The solution of infinitely ill-conditioned weakly-singular problems

It is demonstrated on examples that a weak singularity (i.e., with convergi ng improper integral) may produce in computations (depending on the algorit hm employed) an infinitely ill-conditioned situation when arbitrarily small imprecisions 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 identifi ed and demonstrated in a simple example. Construction of test problems is p roposed to check the applicability of existing software prior to its use fo r the solution of real life problems with weakly-singular equations. It is shown that the application of the integration by parts formula to weakly-si ngular 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. (C) 2000 Elsevier Science Ltd. Al l rights reserved.