We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form (Dwhere k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation D (k) u = −λu, λ > 0, proved to be (under some conditions upon k) continuous on [0, ∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N (E(t)) as a renewal process. Here N (t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.Mathematics Subject Classification (2010). Primary 26A33, 34A08, 35R11; Secondary 60K05.

We consider an evolution equation with the regularized fractional derivative of an order aAð0; 1Þ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated. r

We consider equations of the formis 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.