction of the mentioned singularities in the approximation apparatus (see [1], [4], [11], [22]). The second approach is usually simpler because it is required less than the priori information about the function in question (it is not required to know the exact function characteristics, its asymptotic behavior, orders of the defining multipliers, etc.)…”

“…The second approach is usually simpler because it is required less than the priori information about the function in question (it is not required to know the exact function characteristics, its asymptotic behavior, orders of the defining multipliers, etc.) In [11] the singular splines determined by the kernel of the differential operator in special boundary value problems were used for the collocation method. In the work [22] the singular splines are constructed with an implicit representation of manifold on which coefficient of model elliptic problem are discontinuous.…”