Time-marching numerical schemes for the electric field integral equation on a straight thin wire (Q1895914)

The authors propose numerical algorithms for solving the following system occurring in the problem of computing the current induced on a thin straight wire by a transient electric field: \[ A(\sigma, t)= {1\over 4\pi} \int^{1/2}_{-1/2} {u(s, t- R)\over R} ds,\;{\partial A\over \partial t} (\sigma, t)+ {\partial\Phi\over \partial\sigma} (\sigma, t)= E(\sigma, t),\;{\partial A\over \partial\sigma} (\sigma, t)+ {\partial\Phi\over \partial t} (\sigma, t)= 0, \] \(u(- 1/2, t)= u(1/2, t)= 0\), \(t> 0\), where \(R= |s-\sigma|\) and the functions \(A\), \(\Phi\), \(u\), \(E\) represent the electromagnetic potentials, the current and the transient electric field after the assumption of the straightness of the wire. A detailed analysis of the stability of the proposed algorithms is given.