This paper develops a new numerical method of fundamental solutions for the non-homogeneous convection-diffusion equations with time-dependent heat sources. A summation of the fundamental solutions of the diffusion operator is considered with time-dependent coefficients for the solution of the underlying problem. By a $\theta$-weight discretiztion for the time derivative and selecting the source points and the field points at each time level, the solutions of all time levels are obtained. In addition, the stability of this approach is analyzed by considering $\theta=1$ in numerical results. This method is truly meshless and it is not necessary to discretize any part of the domain or boundary. As a result, this method is easily applicable to higher dimensional problems with irregular domains. In this work, we consider a non-homogeneous convection-diffusion equation (NCDE) in 2D with a regular domain and present some numerical results to show the effectiveness of the proposed method
