Showing papers on "Biorthogonal system published in 1982"

Matched eigenfunction expansions for slow flow over a slot

Abstract: We solve the problem of plane flow of a second-order fluid over a rectangular slot when inertia is neglected by matching biorthogonal eigenfunction expansions in different regions of flow. The method appears to be cheaper and more accurate than direct numerical methods. The effect of normal stresses on pressure measurements at the bottom of the slot is discussed.

Cubature Remainder and Biorthogonal Systems

Abstract: Error estimates for cubature formulas are usually given in terms of higher derivatives (Peano-Sard) or in terms of analyticity properties (Davis-Hammerlin). The approximation method has found little attention. We present the latter method in a generalized and refined form, based on biorthogonal systems (BOGS). The degrees of approximation and coefficient estimates connected with a BOGS lead to rather good and versatile inequalities for the error. More specifically, we consider Chebyshev polynomials, Clenshaw-Curtis procedures and product formulas. Estimates for the employed degrees of approximation are available in the theory of approximation, and these estimates can be supported or refined by numerical computation.

Biorthogonality in Approximation

Abstract: Numerical approximation can be carried out by ascent or descent methods — or in a more explicit way by expansion methods (like truncation, telescoping, pre-iteration). We use general biorthogonal systems (BOGS) to describe procedures of the latter type. This setting leads easily to useful results and provides good insight. The basic task is to improve or shorten a given expression by changing the coefficients. Thereby one employs information comprised in the elements and functionals of the BOGS, More specifically we consider expansions of Fourier (Chebyshev) type in the univariate and bivariate case.

The Shape of Stress-Free Surfaces on a Sheared Block

Abstract: We obtain solutions for the shape of the free surface on the upper and lower boundaries of an initially rectangular, incompressible linearly viscoelastic block when the block is sheared at the vertical sidewall. To solve the problem when the vertical displacement of the sidewall is prescribed we use biorthogonal series of the Fadle–Papkovich type. The series has a point of novelty in that zero is an algebraically double but geometrically simple eigenvalue. To expand arbitrary vector fields with two components it is necessary to include in the series the proper and generalized eigenvectors belonging to zero.

Biorthogonal sequences of solutions of the generalized feller equation

Abstract: The generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.