Properties of a family of singular integral operators on an arbitrary piecewise smooth curve

Singular Integral Equations with a Fixed Hypersingularity on an Arbitrary Piecewise Smooth Curve

“…, r. An open-ended arc L m in L is said to be closed if its endpoints are included in L m . By L we denote any closed part of L without the nodes c 1 In the following, we assume that the functions ϕ(t) and ψ(t) occurring in (1) and (2) belong to the class H * (L), and the coefficients a(t) and b(t) of the singular integral operator belong to the class H 0 (L). Moreover, a 2 (t) − b 2 (t) = 0 everywhere on L. For any interior point of L other than q, the singular integrals…”

“…In the case under consideration, a complex-valued solution of a singular integral equation is sought in the class of functions with integrable singularity at the suction point. We analyze the properties of a family of singular integral operators of the second kind with Cauchy kernel and with a fixed hypersingularity [1,2] at an interior point of the integration line. We show that the problem of solving a singular integral equation with a fixed hypersingularity admits regularization by the solution of the characteristic equation.…”