Applications of variable-order fractional operators: a review
Sansit Patnaik,John P. Hollkamp,Fabio Semperlotti +2 more
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TLDR
This review provides a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems.Abstract:
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.read more
Citations
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Advanced materials modelling via fractional calculus: challenges and perspectives.
TL;DR: Fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials.
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A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19
TL;DR: In this article, a fractional-order SIRD (Susceptible-Infected-Recovered-Dead) model based on the Caputo derivative for incorporating the memory effects (long and short) in the outbreak progress is presented.
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Numerical analysis of a new space-time variable fractional order advection-dispersion equation
TL;DR: This paper considers a new space–time variable fractional order advection–dispersion equation on a finite domain and proposes an implicit Euler approximation for the equation and investigates the stability and convergence of the approximation.
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Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator
TL;DR: In this paper, a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain is considered.
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Variable-order fractional calculus: A change of perspective
TL;DR: The variable-order Scarpi integral and derivative as mentioned in this paper is based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus.
References
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Book
Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
TL;DR: In this article, the authors present a method for computing fractional derivatives of the Fractional Calculus using the Laplace Transform Method and the Fourier Transformer Transform of fractional Derivatives.
Journal ArticleDOI
Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers
TL;DR: In this article, a fractional-order PI/sup/spl lambda/D/sup /spl mu/controller with fractionalorder integrator and fractional order differentiator is proposed.
Journal ArticleDOI
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
Ronald L. Bagley,Peter J. Torvik +1 more
TL;DR: In this article, the authors established a link between molecular theories that predict the macroscopic behavior of certain viscoelastic media and an empirically developed fractional calculus approach to visco-elasticity.
Book
Fractional Calculus in Bioengineering
TL;DR: Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems, which is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research.