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1. Fractional Leibniz Rules in the Setting of Quasi-Banach Function Spaces

   https://doi.org/10.1007/s00041-023-10044-0  

   Hale, Elizabeth; Naibo, Virginia (October 2023, Journal of Fourier Analysis and Applications)

   Fractional Leibniz rules are reminiscent of the product rule learned in calculus classes, offering estimates in the Lebesgue norm for fractional derivatives of a product of functions in terms of the Lebesgue norms of each function and its fractional derivatives. We prove such estimates for Coifman-Meyer multiplier operators in the setting of Triebel-Lizorkin and Besov spaces based on quasi-Banach function spaces. In particular, these include rearrangement invariant quasi-Banach function spaces such as weighted Lebesgue spaces, weighted Lorentz spaces and generalizations, and Orlicz spaces. The method used also yields results in weighted mixed Lebesgue spaces and Morrey spaces, where we present applications to the specific case of power weights, as well as in variable Lebesgue spaces. 

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2. A Symmetric Neural Network to Compute Fractional Derivatives by Training with Integer Derivatives

   https://doi.org/10.1109/SII52469.2022.9708840  

   Chen, Tan; Goodwine, Bill (January 2022, 2022 IEEE/SICE International Symposium on System Integration (SII))

   Fractional calculus is an increasingly recognized important tool for modeling complicated dynamics in modern engineering systems. While, in some ways, fractional derivatives are a straight-forward generalization of integer-order derivatives that are ubiquitous in engineering modeling, in other ways the use of them requires quite a bit of mathematical expertise and familiarity with some mathematical concepts that are not in everyday use across the broad spectrum of engineering disciplines. In more colloquial terms, the learning curve is steep. While the authors recognize the need for fundamental competence in tools used in engineering, a computational tool that can provide an alternative means to compute fractional derivatives does have a useful role in engineering modeling. This paper presents the use of a symmetric neural network that is trained entirely on integer-order derivatives to provide a means to compute fractional derivatives. The training data does not contain any fractional-order derivatives at all, and is composed of only integer-order derivatives. The means by which a fractional derivative can be obtained is by requiring the neural network to be symmetric, that is, it is the composition of two identical sets of layers trained on integer-order derivatives. From that, the information contained in the nodes between the two sets of layers contains half-order derivative information 

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3. Exact results for a fractional derivative of elementary functions

   https://doi.org/10.21468/SciPostPhys.4.6.029  

   Shchedrin, Gavriil; Smith, Nathanael; Gladkina, Anastasia; Carr, Lincoln (January 2018, SciPost Physics)

   We present exact analytical results for the Caputo fractional derivative ofa wide class of elementary functions, including trigonometric andinverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian,quartic Gaussian, Lorentzian, and shifted polynomial functions. Theseresults are especially important for multi-scale physical systems, suchas porous materials, disordered media, and turbulent fluids, in whichtransport is described by fractional partial differential equations. Theexact results for the Caputo fractional derivative are obtained from a singlegeneralized Euler’s integral transform of the generalized hypergeometricfunction with a power-law argument. We present a proof of thegeneralized Euler’s integral transform and directly apply it to theexact evaluation of the Caputo fractional derivative of a broad spectrum offunctions, provided that these functions can be expressed in terms of ageneralized hypergeometric function with a power-law argument. Wedetermine that the Caputo fractional derivative of elementary functions isgiven by the generalized hypergeometric function. Moreover, we show thatin the most general case the final result cannot be reduced toelementary functions, in contrast to both the Liouville-Caputo and Fourier fractionalderivatives. However, we establish that in the infinite limit of theargument of elementary functions, all three definitions of a fractionalderivative - the Coputo, Liouville-Caputo, and Fourier - converge to the same result given by theelementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourierfractional derivatives. 

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4. Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

   https://doi.org/10.1177/1045389X20969847  

   Miles, Paul_R; Pash, Graham_T; Smith, Ralph_C; Oates, William_S (November 2020, Journal of Intelligent Material Systems and Structures)

   Dielectric elastomers are employed for a wide variety of adaptive structures. Many of these soft elastomers exhibit significant rate-dependencies in their response. Accurately quantifying this viscoelastic behavior is non-trivial and in many cases a nonlinear modeling framework is required. Fractional-order operators have been applied to modeling viscoelastic behavior for many years, and recent research has shown fractional-order methods to be effective for nonlinear frameworks. This implementation can become computationally expensive to achieve an accurate approximation of the fractional-order derivative. Accurate estimation of the elastomer’s viscoelastic behavior to quantify parameter uncertainty motivates the use of Markov Chain Monte Carlo (MCMC) methods. Since MCMC is a sampling based method, requiring many model evaluations, efficient estimation of the fractional derivative operator is crucial. In this paper, we demonstrate the effectiveness of using quadrature techniques to approximate the Riemann–Liouville definition for fractional derivatives in the context of estimating the uncertainty of a nonlinear viscoelastic model. We also demonstrate the use of parameter subset selection techniques to isolate parameters that are identifiable in the sense that they are uniquely determined by measured data. For those identifiable parameters, we employ Bayesian inference to compute posterior distributions for parameters. Finally, we propagate parameter uncertainties through the models to compute prediction intervals for quantities of interest. 

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5. On properties of univariate max functions at local maximizers

   https://doi.org/10.1007/s11590-022-01872-y  

   Mitchell, Tim; Overton, Michael L. (March 2022, Optimization Letters)

   Abstract More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that is useful in several applications that we describe. We also investigate whether this smoothness property extends to max functions more generally. We show that the pointwise maximum of a finite set ofq-times continuously differentiable univariate functions must have zero derivative at a maximizer for$$q=1$$ $q=1$, but arbitrarily close to the maximizer, the derivative may not be defined, even when$$q=3$$ $q=3$and the maximizer is isolated. 

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