## 1 Introduction  
The fractional partial differential equations (FPDEs) have become increasingly popular in recent years. The interest in these equations comes from their mathematical structure and from their applications. FPDEs are applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and so on. Several ways to solve them theoretically have been proposed [1; 2; 3], including the Green function method and Laplace and Fourier transform methods [4; 5], the Adomian decomposition [6; 7], and the Homotopy Perturbation methods [8; 9].  
Generally, numerical solution techniques are preferred when dealing with fractional models since the analytical solutions are available for a few simple cases, but also the memory effect of the problem under consideration may lead to arbitrarily wild solutions, as shown in [10]. In recent years, efficient numerical methods have been developed to solve fractional differential equations, including finite difference methods [11; 12; 13; 14; 15], the finite volume method [16], the finite element method [17; 18; 19], and the spectral method [20; 21].  
Generally, Riemann-Liouville derivatives and Caputo derivatives are used for the formulation of fractional differential problems, and often, the Riemann-Liouville formula is approximated by the Caputo one. As for the non-fractional differential equations involving the Caputo derivative, finite difference methods are one of the most important classes of numerical methods for solving FPDEs. [PERSON] and [PERSON] [22] obtained an implicit difference approximation to solve time-fractional diffusion equations. [PERSON] and [PERSON] [21; 23] proposed the numerical solution by finite/difference approximations for a time-fractional diffusion equation. [PERSON] et al. [24] developed an explicit difference method and an implicit difference method for solving a space-time fractional advection dispersion equation on a finite domain. In [25; 26; 27], high order numerical difference schemes were constructed in order to solve the Caputo-type advection-diffusion equations. [PERSON] et al. [28] obtained a finite difference method for FPDEs involving the Caputo derivative on a non-uniform mesh. Recently [PERSON] et al. [29; 30; 31; 32] determined exact and numerical solutions for the time and space fractional advection-diffusion differential equations involving the Riemann-Liouville derivative with a non-linear source term by means of the Lie symmetries. The authors transformed the fractional partial differential equations into a fractional ordinary differential equations, written in terms of Caputo derivatives, which were then solved using implicit finite difference methods.  
The main goal of this paper was to construct an unconditionally-stable, implicit finite difference method for solving the time-fractional advection-diffusion equations (TFADEs) with a non-homogeneous source term involving the Caputo fractional derivative on non-uniform grids. We chose to use a special non-uniform mesh in order to improve the numerical accuracy of the classical discrete fractional formula for the Caputo derivative, since the fractional derivatives are integrals with weakly singular kernels, and the discretization on the uniform mesh may lead to poor accuracy. The consistency, stability, and convergence of the proposed difference method were investigated. Three numerical examples are given to show the reliability and efficiency of the derived difference method.