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1. Identifying Arguments of Space-Time Fractional Diffusion: Data-Driven Approach

   https://doi.org/https://doi.org/10.3389/fams.2020.00014  

   Znaidi Mohamed Ridha, Gupta Gaurav ( May 2020 , Frontiers in applied mathematics and statistics)

   null (Ed.)

   A plethora of complex dynamical systems from disordered media to biological systems exhibit mathematical characteristics (e.g., long-range dependence, self-similar and power law magnitude increments) that are well-fitted by fractional partial differential equations (PDEs). For instance, some biological systems displaying an anomalous diffusion behavior, which is characterized by a non-linear mean-square displacement relation, can be mathematically described by fractional PDEs. In general, the PDEs represent various physical laws or rules governing complex dynamical systems. Since prior knowledge about the mathematical equations describing complex dynamical systems in biology, healthcare, disaster mitigation, transportation, or environmental sciences may not be available, we aim to provide algorithmic strategies to discover the integer or fractional PDEs and their parameters from system's evolution data. Toward deciphering non-trivial mechanisms driving a complex system, we propose a data-driven approach that estimates the parameters of a fractional PDE model. We study the space-time fractional diffusion model that describes a complex stochastic process, where the magnitude and the time increments are stable processes. Starting from limited time-series data recorded while the system is evolving, we develop a fractional-order moments-based approach to determine the parameters of a generalized fractional PDE. We formulate two optimization problems to allow us to estimate the arguments of the fractional PDE. Employing extensive simulation studies, we show that the proposed approach is effective at retrieving the relevant parameters of the space-time fractional PDE. The presented mathematical approach can be further enhanced and generalized to include additional operators that may help to identify the dominant rule governing the measurements or to determine the degree to which multiple physical laws contribute to the observed dynamics. 

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2. 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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3. Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

   https://doi.org/10.3390/ijerph17062014  

   Islam, Md Rafiul ; Peace, Angela ; Medina, Daniel ; Oraby, Tamer ( March 2020 , International Journal of Environmental Research and Public Health)

   In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms. 

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4. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

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5. Analysis of the Postbuckling Response of Nonlocal Plates Via Fractional-Order Continuum Theory

   https://doi.org/10.1115/1.4049224  

   Sidhardh, Sai ; Patnaik, Sansit ; Semperlotti, Fabio ( April 2021 , Journal of Applied Mechanics)

   null (Ed.)

   Abstract We present a comprehensive study on the postbuckling response of nonlocal structures performed by means of a frame-invariant fractional-order continuum theory to model the long-range (nonlocal) interactions. The use of fractional calculus facilitates an energy-based approach to nonlocal elasticity that plays a fundamental role in the present study. The underlying fractional framework enables mathematically, physically, and thermodynamically consistent integral-type constitutive models that, in contrast to the existing integer-order differential approaches, allow the nonlinear buckling and postbifurcation analyses of nonlocal structures. Furthermore, we present the first application of the Koiter’s asymptotic method to investigate postbifurcation branches of nonlocal structures. Finally, the theoretical framework is applied to study the postbuckling behavior of slender nonlocal plates. Both qualitative and quantitative analyses of the influence that long-range interactions bear on postbuckling response are undertaken. Numerical studies are carried out using a 2D fractional-order finite element method (f-FEM) modified to include a combination of the Newton–Raphson and a path-following arc-length iterative methods to solve the system of nonlinear algebraic equations that govern the equilibrium beyond the critical points. The present framework provides a general foundation to investigate the postbuckling response of potentially any type of nonlocal structure. 

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