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1. Inverting the variable fractional order in a variable-order space-fractional diffusion equation with variable diffusivity: analysis and simulation

   https://doi.org/10.1515/jiip-2019-0040  

   Zheng, Xiangcheng; Li, Yiqun; Cheng, Jin; Wang, Hong (April 2021, Journal of Inverse and Ill-posed Problems)

   null (Ed.)

   Abstract Variable-order space-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena, including anomalously superdiffusive transport of solutes in heterogeneous porous media, long-range spatial interactions and other applications, as well as eliminating the nonphysical boundary layers of the solutions to their constant-order analogues.In this paper, we prove the uniqueness of determining the variable fractional order of the homogeneous Dirichlet boundary-value problem of the one-sided linear variable-order space-fractional diffusion equation with some observed values of the unknown solutions near the boundary of the spatial domain.We base on the analysis to develop a spectral-Galerkin Levenberg–Marquardt method and a finite difference Levenberg–Marquardt method to numerically invert the variable order.We carry out numerical experiments to investigate the numerical performance of these methods. 

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2. Mitigating Epilepsy by Stabilizing Linear Fractional-Order Systems

   https://doi.org/10.23919/ACC55779.2023.10156404  

   Emily A. Reed, Guilherme Ramos (May 2023, 2023 American Control Conference (ACC))

   Epilepsy affects approximately 50 million people worldwide. Despite its prevalence, the recurrence of seizures can be mitigated only 70% of the time through medication. Furthermore, surgery success rates range from 30% - 70% because of our limited understanding of how a seizure starts. However, one leading hypothesis suggests that a seizure starts because of a critical transition due to an instability. Unfortunately, we lack a meaningful way to quantify this notion that would allow physicians to not only better predict seizures but also to mitigate them. Hence, in this paper, we develop a method to not only characterize the instability of seizures but also to leverage these conditions to stabilize the system underlying these seizures. Remarkably, evidence suggests that such critical transitions are associated with long-term memory dynamics, which can be captured by considering linear fractional-order systems. Subsequently, we provide for the first time tractable necessary and sufficient conditions for the global asymptotic stability of discrete-time linear fractional-order systems. Next, we propose a method to obtain a stabilizing control strategy for these systems using linear matrix inequalities. Finally, we apply our methodology to a real-world epileptic patient dataset to provide insight into mitigating epilepsy and designing future cyber-neural systems. 

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3. Applications of variable-order fractional operators: a review

   https://doi.org/10.1098/rspa.2019.0498  

   Patnaik, Sansit; Hollkamp, John P.; Semperlotti, Fabio (February 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   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. 

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4. Applications of Distributed-Order Fractional Operators: A Review

   https://doi.org/10.3390/e23010110  

   Ding, Wei; Patnaik, Sansit; Sidhardh, Sai; Semperlotti, Fabio (January 2021, Entropy)

   null (Ed.)

   Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date. 

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5. Quasi-linear Fractional-Order Operators in Lipschitz Domains

   https://doi.org/10.1137/23M1575871  

   Borthagaray, Juan Pablo; Li, Wenbo; Nochetto, Ricardo H (June 2024, SIAM Journal on Mathematical Analysis)