Abstract: In this paper, we will study the time fractional convection-diffusion equation with variable coefficients. First, we use the finite difference method. The time variable is discretized, and the semi-discrete scheme of the equation is obtained. The exact solution $u(x,t_{n})$ of the equation is obtained by using the theory of reproducing kernel method. Then the exact solution $u(x,t_{n})$ is truncated by $m$ term to obtain the approximate solution $u_{m}(x,t_{n})$. By proving, we know that the method is stable. Moreover, $u_{m}^{(i)}(x,t_{n})$ converge uniformly to $u^{(i)}(x,t_{n})$ $(i=0,1,2)$. Finally, we give several numerical examples and compare them with the methods in other literatures, which show that our algorithm is effective.

Key words: Caputo fractional derivative, reproducing kernel method, variable coefficient time fractional convection-diffusion equation, finite difference method