The solution of infinitely ill-conditioned weakly-singular problems

It is demonstrated on examples that a weak singularity (i.e., with converging improper integral) may produce in computations (depending on the algorithm employed) an infinitely ill-conditioned situation when arbitrarily small imprécisions introduced by the algorithm or by a software create divergent approximations for mathematically convergent integrals. The possibility of hidden singularities is shown, and the double error phenomenon is identified and demonstrated in a simple example. Construction of test problems is proposed to check the applicability of existing software prior to its use for the solution of real life problems with weakly-singular equations. It is shown that the application of the integration by parts formula to weakly-singular integrals may create strong singularities (i.e., unbounded terms or divergent improper integrals). Methods of removal of singularities with and without compensation are studied for the numerical solution of infinitely ill-conditioned weakly-singular problems.