Solution of a singular integro-differential equation with the use of asymptotic polynomials (Q2434665)

The authors study the singular integro-differential equation \[ \int_{-1}^{+1}\frac {\varphi'(\tau)} {\tau-t} d\tau+ \int_{-1} ^{+1} h(t,\tau)\varphi (\tau)d \tau=y(t),\quad | t|\leq 1, \] with the boundary conditions \(\varphi(-1)=\varphi (+1)=0\). The kernel \(h(t,\tau)\) is a given positive function, \(\varphi\) is the unknown function. The singular integral is considered in the sense of the Cauchy principal value. The solving of the given equation is based on using of an asymptotic polynomial of the second kind. The remainder term of the approximate solution is expressed as an infinite sum via linear functionals. The convergence of the presented method is proven.