Distributed order calculus and equations of ultraslow diffusion

We consider equations of the form(D(μ)u)(t,x)−Δu(t,x)=f(t,x),t>0,x∈Rn, where D(μ)D(μ) is a distributed order derivative, that isD(μ)φ(t)=∫01(D(α)φ)(t)μ(α)dα,D(α)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.