A New Formulation of Pocklington's Equation for Thin Wires Using the Exact Kernel

Pocklington's integro-differential equation for thin wires with the exact kernel is reformulated using a second derivative formula for improper integrals. This allows for analytical evaluation of the second derivatives, resulting in a pure integral equation, in a similar manner to what is done when using the approximate kernel. However, as opposed to using the approximate kernel, the resulting integral equation developed here is numerically stable even with a simple pulse function/point matching solution. Good convergence for the current is obtained using pulse functions, and the severe unphysical oscillations of the current that are encountered when using pulse functions with the approximate kernel are avoided.