Distributed order calculus and equations of ultraslow diffusion

Abstract

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.

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

“…First of all, we mention here [7] and [8], where the fundamental solutions to the Cauchy problems for both the ordinary and the partial distributed order fractional differential equations have been derived and investigated in detail. In [11], a first attempt to analyze the initial-boundaryvalue problems for the generalized distributed order time-fractional diffusion equation over an open bounded multi-dimensional domain has been undertaken.…”

“…In this paper, we continue the research activities initiated in [7], [8], and [11] and investigate the asymptotic behavior of the solutions to the initial-boundary-value problems for the distributed order time-fractional diffusion equation both for the short and long times. As a byproduct of our investigation, we show the unique existence of solution to the problem under consideration.…”

Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density

“…For the theory of the general linear evolution equations with temporal fractional derivatives of distributed order we refer the reader e.g. to [17] and [36], where some existence and uniqueness results for the initial-value problems for these equations were given. Let us also cite the early paper [30].…”

Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations

“…Similarly, as in Chechkin et al (2002) and Kochubei (2008), rewrite (3.12) in the formĝ ðu; sÞ Z ð N 0 e KpðB u ðsÞCu 2 Þ dp; uR 0; ReB u ðsÞO 0; s 2 C C :…”

“…Similarly, as was carried out by Hanyga (2002b), Kochubei (2008) and Mainardi et al (2008), we use (3.12) to obtain a solution kernel. Three cases can be distinguished, according to the number of spatial dimensions.…”