Finite Difference Scheme for a Parabolic Variational Inequality with Time-fractional Derivative

The ‘‘obstacle problem’’ is posed for a one-dimensional equationwith a monotone quasilinear diffusion operator and a materialfractional derivative of variable order in time (fractionalderivative with respect to the characteristics of the convectionoperator). An implicit finite-difference scheme is constructed andinvestigated. The grid scheme is formulated on the current timelayer in two forms: as a complementarity problem and as aninclusion containing a diagonal multivalued maximal monotoneoperator. The inclusion formulation is used to obtain a prioriestimates of the solution in the maximum norm with a maximum-normor grid \(L_{1}((0;T);L\infty)\) -norm for the right-hand side. Inturn, the formulation in the form of a complementarity problemallows us to construct a grid scheme for projecting onto the gridthe exact solution of the differential problem. The error ofapproximation of the grid scheme on the assumed smooth solution ofthe differential problem is studied. The obtained stability andapproximation estimates imply the accuracy estimate, which turnsout to be the same as in the case of the corresponding equationwith a smooth solution.