Towards a Unified theory of Fractional and Nonlocal Vector Calculus

D’Elia, Marta; Gulian, Mamikon; Olson, Hayley; Karniadakis, George Em

doi:10.1515/fca-2021-0057

Title: Towards a Unified theory of Fractional and Nonlocal Vector Calculus

Journal Article · Thu Oct 28 00:00:00 EDT 2021 · Fractional Calculus and Applied Analysis

DOI:https://doi.org/10.1515/fca-2021-0057· OSTI ID:1897197

D’Elia, Marta ^[1]; Gulian, Mamikon ^[2]; Olson, Hayley ^[3]; Karniadakis, George Em ^[4]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Univ. of Nebraska, Lincoln, NE (United States)
Brown Univ., Providence, RI (United States)
Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green’s identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. Here, the proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus.

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Research Organization:: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States); Brown Univ., Providence, RI (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE National Nuclear Security Administration (NNSA); US Army Research Office (ARO)

Grant/Contract Number:: AC05-76RL01830; NA0003525; W911NF-15-1-0562; SC0019453

OSTI ID:: 1897197

Alternate ID(s):: OSTI ID: 2282020

Report Number(s):: PNNL-SA-178937

Journal Information:: Fractional Calculus and Applied Analysis, Vol. 24, Issue 5; ISSN 1311-0454

Publisher:: de GruyterCopyright Statement

Country of Publication:: United States

Language:: English

References (64)

Fractional vector calculus for fractional advection–dispersion Meerschaert, Mark M.; Mortensen, Jeff; Wheatcraft, Stephen W. Physica A: Statistical Mechanics and its Applications, Vol. 367 https://doi.org/10.1016/j.physa.2005.11.015	journal	July 2006
On a new class of fractional partial differential equations II Shieh, Tien-Tsan; Spector, Daniel E. Advances in Calculus of Variations, Vol. 11, Issue 3 https://doi.org/10.1515/acv-2016-0056	journal	July 2018
Convergence studies in meshfree peridynamic simulations Seleson, Pablo; Littlewood, David J. Computers & Mathematics with Applications, Vol. 71, Issue 11 https://doi.org/10.1016/j.camwa.2015.12.021	journal	June 2016
How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D. Archive for Rational Mechanics and Analysis, Vol. 187, Issue 1 https://doi.org/10.1007/s00205-007-0062-8	journal	November 2007
Multiscaling fractional advection-dispersion equations and their solutions: MULTISCALING FRACTIONAL ADES Schumer, Rina; Benson, David A.; Meerschaert, Mark M. Water Resources Research, Vol. 39, Issue 1 https://doi.org/10.1029/2001WR001229	journal	January 2003
An energy-based coupling approach to nonlocal interface problems Capodaglio, Giacomo; D’Elia, Marta; Bochev, Pavel Computers & Fluids, Vol. 207 https://doi.org/10.1016/j.compfluid.2020.104593	journal	July 2020
A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization Tian, Hao; Ju, Lili; Du, Qiang Computer Methods in Applied Mechanics and Engineering, Vol. 320 https://doi.org/10.1016/j.cma.2017.03.020	journal	June 2017
Two-point stress–strain-rate correlation structure and non-local eddy viscosity in turbulent flows Clark Di Leoni, Patricio; Zaki, Tamer A.; Karniadakis, George Journal of Fluid Mechanics, Vol. 914 https://doi.org/10.1017/jfm.2020.977	journal	March 2021
The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics Metzler, Ralf; Klafter, Joseph Journal of Physics A: Mathematical and General, Vol. 37, Issue 31 https://doi.org/10.1088/0305-4470/37/31/R01	journal	July 2004
Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications Lee, Hwi; Du, Qiang ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 54, Issue 1 https://doi.org/10.1051/m2an/2019053	journal	January 2020
Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver Ainsworth, Mark; Glusa, Christian arXiv https://doi.org/10.48550/arxiv.1708.03912	text	January 2017
Fractional vector calculus and fractional Maxwell’s equations Tarasov, Vasily E. Annals of Physics, Vol. 323, Issue 11 https://doi.org/10.1016/j.aop.2008.04.005	journal	November 2008
Numerical methods for nonlocal and fractional models D’Elia, Marta; Du, Qiang; Glusa, Christian Acta Numerica, Vol. 29 https://doi.org/10.1017/S096249292000001X	journal	May 2020
Fractional Diffusion on Bounded Domains Defterli, Ozlem; D’Elia, Marta; Du, Qiang Fractional Calculus and Applied Analysis, Vol. 18, Issue 2 https://doi.org/10.1515/fca-2015-0023	journal	January 2015
Kinetics of phase transformations in the peridynamic formulation of continuum mechanics Dayal, Kaushik; Bhattacharya, Kaushik Journal of the Mechanics and Physics of Solids, Vol. 54, Issue 9 https://doi.org/10.1016/j.jmps.2006.04.001	journal	September 2006
Fractional calculus and continuous-time finance Scalas, Enrico; Gorenflo, Rudolf; Mainardi, Francesco Physica A: Statistical Mechanics and its Applications, Vol. 284, Issue 1-4 https://doi.org/10.1016/S0378-4371(00)00255-7	journal	September 2000
Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems D’Elia, M.; Gunzburger, M. Applied Mathematics & Optimization, Vol. 73, Issue 2 https://doi.org/10.1007/s00245-015-9300-x	journal	May 2015
A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems Gunzburger, Max; Lehoucq, R. B. Multiscale Modeling & Simulation, Vol. 8, Issue 5 https://doi.org/10.1137/090766607	journal	January 2010
Characterization of function spaces of vector fields and an application in nonlinear peridynamics Mengesha, Tadele; Du, Qiang Nonlinear Analysis, Vol. 140 https://doi.org/10.1016/j.na.2016.02.024	journal	July 2016
Image Recovery via Nonlocal Operators Lou, Yifei; Zhang, Xiaoqun; Osher, Stanley Journal of Scientific Computing, Vol. 42, Issue 2 https://doi.org/10.1007/s10915-009-9320-2	journal	August 2009
Peridynamics and Material Interfaces Alali, Bacim; Gunzburger, Max Journal of Elasticity, Vol. 120, Issue 2 https://doi.org/10.1007/s10659-014-9512-3	journal	January 2015
Boundary Problems for the Fractional and Tempered Fractional Operators Deng, Weihua; Li, Buyang; Tian, Wenyi Multiscale Modeling & Simulation, Vol. 16, Issue 1 https://doi.org/10.1137/17M1116222	journal	January 2018
nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications Pang, G.; D'Elia, M.; Parks, M. Journal of Computational Physics, Vol. 422 https://doi.org/10.1016/j.jcp.2020.109760	journal	December 2020
Data-driven learning of nonlocal physics from high-fidelity synthetic data You, Huaiqian; Yu, Yue; Trask, Nathaniel Computer Methods in Applied Mechanics and Engineering, Vol. 374 https://doi.org/10.1016/j.cma.2020.113553	journal	February 2021
Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver Ainsworth, Mark; Glusa, Christian Computer Methods in Applied Mechanics and Engineering, Vol. 327 https://doi.org/10.1016/j.cma.2017.08.019	journal	December 2017
fPINNs: Fractional Physics-Informed Neural Networks Pang, Guofei; Lu, Lu; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 41, Issue 4 https://doi.org/10.1137/18M1229845	journal	January 2019
Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations Du, Qiang; Tian, Xiaochuan Foundations of Computational Mathematics, Vol. 20, Issue 4 https://doi.org/10.1007/s10208-019-09432-0	journal	August 2019
What is the fractional Laplacian? A comparative review with new results Lischke, Anna; Pang, Guofei; Gulian, Mamikon Journal of Computational Physics, Vol. 404 https://doi.org/10.1016/j.jcp.2019.109009	journal	March 2020
Fractional div-curl quantities and applications to nonlocal geometric equations Mazowiecka, Katarzyna; Schikorra, Armin Journal of Functional Analysis, Vol. 275, Issue 1 https://doi.org/10.1016/j.jfa.2018.03.016	journal	July 2018
Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains Ainsworth, Mark; Glusa, Christian Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan https://doi.org/10.1007/978-3-319-72456-0_2	book	January 2018
A Riesz Basis Galerkin Method for the Tempered Fractional Laplacian Zhang, Zhijiang; Deng, Weihua; Karniadakis, George Em SIAM Journal on Numerical Analysis, Vol. 56, Issue 5 https://doi.org/10.1137/17M1151791	journal	January 2018
Application of a fractional advection-dispersion equation Benson, David A.; Wheatcraft, Stephen W.; Meerschaert, Mark M. Water Resources Research, Vol. 36, Issue 6 https://doi.org/10.1029/2000WR900031	journal	February 2000
Image Denoising Methods. A New Nonlocal Principle Buades, A.; Coll, B.; Morel, J. M. SIAM Review, Vol. 52, Issue 1 https://doi.org/10.1137/090773908	journal	January 2010
Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity Du, Qiang; Gunzburger, Max; Lehoucq, R. B. Journal of Elasticity, Vol. 113, Issue 2 https://doi.org/10.1007/s10659-012-9418-x	journal	November 2012
A Nonlocal Vector Calculus, Nonlocal Volume-Constrained Problems, and Nonlocal Balance laws Du, Qiang; Gunzburger, Max; Lehoucq, R. B. Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 03 https://doi.org/10.1142/S0218202512500546	journal	January 2013
Helmholtz-Hodge Decompositions in the Nonlocal Framework D’Elia, Marta; Flores, Cynthia; Li, Xingjie Journal of Peridynamics and Nonlocal Modeling, Vol. 2, Issue 4 https://doi.org/10.1007/s42102-020-00035-w	journal	July 2020
Fractional vector analysis based on invariance requirements (critique of coordinate approaches) Šilhavý, M. Continuum Mechanics and Thermodynamics, Vol. 32, Issue 1 https://doi.org/10.1007/s00161-019-00797-9	journal	June 2019
A generalized nonlocal vector calculus Alali, Bacim; Liu, Kuo; Gunzburger, Max Zeitschrift für angewandte Mathematik und Physik, Vol. 66, Issue 5 https://doi.org/10.1007/s00033-015-0514-1	journal	March 2015
Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation Mengesha, Tadele; Du, Qiang Journal of Elasticity, Vol. 116, Issue 1 https://doi.org/10.1007/s10659-013-9456-z	journal	August 2013
Stability of peridynamic correspondence material models and their particle discretizations Silling, S. A. Computer Methods in Applied Mechanics and Engineering, Vol. 322 https://doi.org/10.1016/j.cma.2017.03.043	journal	August 2017
Discovering variable fractional orders of advection–dispersion equations from field data using multi-fidelity Bayesian optimization Pang, Guofei; Perdikaris, Paris; Cai, Wei Journal of Computational Physics, Vol. 348 https://doi.org/10.1016/j.jcp.2017.07.052	journal	November 2017
Analysis of a nonlocal-in-time parabolic equation Du, Qiang; Yang, Jiang; Zhou, Zhi Discrete & Continuous Dynamical Systems - B, Vol. 22, Issue 2 https://doi.org/10.3934/dcdsb.2017016	journal	January 2017
Stability of Nonlocal Dirichlet Integrals and Implications for Peridynamic Correspondence Material Modeling Du, Qiang; Tian, Xiaochuan SIAM Journal on Applied Mathematics, Vol. 78, Issue 3 https://doi.org/10.1137/17M1139874	journal	January 2018
On Riesz Derivative Cai, Min; Li, Changpin Fractional Calculus and Applied Analysis, Vol. 22, Issue 2 https://doi.org/10.1515/fca-2019-0019	journal	April 2019
On some composition formulas Horváth, J. Proceedings of the American Mathematical Society, Vol. 10, Issue 3 https://doi.org/10.1090/S0002-9939-1959-0107788-4	journal	January 1959
A peridynamic formulation of pressure driven convective fluid transport in porous media Katiyar, Amit; Foster, John T.; Ouchi, Hisanao Journal of Computational Physics, Vol. 261 https://doi.org/10.1016/j.jcp.2013.12.039	journal	March 2014
The exit-time problem for a Markov jump process Burch, N.; D’Elia, M.; Lehoucq, R. B. The European Physical Journal Special Topics, Vol. 223, Issue 14 https://doi.org/10.1140/epjst/e2014-02331-7	journal	December 2014
Multiscale Dynamics of Heterogeneous Media in the Peridynamic Formulation Alali, Bacim; Lipton, Robert Journal of Elasticity, Vol. 106, Issue 1 https://doi.org/10.1007/s10659-010-9291-4	journal	December 2010
Nonlocal problems with Neumann boundary conditions Dipierro, Serena; Ros-Oton, Xavier; Valdinoci, Enrico Revista Matemática Iberoamericana, Vol. 33, Issue 2 https://doi.org/10.4171/rmi/942	journal	January 2017
Fractional curl operator in electromagnetics Engheta, Nader Microwave and Optical Technology Letters, Vol. 17, Issue 2 https://doi.org/10.1002/(sici)1098-2760(19980205)17:2<86::aid-mop4>3.0.co;2-e	journal	February 1998
The random walk's guide to anomalous diffusion: a fractional dynamics approach Metzler, Ralf; Klafter, Joseph Physics Reports, Vol. 339, Issue 1 https://doi.org/10.1016/S0370-1573(00)00070-3	journal	December 2000
The Dirichlet problem for nonlocal operators Felsinger, Matthieu; Kassmann, Moritz; Voigt, Paul Mathematische Zeitschrift, Vol. 279, Issue 3-4 https://doi.org/10.1007/s00209-014-1394-3	journal	November 2014
Tempered fractional calculus Sabzikar, Farzad; Meerschaert, Mark M.; Chen, Jinghua Journal of Computational Physics, Vol. 293 https://doi.org/10.1016/j.jcp.2014.04.024	journal	July 2015
The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator D’Elia, Marta; Gunzburger, Max Computers & Mathematics with Applications, Vol. 66, Issue 7 https://doi.org/10.1016/j.camwa.2013.07.022	journal	October 2013
Nonlocal Operators with Applications to Image Processing Gilboa, Guy; Osher, Stanley Multiscale Modeling & Simulation, Vol. 7, Issue 3 https://doi.org/10.1137/070698592	journal	January 2009
How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems Cortazar, C.; Elgueta, M.; Rossi, J. D. arXiv https://doi.org/10.48550/arxiv.math/0607058	preprint	January 2006
A meshfree method based on the peridynamic model of solid mechanics Silling, S. A.; Askari, E. Computers & Structures, Vol. 83, Issue 17-18 https://doi.org/10.1016/j.compstruc.2004.11.026	journal	June 2005
A direct O(Nlog2N) finite difference method for fractional diffusion equations Wang, Hong; Wang, Kaixin; Sircar, Treena Journal of Computational Physics, Vol. 229, Issue 21 https://doi.org/10.1016/j.jcp.2010.07.011	journal	October 2010
The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator D'Elia, Marta; Gunzburger, Max arXiv https://doi.org/10.48550/arxiv.1303.6934	preprint	January 2013
Nonlocal Linear Image Regularization and Supervised Segmentation Gilboa, Guy; Osher, Stanley Multiscale Modeling & Simulation, Vol. 6, Issue 2 https://doi.org/10.1137/060669358	journal	January 2007
Eulerian derivation of the fractional advection–dispersion equation Schumer, Rina; Benson, David A.; Meerschaert, Mark M. Journal of Contaminant Hydrology, Vol. 48, Issue 1-2 https://doi.org/10.1016/S0169-7722(00)00170-4	journal	March 2001
Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints Du, Qiang; Gunzburger, Max; Lehoucq, R. B. SIAM Review, Vol. 54, Issue 4 https://doi.org/10.1137/110833294	journal	January 2012
Calculus On Manifolds Spivak, Michael https://doi.org/10.1201/9780429501906	book	January 1965
On a new class of fractional partial differential equations Shieh, Tien-Tsan; Spector, Daniel E. Advances in Calculus of Variations, Vol. 8, Issue 4 https://doi.org/10.1515/acv-2014-0009	journal	May 2014

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