Two analog computing methods are proposed to compute the continuous-time solutions of Maxwell's equations. 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 have been quantified using i) mean squared differences between the results and fully-discrete FDTD simulations, and ii) the noise to signal energy ratio. The proposed methods have been extended to design analog circuits that compute the continuous-time solution of the 1-D and 2-D wave equations. Experimental results from a simplified board-level low-frequency implementation are also presented.