A simple Taylor-series expansion method for a class of second kind integral equations
The authors consider the integral equation \[ x(s)- \lambda \int^1_0 k(s,t)x(t) dt= y(s),\quad s\in [0,1], \] with kernel \(k\) either a continuous and (rapidly) decreasing convolution kernel, or containing a weak (algebraic) singularity. Motivated by certain limitations of the Taylor series expansion method proposed in \textit{M. Perlmutter} and \textit{R. Siegel} [ASME J. Heat Transfer, 85, 55-62 (1963)], they show how the desired Taylor coefficient functions can be found by solving a linear algebraic system which does not involve the use of boundary conditions. There is no convergence analysis, and no error estimates are given. Instead, the method is applied to three examples of integral equations with smooth kernels (arising in radiative heat transfer and electrostatics), and a constructed example with weakly singular kernel (where, not surprisingly, the method does not do well near the endpoints of the interval of integration).
scientific article; zbMATH DE number 1367865