Time-fractional Diffusion of Distributed Order

By Francesco Mainardi, Antonio Mura, Gianni Pagnini, Rudolf Gorenflo. 2007

222 citations recorded by the source corpus.

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville and the Caputo sense. For a general distribution of time orders we provide the fundamental solution, which is a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative, except for the single order case where the two approaches turn out to be equivalent. We consider in some detail two cases of order distribution: Double-order, and uniformly distributed order. Plots of the corresponding fundamental solutions and their variance are provided for these cases, pointing out the remarkable difference between the two approaches for small and large times.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/357142e9221d99b5dfd3ad696e86863d9a025396

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations (2014, unknown): “, [3]-[4], [15], [17], [21] and the references therein).”
General Fractional Calculus, Evolution Equations, and Renewal Processes (2011, background): “2) A recent theory of evolution equations with distributed order derivatives (used for modeling ultraslow relaxation and diffusion processes; see [4, 9, 19, 20] and references therein) is based”
Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density (2013, unknown): “[26], [28], and [35].”
Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method (2017, background): “where the function c(α) acting as weight for the order of differentiation is such that ([30], [45]) c(α) ≥ 0 and ∫ 1”
Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations (2009, methodology): “(b ) Solution kernel of the distributed diffusion-wave equation Similarly, as was carried out by Hanyga (2002b), Kochubei (2008) and Mainardi et al. (2008), we use (3.12) to obtain a solution kernel.”