scholarly journals Error expansion of trapezoidal rule for certain two-dimensional Cauchy principal value integrals

2017 ◽  
Vol 74 (10) ◽  
pp. 2608-2637 ◽  
Author(s):  
Jin Li ◽  
Hongxing Rui
Keyword(s):  
Trapezoidal Rule ◽  
Two Dimensional ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value ◽  
Error Expansion

scholarly journals Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Chunxiao Yu ◽  
Lingling Wei
Keyword(s):  
Singular Integrals ◽  
Riemann Integral ◽  
Piecewise Constant ◽  
Cauchy Principal ◽  
Numerical Examples ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value ◽  
Error Expansion ◽  
Superconvergence Results

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

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