Numerical analysis for distributed-order differential equations

In this paper we present and analyse a numerical method for the solution of a distributed-order differential equation of the general form ∫ 0 m A ( r , D ∗ r u ( t ) ) d r = f ( t ) where m is a positive real number and where the derivative D ∗ r is taken to be a fractional derivative of Caputo type of order r . We give a convergence theory for our method and conclude with some numerical examples.