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.
On the evaluation of some integral operators with Mellin type kernel
LAURITA, Concetta
2014-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
