Distributed Order Derivatives and Relaxation Patterns

We consider equations of the form $(D_{(ρ)}u)(t)=-λu(t)$, $t>0$, where $λ>0$, $D_{(ρ)}$ is a distributed order derivative, that is the Caputo-Dzhrbashyan fractional derivative of order $α$, integrated in $α\in (0,1)$ with respect to a positive measure $ρ$. Such equations are 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 $ρ$.

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