Analytic solutions to Maxwell's equations: sinusoidal steady-state and transient space-time problems in transverse magnetic and transverse electric field analysis

Maxwell's equations form the basis in electromagnetic field theory. The electromagnetic field, if it exists, satisfies Maxwell's equations and the boundary conditions associated. These equations are simple in the form but contain the variations of the field quantities throughout three-dimensional space (rectangular, cylindrical, and spherical coordinate systems are used) and time. The general solution to Maxwell's equations is usually difficult to, find. However, analysis of electromagnetic fields requires one to find the general solution without simplifications and assumptions, and our goal is to obtain explicit analytic solutions to Maxwell's equations with the corresponding boundary conditions. This paper researches methods and reports a straightforward mathematical foundation for solving Maxwell's equations in analysis of transverse magnetic (TM) and transverse electric (TE) fields.