We study the semi-Lagrangian method on curvilinear grids. The classical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time.

In this paper, by using the Dubovitskii-Milyutin theorem, we consider a differential inclusions problem with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. The Atangana-Baleanu fractional derivative of order α in the sense of Caputo with respect to time t, is considered. Existence and uniqueness of solution are proved by means of the Lions-Stampacchia theorem. The existence of solution is obtained for all values of the fractional parameter α ∈ (0, 1). Moreover, by applying control theory to the fractional differential inclusions problem, we obtain an optimality system which has also a unique solution. The controllability of the fractional Dirichlet problem is studied. Some examples are analyzed in detail. MSC: 46C05; 49J20; 93C20

Some new Hilbert-type inequalities involving Fenchel-Legendre transform are introduced. These inequalities give more general forms of some previously proved inequalities.