Distributed order calculus and equations of ultraslow diffusion

By Anatoly N. Kochubei. 2007

359 citations recorded by the source corpus.

Abstract We consider equations of the form ( D ( μ ) u ) ( t , x ) − Δ u ( t , x ) = f ( t , x ) , t > 0 , x ∈ R n , where D ( μ ) is a distributed order derivative, that is D ( μ ) φ ( t ) = ∫ 0 1 ( D ( α ) φ ) ( t ) μ ( α ) d α , D ( α ) is 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.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/3d00d95b719220a99447a618a8cf4059e3b53005

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Citations in the corpus

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.”
Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation (2011, neutral): “Let us mention here that in [15] the solution of the eigenvalue problem for the distributed order fractional derivative has been given.”
The Fractional Poisson Process and the Inverse Stable Subordinator (2010, neutral): “19) in Kochubei [27] P(Jn > t) = E(e −λEν (t)) = λ π ∫ ∞”
Relaxation patterns and semi-Markov dynamics (2015, background): “In [25,26] he also studied distributed-order relaxation patterns, i.”
Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations (2014, methodology): “For a fixed value of λn, this problem was investigated in [7] and [8].”
A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel (2020, background): “Consider the simplest distributed-order derivative [4, 8, 13] with the kernel w ∫”
On fractional Duhamel's principle and its applications (2010, background): “For more information about recent results on the Cauchy problem for abstract fractional differential-operator equations, we refer the reader to [3, 10, 16]; and for a recent mathematical treatment of the distributed fractional order differential equations to papers [18, 23, 35].”
Fractional derivatives and the fundamental theorem of fractional calculus (2020, background): “Thus, the many parameters families of the fractional integrals and derivatives (say, the Erdélyi-Kober operators [19, 38] or their Caputo type modifications [28, 20]), the distributed order fractional derivatives [5, 6, 21, 24], the fractional derivatives with the general kernels [22, 39, 40], and t”
On History of Mathematical Economics: Application of Fractional Calculus (2019, background): “The growth equation is considered in [242] for the special case of a distributed order derivative, where it was proved that a smooth solution exists and is monotone increasing.”
Distributed-order fractional diffusions on bounded domains (2009, background): “6) as the definition of the Caputo derivative, see for example Kochubei [20].”
Well-posedness of general Caputo-type fractional differential equations (2018, background): “Kochubei studied the existence, uniqueness and asymptotic behavior of solutions of distributed order diffusion equations in [9].”
Positivity, monotonicity, and convexity for convolution operators (2020, background): “An account of their main properties can be found in [26].”
General Fractional Calculus, Evolution Equations, and Renewal Processes (2011, background): “For example, it would be sufficient to assume its asymptotic properties found in [14] for the distributed order case (1.”
Asymptotic properties of Brownian motion delayed by inverse subordinators (2013, background): “Another interesting example are distributed order fractional diffusion equations [12,24,38], which correspond to the case f(u) = ∞ 0 (1−e−ux) ν(dx) with ν(t,∞) =”