Present results concern the general scientific tendency dealing with mathematical modeling and analytical study of electromagnetic field phenomena described by the systems of partial differential equations. Specific electrodynamic engineering process with expofunctional influences is simulated by the differential Maxwell system whose effective research is equivalent to the rigorous solution of the general wave partial differential equation regarding all scalar components of electromagnetic field vector intensities. The given equation is solved explicitly in detail using method of integral transforms and irrespectively to the concrete boundary conditions. Specific cases of unexcited vacuum and isotropic homogeneous medium were considered. Proposed approach can be applied to any finite dimensional system of partial differential equations with piece wise constant coefficients and its corresponding scalar equations representing mathematical models in modern electrodynamics. In comparison with the known results, current research is completely thorough and accurate that implies its direct practical application.