Dual integral equations -- revisited.
Efficient iterative solution of linear systems from discretizing singular integral equations.
Existence and uniqueness for non-linear singular integral equations used in fluid mechanics
Non-linear singular integral equations are investigated in connection with some basic applications in two-dimensional fluid mechanics. A general existence and uniqueness analysis is proposed for non-linear singular integral equations defined on a Banach space. Therefore, the non-linear equations are defined over a finite set of contours and the existence of solutions is investigated for two different kinds of equations, the first and the second kind. Moreover, the existence of solutions is further...
Fast solution of periodic integral equations by GMRES.
Galerkin Methods with Splines for Singular Integral Equations over (0, 1).
Hilbert transforms and the Cauchy integral in euclidean space
We generalize the notions of harmonic conjugate functions and Hilbert transforms to higher-dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These harmonic conjugates are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert...
Hypersingular integral equations and applications to porous elastic materials with periodic cracks
In this work a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions, is presented. The first goal of the present study is the development of an efficient analytical and direct numerical collocation method. The second one is the application of the method to the porous elastic materials when a periodic array of co-planar cracks is present. Starting from Cowin- Nunziato model...
Integral equations and boundary value problems for eliptic partial differential equations
Integral repesentations and its applications in Clifford analysis.
Inverse scattering via nonlinear integral equations method for a sound-soft crack with phaseless data
We consider the inverse scattering of time-harmonic plane waves to reconstruct the shape of a sound-soft crack from a knowledge of the given incident field and the phaseless data, and we check the invariance of far field data with respect to translation of the crack. We present a numerical method that is based on a system of nonlinear and ill-posed integral equations, and our scheme is easy and simple to implement. The numerical implementation is described and numerical examples are presented to...
Inversion of the Cauchy integral taken over the double periodic line.
Investigation of a certain system of singular integral equations of arbitrary power
Localization of Singular Integral Operators.
Mathematical Aspects of 'T Hooft's Eigenvalue Problem in Two-Dimensional Quantum Chromodynamics. Part I. A variational approach, and nodal properties of the eigenfunctions.
Mellin pseudodifferential operators with slowly varying symbols and singular integrals on Carleson curves with Muckenhoupt weights.
Midpoint collocation for Cauchy singular integral equations.
Nonlinear singular integral equations involving the Hilbert transform in Clifford analysis.
Numerical solution of Cauchy type singular integral equations by use of the Lobatto-Jacobi numerical integration rule
The Lobatto-Jacobi numerical integration rule can be extended so as to apply to the numerical evaluation of Cauchy type principal value integrals and the numerical solution of singular intergral equations with Cauchy type kernels by reduction to systems of linear equations. To this end, the integrals in such a singular integral equation are replaced by sums, as if they were regular integrals, after the singular integral equation is applied at appropriately selected points of the integration interval....
Numerical solution of integral equations with finite part integrals.
On a 3D-Hypersingular Equation of a Problem for a Crack
MSC 2010: 45DB05, 45E05, 78A45We show that a certain axisymmetric hypersingular integral equation arising in problems of cracks in the elasticity theory may be explicitly solved in the case where the crack occupies a plane circle. We give three different forms of the resolving formula. Two of them involve regular kernels, while the third one involves a singular kernel, but requires less regularity assumptions on the the right-hand side of the equation.