Wavelets explain the stability of Pocklington´s equation

Several of the integral equations in everyday use for electromagnetic computations are "not well-posed" (e.g. they are ill-posed). In most other fields, solutions occur, but with arbitrarily large errors that cause serious practical problems. On the other hand, the specific ill-posed equations used in electromagnetics seem to work just fine in practice. G.C. Hsiao and R.E. Kleinman (see IEEE Trans. Antennas and Propag., vol.45, no.3, p.316-28, 1997) considered the ill-posed problem of the electric field integral equation (EFIE). It has been suggested that Pocklington\´s integral equation is even less well behaved. Our approach is two fold. We re-examine a paper (Canning, F.X. and Scholl, J.F., IEEE Trans. Antennas and Propag., vol.44, no.9, p.1239-46, 1996) which shows how wavelets can be used to improve the condition number (or stability) of the EFIE. We show how these results are equivalent to the results of Hsiao and Kleinman. These wavelet results provide a particularly clear physical picture of what is happening. The stability of the EFIE for its intended purpose becomes clear, in spite of its ill-posed nature. Second, we extend the wavelet related results of Canning and Scholl to Pocklington\´s integral equation, and explicate its stability properties.