This paper presents a regularization technique for the high order efficient numerical evaluation of nearly singular, principal-value, and finite-part Cauchy-type integral operators. By relying on the Cauchy formula, the Cauchy-Goursat theorem, and on-curve Taylor interpolations of the input density, the proposed methodology allows to recast the Cauchy and associated integral operators as smooth contour integrals. As such, they can be accurately evaluated everywhere in the complex plane -- including at problematic points near and on the contour -- by means of elementary quadrature rules. Applications of the technique to the evaluation of the Laplace layer potentials and related integral operators, as well as to the computation conformal mappings, are examined in detail. The former application, in particular, amounts to a significant improvement over the recently introduced harmonic density interpolation method. Spectrally accurate discretization approaches for smooth and piecewise smooth contours are presented. A variety of numerical examples, including the solution of weakly singular and hypersingular Laplace boundary integral equations, and the evaluation of challenging conformal mappings, demonstrate the effectiveness and accuracy of the density interpolation method in this context.
The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing
Gran Sasso Science Institute（格兰萨索科学研究所）；Hankyong National University（韩荣国立大学）；Ho Chi Minh City University of Science（胡志明市科技大学）；Ton Duc Thang University（敦德唐大学）
On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics
Ivan Franko National University of Lviv（利沃夫伊万弗兰科国立大学）；Aston University（阿斯顿大学）；Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences（奥地利科学院约翰·拉德纳计算与应用数学研究所）
