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Zbl 1149.26014
Kochubei, Anatoly N.
Distributed order calculus and equations of ultraslow diffusion.
(English)
[J] J. Math. Anal. Appl. 340, No. 1, 252-281 (2008). ISSN 0022-247X

Summary: We consider equations of the form $$(\bbfD^{(\mu)}u)(t,x)-\Delta u(t,x)=f(t,x), \quad t>0, x\in\bbfR^n$$ where $\bbfD^{(\mu)}$ is a distributed order derivative, that is $$\bbfD^{(\mu)}\varphi (t)=\int_0^1\bbfD^{(\alpha)}\varphi )(t)\mu (\alpha)~d\alpha,$$ $\bbfD^{(\alpha)}$ is the Caputo--Dzhrbashyan fractional derivative of order $\alpha$, $\mu$ 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.
MSC 2000:
*26A33 Fractional derivatives and integrals (real functions)
35K57 Reaction-diffusion equations

Keywords: fractional derivatives and integrals; ultraslow diffusion

Cited in: Zbl 1164.26009

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