General Fractional Calculus, Evolution Equations, and Renewal Processes

By Anatoly N. Kochubei. 2011

368 citations recorded by the source corpus.

We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $${(\mathbb D_{(k)} u)(t)=\frac{d}{dt} \int \nolimits_0^tk(t-\tau )u(\tau )\,d\tau-k(t)u(0)}$$ where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation $${\mathbb D_{(k)} u=-\lambda u}$$, λ > 0, proved to be (under some conditions upon k) continuous on [0, ∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N(E(t)) as a renewal process. Here N(t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/fd148c41a11b2b3ff8ed34dbb1113ad614027fc2

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Citations in the corpus

On History of Mathematical Economics: Application of Fractional Calculus (2019, methodology): “Kochubei, the fractional derivatives and integrals of distributed order are investigated in the works [239,240].”
Models of dielectric relaxation based on completely monotone functions (2016, background): “n [32, 111]). This kind of denition is however interesting because it allows to introduce a regularization of the same type of the regularization (B.8) introduced for the Caputo derivative (see also [71]). As proposed in [31], it is indeed possible to exchange integrals and derivatives in (B.20), th”
Fractional derivatives and the fundamental theorem of fractional calculus (2020, background): “35–37]). These discussions mainly concerned the “new fractional integrals and derivatives” ( [8, 11, 14, 31, 35–37]), the integro-diﬀerential operators of convolution type with some general kernels ( [8,14,22,23,35,39,40]), and the abstract axioms of FC ( [17,30,33]). In this framework, the “classic”
The General Fractional Derivative and Related Fractional Differential Equations (2020, methodology): “The kernels of the GFDs (19) and (21) considered in [10] satisfy the following conditions: (K1) The Laplace transform k̃ of k,”
General Fractional Integrals and Derivatives with the Sonine Kernels (2021, background): “21) and includes, for instance, some functions with the power-logarithmic singularities at the origin ( [19,20]). A completely diﬀerent approach for description of the Sonine kernels was suggested in [8], where a connection between the complete Bernstein functions and the Sonine kernels has been est”