On Fully Discrete Collocation Methods for Solving Weakly Singular Integral Equations

Abstract

A popular class of methods for solving weakly singular integral equations is the class of piecewise polynomial collocation methods. In order to implement those methods one has to compute exactly certain integrals that determine the linear system to be solved. Unfortunately those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids. In this paper fully discrete analogs of the collocation methods, where integrals are replaced by quadrature formulas, are considered, corresponding error estimates are derived.where the integration kernel K is not smooth across the diagonal t = s. If K(t, s) ≡ 0 in the region t > s, then the equations are of Volterra type, otherwise they are of Fredholm type. A popular class of methods for solving such equations is the class of piecewise polynomial collocation methods using *

On Fully Discrete Collocation Methods for Solving Weakly Singular Integro‐differential Equations

“…In a similar way we get the same estimates for ∆ 1 (t) (see [6]). Using (3.1) and (1.4) we obtain (see [6]) min{b,t+s1} max{0,t−s1}…”

“…Repeating the steps of the corresponding part of the proof of Theorem 1 in [6] we can prove the estimates (4.1) also for ν < 0. ⊓ ⊔ We derive sharper estimates for the errors of the approximationsB i J i v of the integrals B i J i v for i = 0, .…”

“…One way of coping with this difficulty is to use for approximating the integral operators additional graded grids that take into account the singularities of the kernels. Such methods for solution of weakly singular integral equations are proposed and investigated in [4,6,11]. In the present paper we propose fully discrete analogs of collocation methods to solve integro-differential equations with weakly singular kernels.…”

“…In the present paper we propose fully discrete analogs of collocation methods to solve integro-differential equations with weakly singular kernels. Here we use results from [6].…”

Redox‐Flow‐Batterien: von metallbasierten zu organischen Aktivmaterialien

“…Kangro erforschte 1949 den ersten mit heutigen Flow‐Batterien vergleichbaren Batterietyp . In diesem System wurde Cr 2 (SO 4 ) 3 als Kathoden‐ und Anodenmaterial in 2 m Schwefelsäure als Hilfselektrolyt eingesetzt und eine Zellspannung von 1.75 V erreicht .…”