Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4645108 | Applied Numerical Mathematics | 2014 | 10 Pages |
Abstract
Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the singular integrals I(1)[u]=â«ab(cotÏ(xât)T)u(x)dx and I(2)[u]=â«ab(csc2Ï(xât)T)u(x)dx, with bâa=T and u(x) a T-periodic continuous function on R. These integrals are not defined in the regular sense, but are defined in the sense of Cauchy Principal Value and Hadamard Finite Part, respectively. With h=(bâa)/n, n=1,2,â¦, the numerical quadrature formulas Qn(1)[u] for I(1)[u] and Qn(2)[u] for I(2)[u] areQn(1)[u]=hâj=1nf(t+jhâh/2),f(x)=(cotÏ(xât)T)u(x), andQn(2)[u]=hâj=1nf(t+jhâh/2)âT2u(t)hâ1,f(x)=(csc2Ï(xât)T)u(x). We provided a complete analysis of the errors in these formulas under the assumption that uâCâ(R) and is T-periodic. We actually showed that,I(1)[u]âQn(1)[u]=O(nâμ)andI(2)[u]âQn(2)[u]=O(nâμ)as nââ,âμ>0. In this note, we analyze the errors in these formulas under the weaker assumption that uâCs(R) for some finite integer s. By first regularizing these integrals, we prove that, if u(s+1) is piecewise continuous, thenI(1)[u]âQn(1)[u]=o(nâsâ1/2)as nââ, if s⩾1,andI(2)[u]âQn(2)[u]=o(nâs+1/2)as nââ, if s⩾2. We also extend these results by imposing different smoothness conditions on u(s+1). Finally, we append suitable numerical examples.
Keywords
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Physical Sciences and Engineering
Mathematics
Computational Mathematics
Authors
Avram Sidi,