  This chapter presents some numerical methods to solve problems in the
fractional calculus of variations and fractional optimal control. Although
there are plenty of methods available in the literature, we concentrate mainly
on approximating the fractional problem either by discretizing the fractional
term or expanding the fractional derivatives as a series involving integer
order derivatives. The former method, as a subclass of direct methods in the
theory of calculus of variations, uses finite differences, Grunwald-Letnikov
definition in this case, to discretize the fractional term. Any quadrature rule
for integration, regarding the desired accuracy, is then used to discretize the
whole problem including constraints. The final task in this method is to solve
a static optimization problem to reach approximated values of the unknown
functions on some mesh points.
  The latter method, however, approximates fractional problems by classical
ones in which only derivatives of integer order are present. Precisely, two
continuous approximations for fractional derivatives by series involving
ordinary derivatives are introduced. Local upper bounds for truncation errors
are provided and, through some test functions, the accuracy of the
approximations are justified. Then we substitute the fractional term in the
original problem with these series and transform the fractional problem to an
ordinary one. Hereafter, we use indirect methods of classical theory, e.g.
Euler-Lagrange equations, to solve the approximated problem. The methods are
mainly developed through some concrete examples which either have obvious
solutions or the solution is computed using the fractional Euler-Lagrange
equation.
