On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation
The Fredholm integral equation of the second kind \(x-Tx=b\) in Banach space \(C[-1,1]\) is considered. Here \[ T:C[-1,1]\to C[-1,1],\quad (Tx)(s):=\int\limits_{-1}^{1}k(s,t)x(t)dt, \] with continuous kernel \(k:[-1,1]^2\to\mathbb{R}\) and \(b\in C[-1,1]\). The suggested numerical method is based on the approximation of the integral operator \(T\) by quasi-interpolating the density function using Gaussian kernels. The authors show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström's method with the trapezoidal rule. For this, a convergence analysis is carried out. The paper also contains some numerical results illustrating the theoretical estimates.
scientific article; zbMATH DE number 5870906