This work presents the analytical solution and temporal moments of one-dimensional advection-diffusion model with variable coefficients. Two case studies along with the two different sets of boundary conditions are considered at the inlet and outlet of the domain. In the first case, a time-dependent solute dispersion in the homogeneous domain along uniform flow is taken into account, whereas in the second case, due to inhomogeneity of domain, velocity is taken spatially dependent and the dispersion is assumed proportional to the square of the velocity. The Laplace transform is used to obtain the analytical solutions. The analytical temporal moments are derived from the Laplace domain solutions. To verify the correctness of the analytical solutions, a high-resolution second-order finite volume scheme is applied. Different case studies are considered and discussed. Both analytical and numerical results are in good agreement with each other.