While the approach used by Barut & Kraus to give a non-perturbative alternative to the usual
approach to QED, a clear and straightforward derivation of their basic integro-differential
equation is not easily found in the literature. One is given here.
A pdf is available here:
https://www.gemarsh.com/wp-content/uploads/DERIVATION-OF-THE-BASIC-INTEGRO.pdf
DERIVATION OF THE BASIC INTEGRODIFFERENTIAL EQUATION FOR BARUT & KRAUS: NON-PERTURBATIVE QUANTUM ELECTRODYNAMICS