Numerical electromagnetic methods are basically classified into two methods, namely, the differential and integral equation based methods. The differential equation based methods need to use the truncation boundary conditions; however, the integral equation based methods can implicitly satisfy the radiation condition. As far as the integral equation based methods are concerned, for the two-dimensional objects, the frequency-domain methods are intensively studied for the perfectly electrically conducting (PEC), homogeneous and inhomogeneous dielectric bodies [Harrington, 1993; Richmond, 1965, 1966]. In the time-domain, the integral equation based methods for PEC and homogeneous objects are often investigated [Bennett, 1968; Mittra, 1976; Vechinski and Rao, 1992], but the methods for the inhomogeneous objects are seldom studied.
 This paper proposes a time-domain volume integral equation (TDVIE) based method for analyzing the scattering from inhomogeneous cylinders. Though this TDVIE based method has been used to analyze the scattering from the nonlinear objects under the transverse magnetic (TM) illumination [Wang et al., 2000], it is not trivial extension of the TM case to the transverse electric (TE) case. For the sake of completeness, this paper presents both the TM and TE cases.
 Consider an inhomogeneous dielectric cylinder in the free space. It is illuminated by an incident plane wave. The dielectric cylinder is assumed to have the same permeability as that in the free space (μ = μ0). The permittivity is assumed to be ε(r). To facilitate the derivation of the integral equation, the total electric field is decomposed into the incident and scattered components as
 By invoking the volume equivalence principle, the scattered field is characterized by an equivalent volume current density J(r, t) that radiates in the unbounded free space and is given by
where D(r, t) = ε(r)E(r, t) denotes the electric flux density, and κ(r) = (ε(r) − ε0)/ε(r) is the contrast ratio. For the nonlinear objects [Wang et al., 2000], ε(r) is a function of the total electric field E(r, t). Here, we only consider the linear inhomogeneous objects in which ε(r) is just dependent on the position r.
 In the following two sections, the TDVIE based method will be presented for the TM and TE cases, respectively. The marching-on-in-time (MOT) based scheme will be proposed to solve the TDVIE. In section 4, some numerical results are presented for inhomogeneous circular and square cylinders. For the sake of comparison, the results obtained using the finite-difference time-domain (FDTD) method are also presented. The results obtained using the TDVIE and FDTD methods are in excellent agreement with each other.