Distributed order derivatives and relaxation patterns

Abstract

We consider equations of the formis the Caputo-Dzhrbashyan fractional derivative of order α, ρ is a positive measure. The above equation is used for modeling anomalous, non-exponential relaxation processes. In this work we study asymptotic behavior of solutions of the above equation, depending on properties of the measure ρ.

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

“…For a fixed value of λ n , this problem was investigated in [7] and [8]. In this part of the section, we follow the analysis employed in [8] but additionally take into consideration the limit relation λ n → ∞ as n → ∞. Our target is to describe a long-time asymptotic behavior of the solution u n (t) to the Cauchy problem (3.2).…”

“…Our target is to describe a long-time asymptotic behavior of the solution u n (t) to the Cauchy problem (3.2). To reach this aim, we need a more detailed analysis compared to one provided in [7] and [8].…”

“…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.…”

“…Let us note here that because we work with the initial-boundary-value problems and not with the Cauchy problems for the distributed order fractional diffusion equations, we need some more precise estimates for the solutions of the corresponding eigenvalue problems compared to ones deduced in [7] and [8]. Still, some techniques suggested in [7] and [8] are employed for our problem, too.…”