Distributed order calculus and equations of ultraslow diffusion

We consider equations of the form D(μ)u (t, x) − u(t, x) = f (t,x), t >0, x ∈ Rn, where D(μ) is a distributed order derivative, that is D(μ)ϕ(t) = 1 0 D(α)ϕ (t)μ(α) dα, D(α) is the Caputo–Dzhrbashyan fractional derivative of order α, μ 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. © 2007 Elsevier Inc. All rights reserved