  For the fractional Laplacian of variable order, an efficient and accurate
numerical evaluation in multi-dimension is a challenge for the nature of a
singular integral. We propose a simple and easy-to-implement finite difference
scheme for the multi-dimensional variable-order fractional Laplacian defined by
a hypersingular integral. We prove that the scheme is of second-order
convergence and apply the developed finite difference scheme to solve various
equations with the variable-order fractional Laplacian. We present a fast
solver with quasi-linear complexity of the scheme for computing variable-order
fractional Laplacian and corresponding PDEs. Several numerical examples
demonstrate the accuracy and efficiency of our algorithm and verify our theory.
