图书馆订阅: Guest
国际多尺度计算工程期刊

每年出版 6 

ISSN 打印: 1543-1649

ISSN 在线: 1940-4352

The Impact Factor measures the average number of citations received in a particular year by papers published in the journal during the two preceding years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) IF: 1.4 To calculate the five year Impact Factor, citations are counted in 2017 to the previous five years and divided by the source items published in the previous five years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) 5-Year IF: 1.3 The Immediacy Index is the average number of times an article is cited in the year it is published. The journal Immediacy Index indicates how quickly articles in a journal are cited. Immediacy Index: 2.2 The Eigenfactor score, developed by Jevin West and Carl Bergstrom at the University of Washington, is a rating of the total importance of a scientific journal. Journals are rated according to the number of incoming citations, with citations from highly ranked journals weighted to make a larger contribution to the eigenfactor than those from poorly ranked journals. Eigenfactor: 0.00034 The Journal Citation Indicator (JCI) is a single measurement of the field-normalized citation impact of journals in the Web of Science Core Collection across disciplines. The key words here are that the metric is normalized and cross-disciplinary. JCI: 0.46 SJR: 0.333 SNIP: 0.606 CiteScore™:: 3.1 H-Index: 31

Indexed in

CLASSICAL, NONLOCAL, AND FRACTIONAL DIFFUSION EQUATIONS ON BOUNDED DOMAINS

卷 9, 册 6, 2011, pp. 661-674
DOI: 10.1615/IntJMultCompEng.2011002402
Get accessGet access

摘要

The purpose of this paper is to compare the solutions of one-dimensional boundary value problems corresponding to classical, fractional, and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion when Fick's first law is an inaccurate model. In the case of nonlocal diffusion, a generalization of Fick's first law in terms of a nonlocal flux is demonstrated to hold. A relationship between nonlocal and fractional diffusion is also reviewed, where the order of the fractional Laplacian can lie in the interval (0, 2]. The contribution of this paper is to present boundary value problems for nonlocal diffusion including a variational formulation that leads to a conforming finite-element method using piecewise discontinuous shape functions. The nonlocal Dirichlet and Neumann boundary conditions used represent generalizations of the classical boundary conditions. Several examples are given where the effect of nonlocality is studied. The relationship between nonlocal and fractional diffusion explains that the numerical solution of boundary value problems, where the order of the fractional Laplacian can lie in the interval (0, 2], is possible.

参考文献
  1. Andreu, F., Mazon, J., Rossi, J., and Toledo, J., A nonlocal <i>p</i>-Laplacian evolution equation with Neumann boundary conditions. DOI: 10.1016/j.matpur.2008.04.003

  2. Andreu, F., Mazon, J. M., Rossi, J. D., and Toledo, J., A nonlocal <i>p</i>-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions. DOI: 10.1137/080720991

  3. Applebaum, D., L&eacute;vy Processes and Stochastic Calculus. DOI: 10.1017/CBO9780511755323

  4. Aranovich, G. and Donohue, M., Diffusion in fluids with large mean free paths: Non-classical behavior between Knudsen and Fickian limits. DOI: 10.1016/j.physa.2009.05.012

  5. Bhowmik, S., Stability and convergence analysis of a one step approximation of a linear partial integro-differential equation. DOI: 10.1002/num.20576

  6. Bobaru, F. and Duangpanya, M., The peridynamic formulation for transient heat conduction. DOI: 10.1016/j.ijheatmasstransfer.2010.05.024

  7. Chasseigne, E., Chaves, M., and Rossi, J., Asymptotic behavior for nonlocal diffusion equations. DOI: 10.1016/j.matpur.2006.04.005

  8. Ervin, V. J. and Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation. DOI: 10.1002/num.20112

  9. Gunzburger, M. and Lehoucq, R. B., A nonlocal vector calculus with application to nonlocal boundary value problems. DOI: 10.1137/090766607

  10. Ignat, L. I. and Rossi, J. D., Decay estimates for nonlocal problems via energy methods. DOI: 10.1016/j.matpur.2009.04.009

  11. Lehoucq, R. B. and von Lilienfeld, O. A., Translation of Walter Noll's ''Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics''. DOI: 10.1007/s10659-010-9246-9

  12. Meerschaert, M., Mortensen, J., and Wheatcraft, S., Fractional vector calculus for fractional advection-dispersion. DOI: 10.1016/j.physa.2005.11.015

  13. Neuman, S. P. and Tartakovsky, D. M., Perspective on theories of non-Fickian transport in heterogeneous media. DOI: 10.1016/j.advwatres.2008.08.005

  14. Noll, W., Die herleitung der grundgleichungen der thermomechanik der kontinua aus der statistischen mechanik. DOI: 10.1512/iumj.1955.4.54022

  15. Paradisi, P., Cesari, R., Mainardi, F.,Maurizi, A., and Tampieri, F., A generalized Fick's law to describe non-local transport effects. DOI: 10.1016/S1464-1909(01)00006-5

  16. P&eacute;rez-Llanos, M. and Rossi, J., Numerical approximations for a nonlocal evolution equation. DOI: 10.1137/110823559

  17. Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., and Jara, B. M. V., Matrix approach to discrete fractional calculus II: Partial fractional differential equations. DOI: 10.1016/j.jcp.2009.01.014

  18. Schumer, R., Benson, D. A., Meerschaert, M. M., and Baeumer, B., Multiscaling fractional advection-dispersion equations and their solutions. DOI: 10.1029/2001WR001229

对本文的引用
  1. Du Qiang, Gunzburger Max, Lehoucq R. B., Zhou Kun, Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints, SIAM Review, 54, 4, 2012. Crossref

  2. Tian Xiaochuan, Du Qiang, Analysis and Comparison of Different Approximations to Nonlocal Diffusion and Linear Peridynamic Equations, SIAM Journal on Numerical Analysis, 51, 6, 2013. Crossref

  3. Seleson Pablo, Gunzburger Max, Parks Michael L., Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains, Computer Methods in Applied Mechanics and Engineering, 266, 2013. Crossref

  4. Henke Steven F., Shanbhag Sachin, Mesh sensitivity in peridynamic simulations, Computer Physics Communications, 185, 1, 2014. Crossref

  5. D’Elia Marta, Gunzburger Max, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Computers & Mathematics with Applications, 66, 7, 2013. Crossref

  6. Li Xiantao, Heat conduction in nanoscale materials: A statistical-mechanics derivation of the local heat flux, Physical Review E, 90, 3, 2014. Crossref

  7. Office Editorial, Retraction by the Editorial Office, Discrete & Continuous Dynamical Systems - B, 19, 4, 2014. Crossref

  8. Gao T., Duan J., Li X., Song R., Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises, SIAM Journal on Scientific Computing, 36, 3, 2014. Crossref

  9. Seleson Pablo, Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations, Computer Methods in Applied Mechanics and Engineering, 282, 2014. Crossref

  10. Tian Xiaochuan, Du Qiang, Asymptotically Compatible Schemes and Applications to Robust Discretization of Nonlocal Models, SIAM Journal on Numerical Analysis, 52, 4, 2014. Crossref

  11. Du Qiang, Huang Zhan, B. Lehoucq Richard, Nonlocal convection-diffusion volume-constrained problems and jump processes, Discrete & Continuous Dynamical Systems - B, 19, 2, 2014. Crossref

  12. D’Elia M., Gunzburger M., Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems, Applied Mathematics & Optimization, 73, 2, 2016. Crossref

  13. Tian Xiaochuan, Du Qiang, Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems, SIAM Journal on Numerical Analysis, 53, 2, 2015. Crossref

  14. Zhao Yanxiang, Wang Jiakou, Ma Yanping, Du Qiang, Generalized local and nonlocal master equations for some stochastic processes, Computers & Mathematics with Applications, 71, 11, 2016. Crossref

  15. D’Elia Marta, Perego Mauro, Bochev Pavel, Littlewood David, A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions, Computers & Mathematics with Applications, 71, 11, 2016. Crossref

  16. Tian Xiaochuan, Du Qiang, A Class of High Order Nonlocal Operators, Archive for Rational Mechanics and Analysis, 222, 3, 2016. Crossref

  17. Du Qiang, Yang Jiang, Asymptotically Compatible Fourier Spectral Approximations of Nonlocal Allen--Cahn Equations, SIAM Journal on Numerical Analysis, 54, 3, 2016. Crossref

  18. Radu Petronela, Todorova Grozdena, Yordanov Borislav, The Generalized Diffusion Phenomenon and Applications, SIAM Journal on Mathematical Analysis, 48, 1, 2016. Crossref

  19. Zhang Wei, Yang Jiang, Zhang Jiwei, Du Qiang, Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain, Communications in Computational Physics, 21, 1, 2017. Crossref

  20. Du Qiang, Yang Jiang, Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications, Journal of Computational Physics, 332, 2017. Crossref

  21. Lees Eitan, Rokkam Srujan, Shanbhag Sachin, Gunzburger Max, The electroneutrality constraint in nonlocal models, The Journal of Chemical Physics, 147, 12, 2017. Crossref

  22. D’Elia Marta, Du Qiang, Gunzburger Max, Lehoucq Richard, Nonlocal Convection-Diffusion Problems on Bounded Domains and Finite-Range Jump Processes, Computational Methods in Applied Mathematics, 17, 4, 2017. Crossref

  23. Zhang Xiaoping, Wu Jiming, Ju Lili, An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems, Applied Numerical Mathematics, 133, 2018. Crossref

  24. References, in Handbook of Structural Life Assessment, 2017. Crossref

  25. Tao Yunzhe, Tian Xiaochuan, Du Qiang, Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations, Applied Mathematics and Computation, 305, 2017. Crossref

  26. Drapaca Corina, Sivaloganathan Siv, Brief Review of Continuum Mechanics Theories, in Mathematical Modelling and Biomechanics of the Brain, 37, 2019. Crossref

  27. Bond Stephen D., Lehoucq Richard B., Rowe Stephen T., A Galerkin Radial Basis Function Method for Nonlocal Diffusion, in Meshfree Methods for Partial Differential Equations VII, 100, 2015. Crossref

  28. Du Qiang, Yin Xiaobo, A Conforming DG Method for Linear Nonlocal Models with Integrable Kernels, Journal of Scientific Computing, 80, 3, 2019. Crossref

  29. You Huaiqian, Lu XinYang, Task Nathaniel, Yu Yue, An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems, ESAIM: Mathematical Modelling and Numerical Analysis, 54, 4, 2020. Crossref

  30. Alali Bacim, Albin Nathan, Fourier Spectral Methods for Nonlocal Models, Journal of Peridynamics and Nonlocal Modeling, 2, 3, 2020. Crossref

  31. You Huaiqian, Yu Yue, Kamensky David, An asymptotically compatible formulation for local-to-nonlocal coupling problems without overlapping regions, Computer Methods in Applied Mechanics and Engineering, 366, 2020. Crossref

  32. Lee Hwi, Du Qiang, Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications, ESAIM: Mathematical Modelling and Numerical Analysis, 54, 1, 2020. Crossref

  33. Tian Xiaochuan, Du Qiang, Asymptotically Compatible Schemes for Robust Discretization of Parametrized Problems with Applications to Nonlocal Models, SIAM Review, 62, 1, 2020. Crossref

  34. Alali Bacim, Albin Nathan, Fourier multipliers for nonlocal Laplace operators, Applicable Analysis, 100, 12, 2021. Crossref

  35. You Huaiqian, Lu Xin Yang, Trask Nathaniel, Yu Yue, An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems, ESAIM: Mathematical Modelling and Numerical Analysis, 55, 2021. Crossref

  36. Gadikar Pranav, Diehl Patrick, Jha Prashant K., Load balancing for distributed nonlocal models within asynchronous many-task systems, 2021 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW), 2021. Crossref

  37. Čiegis Raimondas, Čiegis Remigijus, Dapšys Ignas, A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators, Mathematics, 9, 12, 2021. Crossref

  38. Fan Yiming, Tian Xiaochuan, Yang Xiu, Li Xingjie, Webster Clayton, Yu Yue, An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems, Journal of Computational Physics, 465, 2022. Crossref

  39. Umarov Sabir, Fractional Fokker-Planck-Kolmogorov equations associated with SDES on a bounded domain, Fractional Calculus and Applied Analysis, 20, 5, 2017. Crossref

Begell Digital Portal Begell 数字图书馆 电子图书 期刊 参考文献及会议录 研究收集 订购及政策 Begell House 联系我们 Language English 中文 Русский Português German French Spain