  Anomalous diffusion is a phenomenon that cannot be modeled accurately by
second-order diffusion equations, but is better described by fractional
diffusion models. The nonlocal nature of the fractional diffusion operators
makes substantially more difficult the mathematical analysis of these models
and the establishment of suitable numerical schemes. This paper proposes and
analyzes the first finite difference method for solving {\em
variable-coefficient} fractional differential equations, with two-sided
fractional derivatives, in one-dimensional space. The proposed scheme combines
first-order forward and backward Euler methods for approximating the left-sided
fractional derivative when the right-sided fractional derivative is
approximated by two consecutive applications of the first-order backward Euler
method. Our finite difference scheme reduces to the standard second-order
central difference scheme in the absence of fractional derivatives. The
existence and uniqueness of the solution for the proposed scheme are proved,
and truncation errors of order $h$ are demonstrated, where $h$ denotes the
maximum space step size. The numerical tests illustrate the global $O(h)$
accuracy of our scheme, except for nonsmooth cases which, as expected, have
deteriorated convergence rates.
