Abstract
The classical composite rectangle (constant) rule for the computation of Cauchy principle value integral with the singular kernel is discussed. We show that the superconvergence rate of the composite midpoint rule occurs at certain local coodinate of each subinterval and obtain the corresponding superconvergence error estimate. Then collation methods are presented to solve certain kind of Hilbert singular integral equation. At last, some numerical examples are provided to validate the theoretical analysis.
Highlights
Consider the Cauchy principle integral I= ( f ; s) ∫c+2π cot c x−s 2 f= ( x)dx g (s), s ∈ (0, 2π) (1) where c+2π∫ c denotes a Cauchy principle value integral and s is the singular point.There are several different definitions which can be proved such as the definition of subtraction of the singularity, regularity definition, direct definition and so on
The main reason for this interest is probably due to the fact that integral equations with Cauchy principal value integrals have shown to be an adequate tool for the modeling of many physical situations, such as acoustics, fluid mechanics, elasticity, fracture mechanics and electromagnetic scattering problems and so on
It is the aim of this paper to investigate the superconvergence phenomenon of rectangle rule for it and, in particular, to derive error estimates
Summary
∫ c denotes a Cauchy principle value integral and s is the singular point. There are several different definitions which can be proved such as the definition of subtraction of the singularity, regularity definition, direct definition and so on. The main reason for this interest is probably due to the fact that integral equations with Cauchy principal value integrals have shown to be an adequate tool for the modeling of many physical situations, such as acoustics, fluid mechanics, elasticity, fracture mechanics and electromagnetic scattering problems and so on It is the aim of this paper to investigate the superconvergence phenomenon of rectangle rule for it and, in particular, to derive error estimates. This idea was firstly presented by Linz [16] in the paper to calculated the hypersingular integral on interval He used the trapezoidal rule and Simpson rule to approximate the density function f(x) and the convergence rate was ( ) O hk , k = 1, 2 when the singular point was always located at the middle of certain subinterval. Several numerical examples are provided to validate our analysis
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