Two analog computing methods are proposed to compute the continuous-time solutions of one-dimensional (1-D) Maxwell’s equations. In the first method, the spatial domain partial derivatives in the governing partial differential equation (PDE) are approximated using discrete finite differences while applying the Laplace transformation along the time dimension. The resulting spatially discrete time-continuous update equation is utilized to design an analog circuit that can compute the continuous-time solution. The second method replaces the discrete-time difference operators in the standard finite difference time domain (FDTD) cell (Yee cell) using continuous-time delay operators, which can be realized using analog all-pass filters. Both methods have been simulated using ideal analog circuits in Cadence Spectre for the Dirichlet, Neumann, and radiation boundary conditions. The performance of the proposed methods has been quantified using: i) mean squared differences between the results and fully discrete FDTD simulations and ii) the noise to signal energy ratio. Both methods have been extended to design the analog circuits that compute the continuous-time solution of the 1-D and 2-D wave equations. The 1-D wave equation solver is simulated with a dominant-pole model (which better approximates the non-ideal circuit behavior) along with a propagation delay compensation technique. The experimental results from a simplified board-level low-frequency implementation are also presented. The key challenges toward CMOS implementations of the proposed solvers are identified and briefly discussed with possible solutions.