TY  - EJOU
AU  - Li, Jin 
AU  - Yu, De-hao 

TI  - Collocation Methods to Solve Certain Hilbert Integral Equation with Middle Rectangle Rule
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2014
VL  - 102
IS  - 2
SN  - 1526-1506

AB  - The generalized composite middle rectangle rule for the computation of Hilbert integral is discussed. The pointwise superconvergence phenomenon is presented, i.e., when the singular point coincides with some a priori known point, the convergence rate of the rectangle rule is higher than what is global possible. We proved that the superconvergence rate of the composite middle rectangle rule occurs at certain local coordinate of each subinterval and the corresponding superconvergence error estimate is obtained. By choosing the superconvergence point as the collocation points, a collocation scheme for solving the relevant Hilbert integral equation is presented and an error estimate is established. At last, some numerical examples are provided to validate the theoretical analysis.
KW  - Hilbert integral
KW  -  Composite middle rectangle rule
KW  -  Boundary integral equation
KW  -  Superconvergence
KW  -  Error expansion
KW  -  Collocation methods

DO  - 10.3970/cmes.2014.102.103