  In this paper, we are interested in the study of a problem with fractional
derivatives having boundary conditions of integral types. The problem
represents a Caputo type advection-diffusion equation where the fractional
order derivative with respect to time with $1<\alpha <2$. The method of the
energy inequalities is used to prove the existence and the uniqueness of
solutions of the problem. The finite difference method is also introduced to
study the problem numerically in order to find an approximate solution of the
considered problem. Some numerical examples are presented to show satisfactory
results.
