On some generalizations of the implicit Euler method for discontinuous fractional differential equations

We discuss the numerical solution of differential equations of fractional order with discontinuous right–hand side. Problems of this kind arise, for instance, in sliding mode control. After applying a set–valued regularization, the behavior of some generalizations of the implicit Euler method is investigated. We show that the scheme in the family of fractional Adams methods possesses the same chattering–free property of the implicit Euler method in the integer case. A test problem is considered to discuss in details some implementation issues and numerical experiments are presented.