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A numerical method for the solution of integral equations of Mellin type
Full text linkWe 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
Condition numbers for singular integral equations in weighted L2 spaces
No full textAbstractThe 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 quadrature method for Cauchy Singular Integral Equations with index -1
No full textIn this paper a quadrature method for Cauchy singular integral equations having constant coefficients and index equal to -1 is proposed. A polynomial approximation of the solution is constructed by solving a determined and well conditioned linear system. Error estimates and numerical tests are also included
A numerical method for the solution of exterior Neumann problems for the Laplace equation in domains with corners
Full text linkIn 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
A Nyström method for solving the exterior Neumann problem on planar domains with corners
No full textThis 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
A numerical method for the Dirichlet problem on domains with corners
No full textThis talk deals with the numerical solution of the interior Dirichlet 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 double layer boundary integral equation.
The convergence and stability of the method are proved and some numerical tests are shown
On the evaluation of some integral operators with Mellin type kernel
No full textWe 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 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 . \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 , .
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 is very close to the endpoint . Then, such algorithm is applied to the numerical solution of second kind integral equations of Mellin type
A modified Nystrom method for integral equations with Mellin type kernels
No full textThe aim of this paper is to propose a new modified Nyström method for the approximation of the solutions of second kind integral equations with fixed singularities of Mellin convolution type. The stability and the convergence are proved in L^2 spaces and error estimates in L^2 norm are given. Finally, numerical tests showing the effectiveness of the method are presented
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