Certain Comments on the Numeric Solutions of Integral Equations

Summary

In this work we consider the algorithm of solving integral equations of the second kind and of Fredholm type with a continuous kernel for the functions of one independent variable that is based on replacement of the integral by a sum. The possibility of this replacement is established using the theorem on a regular approximation of completely continuous operators (the strong convergence for uniform complete continuity of the approximating operators). We introduce a definition of closure of the computational algorithm, and indicate the possibility of the loss of significant digits in computations in the case when the algorithm is irregularly closed. We also give other applications of the closure of computational algorithms (see [1,2]).

Izv. Akad. Nauk SSSR, Ser. Mat., 20, 413–436 (1956)