Analytical Solutions of 3D Maxwell's Equations via Infinite-Order Curl-Operator Expansions

Abstract

Solving Maxwell's equations to obtain explicit analytical representations of electromagnetic fields in open-space settings with general initial conditions and source terms remains a fundamental challenge. We address this problem through an operator-based construction, developing an infinite-order curl-operator expansion that yields analytical solutions as function-to-function mappings from initial data and source terms to electromagnetic fields. The framework is applied to representative cases, including Gaussian initial conditions, Gaussian sources, harmonic sources, and Ricker wavelet sources. In these applications, a family of deformed sine and cosine functions emerges naturally, enabling explicit analytical representations of the solutions. The analysis further shows that these deformed trigonometric functions are intrinsic to the general solution formulas, with their structure determined by the initial conditions and source terms. The results are compared with numerical simulations based on the FDTD method and further applied to a systematic study of mesh refinement effects. This study shows that, for the problems considered, the error measures approach limiting values as the mesh is refined, indicating that further step size reduction alone may produce little additional improvement once the mesh is sufficiently fine. The results demonstrate that the analytical solutions provide theoretical insight into electromagnetic field behavior while serving as practical benchmark solutions for computational electromagnetics.

54 pages, 72 figures. Major revision: added convergence conditions in the solution theorems; separated the uniqueness theorem into two theorems with independent proofs; added sections on deformed trigonometric functions and comparison with existing analytical methods; revised numerical results with a new FDTD initial-value finding and reorganized presentation