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1. Optimal Control, Numerics, and Applications of Fractional PDEs

   https://doi.org/10.1016/bs.hna.2021.12.003  

   Antil, H. ; Brown T. ; Khatri, R. ; Onwunta, A. ; Verma, D. ; Warma, M. ( February 2022 , Handbook of numerical analysis)

   Trélat, E. ; Zuazua, E. (Ed.)

   This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems.
2. Analysis of an approximation to a fractional extension problem

   https://doi.org/10.1007/s10543-019-00787-y  

   Padgett, Joshua L. ( November 2019 , BIT Numerical Mathematics)

   The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results.
3. Bilinear Systems With Initial Gaps Involving Inelastic Collision: Forced Response Experiments and Simulations

   https://doi.org/10.1115/1.4051493  

   Noguchi, Kohei ; Saito, Akira ; Tien, Meng-Hsuan ; D’Souza, Kiran ( April 2022 , Journal of Vibration and Acoustics)

   Abstract In this paper, the forced response of a two degrees-of-freedom (DOF) bilinear oscillator with initial gaps involving inelastic collision is discussed. In particular, a focus is placed upon the experimental verification of the generalized bilinear amplitude approximation (BAA) method, which can be used for the accurate estimation of forced responses for bilinear systems with initial gaps. Both experimental and numerical investigations on the system have been carried out. An experimental setup that is capable of representing the dynamics of a 2DOF oscillator has been developed, and forced response tests have been conducted under swept-sine base excitation for different initial gap sizes. The steady-state response of the system under base excitation was computed by both traditional time integration and BAA. It is shown that the results of experiments and numerical predictions are in good agreement especially at resonance. However, slight differences in the responses obtained from both numerical methods are observed. It was found that the time duration where the DOFs are in contact with each other predicted by BAA is longer than that predicted by time integration. Spectral analyses have also been conducted on both experimental and numerical results. It was observed that in a frequency range where intermittentmore »contact between the masses occurs, super-harmonic components of the excitation frequency are present in the spectra. Moreover, as the initial gap size increases, the frequency band where the super-harmonic components are observed decreases.« less
4. Global existence of weak solutions to unsaturated poroelasticity

   https://doi.org/10.1051/m2an/2021063  

   Both, Jakub Wiktor ; Pop, Iuliu Sorin ; Yotov, Ivan ( November 2021 , ESAIM: Mathematical Modelling and Numerical Analysis)

   We study unsaturated poroelasticity, i.e. , coupled hydro-mechanical processes in variably saturated porous media, here modeled by a non-linear extension of Biot’s well-known quasi-static consolidation model. The coupled elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media, obtained under similar assumptions as usually considered for Richards’ equation. In this work, existence of weak solutions is established in several steps involving a numerical approximation of the problem using a physically-motivated regularization and a finite element/finite volume discretization. Eventually, solvability of the original problem is proved by a combination of the Rothe and Galerkin methods, and further compactness arguments. This approach in particular provides the convergence of the numerical discretization to a regularized model for unsaturated poroelasticity. The final existence result holds under non-degeneracy conditions and natural continuity properties for the constitutive relations. The assumptions are demonstrated to be reasonable in view of geotechnical applications.
5. Log orthogonal functions: approximation properties and applications

   https://doi.org/10.1093/imanum/draa087  

   Chen, Sheng ; Shen, Jie ( December 2020 , IMA Journal of Numerical Analysis)

   Abstract We present two new classes of orthogonal functions, log orthogonal functions and generalized log orthogonal functions, which are constructed by applying a $\log $ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions, and apply them to solve several typical fractional differential equations whose solutions exhibit weak singularities. Our error analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions that have weak singularities at one endpoint and can lead to exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used.