Inversion of singular integrals with multiplicative Cauchy kernel and infinite integration domain

We consider the simplest singular integral equations of the first kind with multiplicative Cauchy kernel and with integration either over the three-dimensional domain (0,+∞) × (0,+∞) × (0,+∞) or over the entire three-dimensional Euclidean space. For these equations, we construct general solutions in the Hölder class and find statements of uniquely solvable problems.     

Explicit solutions of Cauchy singular integral equations with weighted Carleman shift

Existence and uniqueness of solutions, as well as their explicit representations, are obtained for singular integral equations with weighted Carleman shift which cannot be reduced to binomial boundary value problems.     

Midpoint collocation for Cauchy singular integral equations

Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems     

On solutions of an infinite system of singular integral equations

Using the classical Schauder fixed point principle we prove an existence result concerning an infinite system of singular integral equations. The obtained result is applied to establish the solvability of an infinite system of differential equations of fractional order.     

Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel

We construct an approximate solution of the singular integral equation with the Cauchy kernel with the use of Faber polynomials, prove its convergence to the exact solution, and indicate the order of the convergence rate.     

Solution of singular integral equations with logarithmic and Cauchy kernels

A direct method of solution is presented for singular integral equations of the first kind, involving the combination of a logarithmic and a Cauchy type singularity. Two typical cases are considered, in one of which the range of integration is a single finite interval and, in the other, the range of integration is a union of disjoint finite intervals. More such general equations associated with a finite number (greater than two) of finite, disjoint, intervals can also be handled by the technique employed here.     

Cauchy problem for singular evolution equations with pseudo-Bessel operators of infinite order

We prove the well-posed solvability of the Cauchy problem for evolution equations with pseudo-Bessel operators of infinite order and with initial data in the space of distributions.     

On polynomial collocation for Cauchy singular integral equations with fixed singularities

In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weightedL ² spaces, we derive necessary and sufficient conditions.     

Nyström method for Cauchy singular integral equations with negative index

In this paper, the authors propose a Nyström method to approximate the solutions of Cauchy singular integral equations with constant coefficients having a negative index. They consider the equations in spaces of continuous functions with weighted uniform norm. They prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates.     

Convergence in the integral metric of the general projective method for solving singular integral equations of the first kind with the Cauchy kernel

We study a projective method for solving singular integral equations of the first kind with the Cauchy kernel. Depending on the index of the equation, we introduce pairs of weight spaces which represent a restriction of the space of summable functions. We prove the correctness of the stated problem. We obtain sufficient conditions for the convergence of the projective method in the integral metric.     

Methods for solving a singular integral equation with Cauchy kernel on the real line

We study exact and approximate methods for solving a singular integral equation with Cauchy kernel on the real line. On the basis of the theory of positive operators, we prove an existence and uniqueness theorem for this equation in the space of Lebesgue square integrable functions. This theorem is then used to give a theoretical justification of general projection and projection-iteration methods as well as an iteration method for solving this equation.     

Computing some classes of Cauchy type singular integrals with Mathematica software

We constructed an algorithm, [SInt], for computing some classes of Cauchy type singular integrals on the unit circle. The design of [SInt] was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithm. Furthermore, we show how the factorization algorithm described in Conceição et al. (2010) allowed us to construct and implement the [SIntAFact] algorithm for calculating several interesting singular integrals that cannot be computed by [SInt]. All the above techniques were implemented using the symbolic computation capabilities of the computer algebra system Mathematica. The corresponding source code of [SInt] is made available in this paper. Several examples of nontrivial singular integrals computed with both algorithms are presented.     

An automatic augmented galerkin method for singular integral equations with Hilbert kernel

In [1, 2], described a Chebyshev series method for the numerical solution of integral equations with three automatic algorithms for computing two regularization parameters,C _f andr. Here we describe a Fourier series expansion method for a class of singular integral equations with Hilbert kernel and constant coefficients with using a new automatic algorithm.     

The Cauchy problem for some classes of quasi-linear parabolic equations

Theorems are established concerning the solubility in the large of the Cauchy problem for quasi-linear parabolic second-order equations.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 295–300, September, 1969.     

A new interpretation of Cauchy type singular integrals with an application to singular integral equations

Cauchy type integrals were given the interpretation of the principal value for points inside the integration interval. Here this interpretation is modified and generalized in a very simple manner. The new interpretation in general is not equivalent to the classical one. The relationship between the new interpretation and the classical one is investigated and various applications of the new interpretation (to the Plemelj formulas, the Riemann-Hilbert boundary value problem, singular integral equations, the inversion formula, quadrature rules and interface crack problems) are presented.     

Decay integral solutions for neutral fractional differential equations with infinite delays

Our aim in this work is to find decay integral solutions for a class of neutral fractional differential equations in Banach spaces involving unbounded delays. By constructing a suitable measure of noncompactness on the space of solutions and establishing new estimates for fractional resolvent operators, we prove the existence of a compact set of decay integral solutions to the mentioned problem. Copyright © 2014 John Wiley & Sons, Ltd.