The numerical solution of Cauchy singular integral equations with additional fixed singularities

In this paper we propose a quadrature method for the numerical solution of Cauchy singular integral equations with additional fixed singularities. The unknown function is approximated by a weighted polynomial which is the solution of a finite dimensional equation obtained discretizing the involved integral operators by means of a Gauss-Jacobi quadrature rule. Stability and convergence results for the proposed procedure are proved. Moreover, we prove that the linear systems one has to solve, in order to determine the unknown coefficients of the approximate solutions, are well conditioned. The efficiency of the proposed method is shown through some numerical examples

Condition numbers for singular integral equations in weighted L2 spaces

AbstractThe convergence and stability of a discrete collocation method for Cauchy singular integral equations in some weighted Besov spaces are studied. This numerical method results in solving a linear system in order to determine the unknown coefficients of the approximate solution. The author proves that this linear system is well conditioned

A numerical method for the solution of exterior Neumann problems for the Laplace equation in domains with corners

In this paper we propose a new boundary integral method for the numerical solution of Neumann problems for the Laplace equation, posed in exterior planar domains with piecewise smooth boundaries. Using the single layer representation of the potential, the differential problem is reformulated as a classical boundary integral equation. The use of a smoothing transformation and the introduction of a modified Gauss–Legendre quadrature formula for the approximation of the singular integrals, which turns out to be convergent, leadsus to apply a Nyström type method for the numerical solution of the integral equation. We solve some test problems and present the numerical results in order to show the efficiency of the proposed procedure

On the evaluation of some integral operators with Mellin type kernel

We consider the numerical evaluation of integral transform of the form \begin{equation} \label{integraloperator} ({\mathcal K}f)(y)=\int_0^1\frac{1}{x}k\left(\frac{y}{x}\right)f(x)dx, \quad y \in (0,1], \end{equation} for some given function $k:[0,\infty)\rightarrow [0,\infty)$ satisfying suitable assumptions. These operators of Mellin convolution type are not compact and their kernels are not smooth but contain a fixed strong singularity at $x=y=0$. \newline The mathematical formulation of many problems in physics and engineering gives rise to the solution of second kind integral equations involving operators of the form (\ref{integraloperator}). When we are interested in the numerical solution of such equations by means of Nystr\"om or discrete collocation methods, efficient quadrature formulas are necessary, in order to approximate the integrals $({\mathcal K}f)(y)$, $y\in (0,1]$. The aim of this talk is to propose an algorithm for the evaluation of these integrals, since the fixed singularity of the Mellin kernel at the origin makes inefficient the use of the classical Gaussian rules when $y$ is very close to the endpoint $0$. Then, such algorithm is applied to the numerical solution of second kind integral equations of Mellin type

A numerical method for the solution of integral equations of Mellin type

We are interested in the numerical solution of second kind integral equations of Mellin convolution type. We describe a modified Nyström method based on the Gauss–Lobatto or Gauss–Radau quadrature rule. Under certain assumptions on the Mellin kernel, we prove the stability and the convergence of the proposed procedure and also derive error estimates. Finally, some test problems are solved and the numerical results showing the effectiveness of our method are presented

A Nyström method for integral equations of the second kind with fixed singularities based on a Gauss-Jacobi-Lobatto quadrature rule

The Gauss-Lobatto quadrature rule for integration over the interval [−1,1], relative to a Jacobi weight function w^{α,β} (t) = (1−t)^α(1+t)^β , α,β > −1, is considered and an error estimate for functions belonging to some Sobolev-type subspaces of the weighted space L^1_w^{α,β} ([−1,1]) is proved. Then, a Nyström type method based on a modified version of this quadrature formula is proposed for the numerical solution of integral equations of the second kind with kernels having fixed singularities at the endpoints of the integration interval and satisfying proper assumptions. The stability and the convergence of the proposed modified Nyström method in suitable weighted spaces are proved and confirmed through some numerical tests

A new stable numerical method for Mellin integral equations in weighted spaces

In this paper a new modified Nyström method is proposed to solve linear integral equations of the second kind with fixed singularities of Mellin convolution type. It is based on the Gauss-Radau quadrature formula with a suitable Jacobi weight. The stability and convergence of the method is proved in weighted spaces with uniform norm. Moreover, an error estimate of the numerical solution is given under certain assumptions on the Mellin kernel. The efficiency of the method is shown through some examples. The numerical results also confirm that the error estimate is sharp

A Nyström method for solving the exterior Neumann problem on planar domains with corners

This talk deals with the numerical solution of the exterior Neumann problem for Laplace's equation on planar domains with corners. The Authors propose a numerical method of Nyström type, based on a Lobatto quadrature rule, in order to approximate the solution of the corresponding boundary integral equation of the direct type. The convergence and stability of the method are proved and some numerical tests are shown