Approximation methods for the distributed order calculus using the convolution quadrature

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