Distributed order calculus and equations of ultraslow diffusion

Abstract

We consider equations of the form ( D ( μ ) u ) ( t , x ) − Δ u ( t , x ) = f ( t , x ) , t > 0 , x ∈ R n , where D ( μ ) is a distributed order derivative, that is D ( μ ) φ ( t ) = ∫ 0 1 ( D ( α ) φ ) ( t ) μ ( α ) 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.