This paper derives the time-dependent fundamental solution of the transient convection-diffusion problem by employing the exponential variable and Fourier transformations. A singular boundary method (SBM) formulation using this time-dependent fundamental solution is first applied in the simulation of the transient convection-diffusion problems. Accurate formulas are derived to evaluate the origin intensity factors in the SBM. The proposed method is mathematically simple, matrix-free and fully explicit. Furthermore, this scheme is computationally fast, stable, and requires low memory, since it does not need to solve any algebraic equations. Three benchmark examples, including three-dimensional cases, are presented to verify this time-dependent SBM strategy. (C) 2017 Elsevier Ltd. All rights reserved.