Distributed order calculus and equations of ultraslow diffusion

We consider equations of the form (D((mu))u) (t, x) - Delta u(t, x) = f (t, x), t > 0, x is an element of R-n where D(mu) is a distributed order derivative, that is D-(mu)phi(t)=integral(1)(0)(D-(alpha) phi) (t)mu(alpha)d alpha, D(alpha), is the Caputo - Dzhrbashyan fractional derivative of order alpha, mu is a positive weight function. The above equation is 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. (c) 2007 Elsevier Inc. All rights reserved.