Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo-Fabrizio fractional derivative

We consider distributed-order partial differential equations with time fractional derivative proposed by Caputo and Fabrizio in a one-dimensional space. Two finite difference schemes are established via Grunwald formula. We show that these two schemes are unconditionally stable with convergence rates O(tau 2+h2+Delta alpha 2) and O(tau 2+h4+Delta alpha 4) in discrete L2, respectively, where Delta alpha, h, and tau are step sizes for distributed-order, space, and time variables, respectively. Finally, the performance of difference schemes is illustrated via numerical examples.