Variable Order and Distributed Order Fractional Operators

By Carl F. Lorenzo, Tom T. Hartley. 2002

852 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Control and Systems Engineering, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/2d34c2fe2a699da0ffee1529834285526bb7e01a

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In Civil and Structural Engineering

In Control and Systems Engineering

In Modeling and Simulation

Citations in the corpus

On fractional Duhamel's principle and its applications (2010, background): “In the study of diffusion processes in complex heterogeneous media with several distinct diffusion modes, even without an external force, the function h(t) embodies memory of the past [22, 38].”
Variable Order Fractional Controllers (2013, background): “Definition (4) below was studied in the 1990s (in [10] and in other papers, some listed therein), but the seminal paper on the comparison of different possible definitions is [8], which addresses mainly variable negative orders, considering positive orders only in passing: these are considered at le”
Generalized fractional derivatives and Laplace transform (2020, background): “Despite of the fact that this calculus is old, it has gained popularity in the few decades because of the interesting results obtained when this calculus was applied to model some real world problems [25, 19, 24, 22, 13, 21].”
A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications (2019, methodology): “(4.2) Moreover, Lorenzo [66] proposed the derivation of the Laplace transform of the VO integral as follows L { 0D −α(t) t f(t) } = ∫ ∞ 0 e−st (∫ t 0 (t− τ)α(t−τ)−1 Γ(α(t− τ)) f(τ)dτ ) dt, α(t) > 0, t > 0.”
Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C0-semigroups (2013, background): “The logic of our definitions starts from the observation of the fractional derivative of order α ∈ (0, 1) (in the Riemann-Liouville sense) can be considered the first-order derivative of the Laplace convolution u(t) ∗ t−α/Γ(1−α) (see Kilbas et al. [15]) dα dtα u(t) = 1 Γ(1− α) d dt ∫ t 0 u(s) (t− s)”
Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems (2016, methodology): “The Riemann-Liouville type FD of variable-order was proposed in [11, 12] and its extended versions were considered in [13].”
Analysis of some generalized ABC – Fractional logistic models (2019, result): “As a matter of fact, this type of calculus, although as old as the classic calculus, it has attracted the attention of researchers working on different disciplines because of the astounding results obtained when some of these researchers exploited fractional operators in modeling some real world pro”
A new fractional operator of variable order: Application in the description of anomalous diffusion (2016, methodology): “Motivated by this idea [27,28], in this paper the new FOD is applied in the modeling of anomalous diffusion problems.”
Numerical studies for the variable-order nonlinear fractional wave equation (2012, background): “Similar to the state of affairs in fractional calculus, multiple definitions of a variable order derivative have been suggested ([7], [12]), each preferred for different reasons [17].”