  The second and all higher order moments of the $\beta$-stable L\'{e}vy
process diverge, the feature of which is sometimes referred to as shortcoming
of the model when applied to physical processes. So, a parameter $\lambda$ is
introduced to exponentially temper the L\'{e}vy process. The generator of the
new process is tempered fractional Laplacian $(\Delta+\lambda)^{\beta/2}$ [W.H.
Deng, B.Y. Li, W.Y. Tian, and P.W. Zhang, Multiscale Model. Simul., in press,
2017]. In this paper, we first design the finite difference schemes for the
tempered fractional Laplacian equation with the generalized Dirichlet type
boundary condition, their accuracy depending on the regularity of the exact
solution on $\bar{\Omega}$. Then the techniques of effectively solving the
resulting algebraic equation are presented, and the performances of the schemes
are demonstrated by several numerical examples.
