an:06147979
Zbl 1274.30146
Drobek, Jaroslav
Approximations by the Cauchy-type integrals with piecewise linear densities
EN
Appl. Math., Praha 57, No. 6, 627-640 (2012).
00316048
2012
j
30E10 30E20 65N12 65N38
Cauchy-type integral; Dini continuous density; piecewise linear interpolation; uniform convergence; complex variable boundary element method
Summary: The paper is a contribution to the complex variable boundary element method, shortly CVBEM. It is focused on Jordan regions having piecewise regular boundaries without cusps. Dini continuous densities whose modulus of continuity \(\omega (\cdot )\) satisfies
\[
\lim \sup _{s\downarrow 0}\omega (s)\ln \frac {1}{s}=0
\]
are considered on these boundaries. Functions satisfying the H??lder condition of order \(\alpha \), \(0<\alpha \leq 1\), belong to them. The statement that any Cauchy-type integral with such a density can be uniformly approximated by a Cauchy-type integral whose density is a piecewise linear interpolant of the original one is proved under the assumption that the mesh of the interpolation nodes is sufficiently fine and uniform. This result ensures the existence of approximate CVBEM solutions of some planar boundary value problems, especially of the Dirichlet ones.