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In this work, we consider boundary value problems involving Caputo andRiemann-Liouville fractional derivatives of order $\alpha\in(1,2)$ on the unitinterval $(0,1)$. These fractional derivatives lead to non-symmetric boundaryvalue problems, which are investigated from a variational point of view. Thevariational problem for the Riemann-Liouville case is coercive on the space$H_0^{\alpha/2}(0,1)$ but the solutions are less regular, whereas that for theCaputo case involves different test and trial spaces. The numerical analysis ofthese problems requires the so-called shift theorems which show that thesolutions of the variational problem are more regular. The regularity pickupenables one to establish convergence rates of the finite elementapproximations. Finally, numerical results are presented to illustrate theerror estimates.

Variational formulation of problems involving fractional order differential operators