Efficient solution of time-fractional differential equations with a new adaptive multi-term discretization of the generalized Caputo–Dzherbashyan derivative

In this paper, we deal with both the discretization and the efficient solution of initial and initial-boundary value problems with a time derivative of distributed order. A new discretization based on an adaptive Gauss quadrature and product integral formulas is introduced and analyzed. The efficient solution of the resulting sequence of linear systems by Krylov iterative methods and approximate inverse preconditioning is discussed, together with the spectral analysis of the relative matrix sequences. Several numerical examples showing the effectiveness of the approach are included.