Spline collocation for singular integro-differential equations over (0.1)

Abstract

Summary. This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0, I). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in Sobolev norms. We give asymptotic error estimates and some numerical results.

The effect of a curvature-dependent surface tension on the singularities at the tips of a straight interface crack

“…Thus, the approximations (6.10) contain 6N + 2 unknowns w j k . To be able to differentiate the approximations of the functions ψ 1 (x), ψ 2 (x) and ψ 0 (x) twice it is necessary to take the higher order approximations of these functions which can be achieved by approximating the functions ψ 1 (x), ψ 2 (x) and ψ 0 (x) by cubic splines (27):…”

A New Approach to the Modeling and Analysis of Fracture through Extension of Continuum Mechanics to the Nanoscale

“…In essence, formula (37) gives an inverse of the finite Hilbert transform operator (36). Using this, we conclude that R 1 []2x3, restricted to the space of functions with 1 61 2x3 dx 2 0, has a trivial null space.…”

“…To find a numerical solution to problem (27), subject to ( 29) and ( 30), we employ a spline collocation method, similar to the one introduced by Samo2 3lova in [36], where a first-order singular integro-differential equation (SIDE) is solved. Spline collocation methods for SIDEs were considered by many others, including Schmidt [37]. Using (30), it suffices to solve the problem on 207 13.…”

Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary

“…Therefore, it is customary to solve numerically the initial system of the singular integro-differential equations (3.23), (3.27) together with the conditions (5.1)-(5.3). There are multiple ways to solve the systems of this type numerically, such as spline collocation methods and representations of unknowns with different special functions [31], [36]. In this work we will follow the approach adopted in [52].…”