Approximation methods for the distributed order calculus using the convolution quadrature

Abstract

In this article we generalize the convolution quadrature (CQ) method, which aims at approximating the fractional calculus, to the case for the distributed order calculus. Our method is a natural expansion that the approximation formulas, convergence results and correction technique reduce to the cases for the CQ method if the weight function \begin{document}$ \mu(\alpha) $\end{document} is defined by \begin{document}$ \delta(\alpha-\alpha_0) $\end{document} . Further, we explore a new structure of the solution of an ODE with the distributed order fractional derivative, which differs from those of the solutions of traditional fractional ODEs, and propose a new correction technique for this new structure to restore the optimal convergence rate. Numerical tests with smooth and nonsmooth solutions confirm our theoretical results and the efficiency of our correction technique.