  We consider diffusion type equations with a distributed order derivative in
the time variable. This derivative is defined as the integral in $\alpha$ of
the Caputo-Dzhrbashian fractional derivative of order $\alpha \in (0,1)$ with a
certain positive weight function. Such equations are used in physical
literature for modeling diffusion with a logarithmic growth of the mean square
displacement. In this work we develop a mathematical theory of such equations,
study the derivatives and integrals of distributed order.
