Compute Distance To:
 Documents Indexed: 317 Publications since 1986, including 7 Books 6 Contributions as Editor Co-Authors: 210 Co-Authors with 317 Joint Publications 5,609 Co-Co-Authors
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### Co-Authors

 6 single-authored 16 Venturi, Daniele 16 Wan, Xiaoliang 16 Zhang, Zhongqiang 15 Mao, Zhiping 14 Sherwin, Spencer J. 14 Zayernouri, Mohsen 13 Li, Zhen 12 Perdikaris, Paris G. 12 Xiu, Dongbin 11 Triantafyllou, Michael S. 11 Zeng, Fanhai 10 Meng, Xuhui 10 Raissi, Maziar 10 Wang, Zhi Cheng 9 Caswell, Bruce 9 Kirby, Robert M. II 9 Lu, Lu 9 Lucor, Didier 9 Song, Fangying 9 Su, Chau-Hsing 8 Dong, Suchuan 8 Lin, Guang 8 Maxey, Martin R. 7 Deng, Mingge 7 Evangelinos, Constantinos 7 Grinberg, Leopold 7 Lomtev, Igor 7 Orszag, Steven Alan 7 Pang, Guofei 7 Rozovskii, Boris L. 7 Yu, Yue 6 Cho, Heyrim 6 Choi, Minseok 6 Fedosov, Dmitry A. 6 Giannakouros, John G. 6 Jagtap, Ameya D. 6 Kharazmi, Ehsan 6 Sirisup, Sirod 6 Zhang, Dongkun 5 Babaee, Hessam 5 Bian, Xin 5 Cai, Wei 5 Constantinides, Yiannis 5 Henderson, Ronald D. 5 Ma, Xia 5 Sidilkover, David 5 Warburton, Timothy C. E. 5 Zheng, Xiaoning 4 Baek, Hyoungsu 4 Bittencourt, Marco Lúcio 4 Cai, Shengze 4 Cao, Wanrong 4 Guo, Ling 4 Lei, Huan 4 Patera, Anthony T. 4 Sapsis, Themistoklis P. 4 Symeonidis, Vasileios 4 Tang, Yuhang 4 Xu, Jin 4 Yang, Xiu 4 Yin, Minglang 3 Ainsworth, Mark 3 Beskok, Ali 3 Chryssostomidis, Chryssostomos 3 Foo, Jasmine 3 Gulian, Mamikon 3 Kawaguchi, Kenji 3 Kevrekidis, Ioannis George 3 Kim, Changho 3 Rockwell, Donald 3 Shu, Chi-Wang 3 Tang, Yifa 3 Tretyakov, Michael V. 3 Triantafyllou, George S. 3 Yazdani, Alireza 3 Zaki, Tamer A. 3 Zheng, Mengdi 2 Batcho, Paul F. 2 Blumers, Ansel L. 2 Bourguet, Rémi 2 Burrage, Kevin 2 Chen, Xiaoli 2 Cockburn, Bernardo 2 D’Elia, Marta 2 Du, Yiqing 2 Duan, Jinqiao 2 Fan, Dixia 2 Goswami, Somdatta 2 Jin, Pengzhan 2 Kaiktsis, Lambros 2 Karamanos, G.-S. 2 Li, Xuejin 2 Lischke, Anna 2 Luo, Xian 2 Newman, David J. 2 Ouyang, Jie 2 Parks, Michael L. 2 Pivkin, Igor V. 2 Rønquist, Einar M. 2 Shin, Yeonjong ...and 110 more Co-Authors
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### Serials

 97 Journal of Computational Physics 40 Computer Methods in Applied Mechanics and Engineering 38 SIAM Journal on Scientific Computing 31 Journal of Fluid Mechanics 8 Communications in Computational Physics 7 International Journal for Numerical Methods in Engineering 7 Journal of Scientific Computing 6 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 4 SIAM Journal on Numerical Analysis 3 International Journal for Numerical Methods in Fluids 3 Applied Numerical Mathematics 3 Computational Mechanics 3 Numerical Mathematics and Scientific Computation 2 Computers and Fluids 2 International Journal of Heat and Mass Transfer 2 Neural Networks 2 Proceedings of the National Academy of Sciences of the United States of America 2 Fractional Calculus & Applied Analysis 2 European Journal of Mechanics. B. Fluids 2 Multiscale Modeling & Simulation 2 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Computers & Mathematics with Applications 1 Computers and Structures 1 Journal of Engineering Mathematics 1 Journal of Statistical Physics 1 Physics of Fluids, A 1 Fluid Dynamics Research 1 Theoretical and Computational Fluid Dynamics 1 Chaos, Solitons and Fractals 1 Bulletin of the Polish Academy of Sciences. Technical Sciences 1 Numerische Mathematik 1 Physica D 1 European Journal of Applied Mathematics 1 Journal of Non-Newtonian Fluid Mechanics 1 SIAM Review 1 Physics of Fluids 1 Electronic Journal of Differential Equations (EJDE) 1 Parallel Algorithms and Applications 1 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Flow, Turbulence and Combustion 1 HERMIS-$$\mu\pi$$. Hellenic European Research on Mathematics and Informatics Science 1 Applied Mathematical Sciences 1 Interdisciplinary Applied Mathematics 1 Lecture Notes in Computational Science and Engineering 1 Science 1 International Journal for Uncertainty Quantification 1 Journal of Theoretical Biology 1 Transactions of Mathematics and its Applications 1 Handbook of Fractional Calculus with Applications 1 Communications on Applied Mathematics and Computation
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### Fields

 174 Fluid mechanics (76-XX) 144 Numerical analysis (65-XX) 79 Partial differential equations (35-XX) 55 Probability theory and stochastic processes (60-XX) 39 Computer science (68-XX) 37 Mechanics of deformable solids (74-XX) 25 Ordinary differential equations (34-XX) 22 Biology and other natural sciences (92-XX) 17 Statistics (62-XX) 11 Statistical mechanics, structure of matter (82-XX) 10 Geophysics (86-XX) 9 Approximations and expansions (41-XX) 8 Dynamical systems and ergodic theory (37-XX) 6 General and overarching topics; collections (00-XX) 6 Classical thermodynamics, heat transfer (80-XX) 5 Real functions (26-XX) 4 Optics, electromagnetic theory (78-XX) 3 Mechanics of particles and systems (70-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Special functions (33-XX) 1 Integral equations (45-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Convex and discrete geometry (52-XX) 1 Quantum theory (81-XX)

### Citations contained in zbMATH Open

276 Publications have been cited 8,534 times in 4,962 Documents Cited by Year
The Wiener–Askey polynomial chaos for stochastic differential equations. Zbl 1014.65004
2002
High-order splitting methods for the incompressible Navier-Stokes equations. Zbl 0738.76050
Karniadakis, George Em; Israeli, Moshe; Orszag, Steven A.
1991
Spectral/$$hp$$ element methods for computational fluid dynamics. 2nd ed. Zbl 1116.76002
Karniadakis, George Em; Sherwin, Spencer J.
2005
Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Zbl 1415.68175
Raissi, M.; Perdikaris, P.; Karniadakis, G. E.
2019
The development of discontinuous Galerkin methods. Zbl 0989.76045
Cockburn, Bernardo; Karniadakis, George E.; Shu, Chi-Wang
2000
Modeling uncertainty in flow simulations via generalized polynomial chaos. Zbl 1047.76111
2003
Spectral/hp element methods for CFD. Zbl 0954.76001
Karniadakis, George Em; Sherwin, Spencer J.
1999
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. Zbl 1078.65008
2005
Microflows and nanoflows. Fundamentals and simulation. Foreword by Chih-Ming Ho. Zbl 1115.76003
Karniadakis, George; Beskok, Ali; Narayan, Aluru
2005
Multi-element generalized polynomial chaos for arbitrary probability measures. Zbl 1128.65009
2006
Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Zbl 1016.65001
2002
Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. Zbl 1349.34095
2013
Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Zbl 0746.76021
Deane, A. E.; Kevrekidis, I. G.; Karniadakis, G. E.; Orszag, S. A.
1991
Fractional spectral collocation method. Zbl 1294.65097
2014
Hidden physics models: machine learning of nonlinear partial differential equations. Zbl 1381.68248
2018
Spectral/$$hp$$ element methods for computational fluid dynamics. Reprint of the 2nd hardback ed. (2005). Zbl 1256.76003
Karniadakis, George Em; Sherwin, Spencer J.
2013
A spectral vanishing viscosity method for large eddy simulations. Zbl 0984.76036
Karamanos, G. S.; Karniadakis, G. E.
2000
Exponentially accurate spectral and spectral element methods for fractional ODEs. Zbl 1349.65257
2014
Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. Zbl 1349.65531
2015
A semi-Lagrangian high-order method for Navier-Stokes equations. Zbl 1028.76026
2001
Second-order approximations for variable order fractional derivatives: algorithms and applications. Zbl 1349.65092
Zhao, Xuan; Sun, Zhi-zhong; Karniadakis, George Em
2015
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications. Zbl 1153.65008
Foo, Jasmine; Wan, Xiaoliang; Karniadakis, George Em
2008
A spectral viscosity method for correcting the long-term behavior of POD models. Zbl 1136.76412
2004
Multi-element probabilistic collocation method in high dimensions. Zbl 1181.65014
2010
Machine learning of linear differential equations using Gaussian processes. Zbl 1380.68339
Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
2017
A triangular spectral element method; applications to the incompressible Navier-Stokes equations. Zbl 1075.76621
Sherwin, S. J.; Karniadakis, G. E.
1995
Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. Zbl 0754.76043
Karniadakis, George Em; Triantafyllou, George S.
1992
A low-dimensional model for simulating three-dimensional cylinder flow. Zbl 1001.76043
2002
A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. Zbl 1339.65197
Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2015
Inferring solutions of differential equations using noisy multi-fidelity data. Zbl 1382.65229
Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
2017
A new triangular and tetrahedral basis for high-order $$(hp)$$ finite element methods. Zbl 0837.73075
Sherwin, Spencer J.; Karniadakis, George Em.
1995
A unified Petrov-Galerkin spectral method for fractional PDEs. Zbl 1425.65127
Zayernouri, Mohsen; Ainsworth, Mark; Karniadakis, George Em.
2015
Long-term behavior of polynomial chaos in stochastic flow simulations. Zbl 1123.76058
2006
De-aliasing on non-uniform grids: algorithms and applications. Zbl 1161.76534
Kirby, Robert M.; Karniadakis, George Em
2003
Discontinuous Galerkin methods. Theory, computation and applications. 1st international symposium on DGM, Newport, RI, USA, May 24–26, 1999. Zbl 0935.00043
2000
Discontinuous spectral element methods for time- and space-fractional advection equations. Zbl 1304.35757
2014
Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. Zbl 1386.65030
Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
2018
Reweighted $$\ell_1$$ minimization method for stochastic elliptic differential equations. Zbl 1349.60113
2013
What is the fractional Laplacian? A comparative review with new results. Zbl 1453.35179
Lischke, Anna; Pang, Guofei; Gulian, Mamikon; Song, Fangying; Glusa, Christian; Zheng, Xiaoning; Mao, Zhiping; Cai, Wei; Meerschaert, Mark M.; Ainsworth, Mark; Karniadakis, George Em
2020
Frequency selection and asymptotic states in laminar wakes. Zbl 0659.76043
Karniadakis, George Em; Triantafyllou, George S.
1989
Dynamics and low-dimensionality of a turbulent near wake. Zbl 0987.76041
Ma, X.; Karamanos, G.-S.; Karniadakis, G. E.
2000
A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. Zbl 0956.76046
Lomtev, I.; Kirby, R. M.; Karniadakis, G. E.
1999
A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. Zbl 1422.65428
2018
Micro flows. Fundamentals and simulation. Zbl 0998.76002
2002
Time-dependent generalized polynomial chaos. Zbl 1201.65216
Gerritsma, Marc; van der Steen, Jan-Bart; Vos, Peter; Karniadakis, George
2010
Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step. Zbl 0728.76057
Kaiktsis, Lambros; Karniadakis, George Em; Orszag, Steven A.
1991
Adaptive ANOVA decomposition of stochastic incompressible and compressible flows. Zbl 1408.76428
Yang, Xiu; Choi, Minseok; Lin, Guang; Karniadakis, George Em
2012
A new stochastic approach to transient heat conduction modeling with uncertainty. Zbl 1038.80003
2003
Tetrahedral spectral elements for CFD. Zbl 0857.76044
Sherwin, S. J.; Karniadakis, G. E.
1995
A combined direct numerical simulation–particle image velocimetry study of the turbulent near wake. Zbl 1177.76156
Dong, S.; Karniadakis, G. E.; Ekmekci, A.; Rockwell, D.
2006
A discontinuous Galerkin method for the viscous MHD equations. Zbl 0954.76051
Warburton, T. C.; Karniadakis, G. E.
1999
Beyond Wiener-Askey expansions: handling arbitrary PDFs. Zbl 1102.65006
2006
DeepXDE: a deep learning library for solving differential equations. Zbl 1459.65002
Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em
2021
FPINNs: fractional physics-informed neural networks. Zbl 1420.35459
Pang, Guofei; Lu, Lu; Karniadakis, George Em
2019
Spectral polynomial chaos solutions of the stochastic advection equation. Zbl 1001.76084
Jardak, M.; Su, C.-H.; Karniadakis, G. E.
2002
Tempered fractional Sturm-Liouville eigenproblems. Zbl 1323.34012
Zayernouri, Mohsen; Ainsworth, Mark; Karniadakis, George Em
2015
A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains. Zbl 1349.76569
Dong, S.; Karniadakis, G. E.; Chryssostomidis, C.
2014
Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. Zbl 1355.65104
Cao, Wanrong; Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2016
Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. Zbl 1352.65278
Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2016
Mechanisms of transverse motions in turbulent wall flows. Zbl 1039.76028
2003
A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Zbl 1423.76102
Song, Fangying; Xu, Chuanju; Karniadakis, George Em
2016
Petrov-Galerkin and spectral collocation methods for distributed order differential equations. Zbl 1367.65113
Kharazmi, Ehsan; Zayernouri, Mohsen; Karniadakis, George Em
2017
A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. Zbl 1431.65193
Zeng, Fanhai; Mao, Zhiping; Karniadakis, George Em
2017
A direct numerical simulation study of flow past a freely vibrating cable. Zbl 0901.76062
Newman, David J.; Karniadakis, George Em
1997
Unstructured spectral element methods for simulation of turbulent flows. Zbl 0840.76070
Henderson, Ronald D.; Karniadakis, George Em
1995
Spectral element-Fourier methods for incompressible turbulent flows. Zbl 0722.76053
1990
Physics-informed neural networks for high-speed flows. Zbl 1442.76092
Mao, Zhiping; Jagtap, Ameya D.; Karniadakis, George Em
2020
Spectral and discontinuous spectral element methods for fractional delay equations. Zbl 1314.34159
Zayernouri, Mohsen; Cao, Wanrong; Zhang, Zhongqiang; Karniadakis, George Em
2014
Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step. Zbl 0875.76111
Kaiktsis, Lambros; Karniadakis, George Em; Orszag, Steven A.
1996
Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling. Zbl 1407.62252
Perdikaris, P.; Raissi, M.; Damianou, A.; Lawrence, N. D.; Karniadakis, G. E.
2017
Equation-free/Galerkin-free POD-assisted computation of incompressible flows. Zbl 1213.76146
Sirisup, Sirod; Karniadakis, George Em; Xiu, Dongbin; Kevrekidis, Ioannis G.
2005
Drag reduction in wall-bounded turbulence via a transverse travelling wave. Zbl 1112.76373
Du, Yiqing; Symeonidis, V.; Karniadakis, G. E.
2002
Tetrahedral $$hp$$ finite elements: Algorithms and flow simulations. Zbl 0847.76038
Sherwin, S. J.; Karniadakis, G. E.
1996
Spectral/hp methods for viscous compressible flows on unstructured 2D meshes. Zbl 0929.76095
Lomtev, I.; Quillen, C. B.; Karniadakis, G. E.
1998
Optimal error estimates of spectral Petrov-Galerkin and collocation methods for initial value problems of fractional differential equations. Zbl 1326.65100
Zhang, Zhongqiang; Zeng, Fanhai; Karniadakis, George Em
2015
Numerical methods for stochastic partial differential equations with white noise. Zbl 1380.65021
2017
Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Zbl 1439.65081
Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2017
Systematic coarse-graining of spectrin-level red blood cell models. Zbl 1231.74311
Fedosov, Dmitry A.; Caswell, Bruce; Karniadakis, George Em
2010
Dynamics and flow structures in the turbulent wake of rigid and flexible cylinders subject to vortex-induced vibrations. Zbl 0983.76029
1999
Generalized polynomial chaos and random oscillators. Zbl 1060.70515
Lucor, D.; Su, C.-H.; Karniadakis, G. E.
2004
Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems. Zbl 1442.92002
Jagtap, Ameya D.; Kharazmi, Ehsan; Karniadakis, George Em
2020
Spectral element simulations of laminar and turbulent flows in complex geometries. Zbl 0678.76050
1989
Basis functions for triangular and quadrilateral high-order elements. Zbl 0930.35016
Warburton, T. C.; Sherwin, S. J.; Karniadakis, G. E.
1999
A new method to impose no-slip boundary conditions in dissipative particle dynamics. Zbl 1177.76325
Pivkin, Igor V.; Karniadakis, George Em
2005
A discontinuous Galerkin method for the Navier-Stokes equations. Zbl 0951.76041
1999
Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. Zbl 1454.65008
Zhang, Dongkun; Lu, Lu; Guo, Ling; Karniadakis, George Em
2019
Exact PDF equations and closure approximations for advective-reactive transport. Zbl 1349.35068
Venturi, D.; Tartakovsky, D. M.; Tartakovsky, A. M.; Karniadakis, G. E.
2013
A convergence study of a new partitioned fluid-structure interaction algorithm based on fictitious mass and damping. Zbl 1426.76496
2012
Deep learning of vortex-induced vibrations. Zbl 1415.76177
Raissi, Maziar; Wang, Zhicheng; Triantafyllou, Michael S.; Karniadakis, George Em
2019
Stability and accuracy of periodic flow solutions obtained by a POD-penalty method. Zbl 1070.35024
2005
Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms. Zbl 1415.74049
Yu, Yue; Perdikaris, Paris; Karniadakis, George Em
2016
Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems. Zbl 1371.60119
2014
Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. Zbl 1380.65357
Song, Fangying; Xu, Chuanju; Karniadakis, George Em
2017
Galerkin and discontinuous Galerkin spectral/$$hp$$ methods. Zbl 0924.76078
Warburton, T. C.; Lomtev, I.; Du, Y.; Sherwin, S. J.; Karniadakis, G. E.
1999
A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. Zbl 1454.76006
2020
A Petrov-Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. Zbl 1367.65108
Lischke, Anna; Zayernouri, Mohsen; Karniadakis, George Em
2017
Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Zbl 1478.76057
Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em.
2020
A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems. Zbl 1365.92023
Venturi, D.; Sapsis, T. P.; Cho, H.; Karniadakis, G. E.
2012
Gappy data and reconstruction procedures for flow past a cylinder. Zbl 1065.76159
2004
Spectral element methods for elliptic problems in nonsmooth domains. Zbl 0844.65082
1995
DeepXDE: a deep learning library for solving differential equations. Zbl 1459.65002
Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em
2021
hp-VPINNs: variational physics-informed neural networks with domain decomposition. Zbl 07338008
Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M.
2021
Two-point stress-strain-rate correlation structure and non-local eddy viscosity in turbulent flows. Zbl 1461.76181
Clark Di Leoni, Patricio; Zaki, Tamer A.; Karniadakis, George; Meneveau, Charles
2021
Non-invasive inference of thrombus material properties with physics-informed neural networks. Zbl 07340448
Yin, Minglang; Zheng, Xiaoning; Humphrey, Jay D.; Karniadakis, George Em
2021
Towards a unified theory of fractional and nonlocal vector calculus. Zbl 07443850
D&rsquo;Elia, Marta; Gulian, Mamikon; Olson, Hayley; Karniadakis, George Em
2021
Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks. Zbl 1461.76417
Cai, Shengze; Wang, Zhicheng; Fuest, Frederik; Jeon, Young Jin; Gray, Callum; Karniadakis, George Em
2021
Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks. Zbl 1480.35377
Chen, Xiaoli; Yang, Liu; Duan, Jinqiao; Karniadakis, George Em
2021
Multiscale parareal algorithm for long-time mesoscopic simulations of microvascular blood flow in zebrafish. Zbl 1479.76123
Blumers, Ansel L.; Yin, Minglang; Nakajima, Hiroyuki; Hasegawa, Yosuke; Li, Zhen; Karniadakis, George Em
2021
Learning and meta-learning of stochastic advection-diffusion-reaction systems from sparse measurements. Zbl 07441294
Chen, Xiaoli; Duan, Jinqiao; Karniadakis, George Em
2021
What is the fractional Laplacian? A comparative review with new results. Zbl 1453.35179
Lischke, Anna; Pang, Guofei; Gulian, Mamikon; Song, Fangying; Glusa, Christian; Zheng, Xiaoning; Mao, Zhiping; Cai, Wei; Meerschaert, Mark M.; Ainsworth, Mark; Karniadakis, George Em
2020
Physics-informed neural networks for high-speed flows. Zbl 1442.76092
Mao, Zhiping; Jagtap, Ameya D.; Karniadakis, George Em
2020
Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems. Zbl 1442.92002
Jagtap, Ameya D.; Kharazmi, Ehsan; Karniadakis, George Em
2020
A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. Zbl 1454.76006
2020
Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Zbl 1478.76057
Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em.
2020
Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. Zbl 1453.68165
Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em
2020
Learning in modal space: solving time-dependent stochastic PDEs using physics-informed neural networks. Zbl 1440.60067
Zhang, Dongkun; Guo, Ling; Karniadakis, George Em
2020
Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Zbl 07419158
Jagtap, Ameya D.; Karniadakis, George Em
2020
Physics-informed generative adversarial networks for stochastic differential equations. Zbl 1440.60065
Yang, Liu; Zhang, Dongkun; Karniadakis, George Em
2020
On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Zbl 1473.65349
Shin, Yeonjong; Darbon, Jérôme; Karniadakis, George Em
2020
PPINN: parareal physics-informed neural network for time-dependent PDEs. Zbl 07337119
Meng, Xuhui; Li, Zhen; Zhang, Dongkun; Karniadakis, George Em
2020
A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations. Zbl 1453.65370
Zhang, Hui; Jiang, Xiaoyun; Zeng, Fanhai; Karniadakis, George Em
2020
Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. Zbl 1472.68175
Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em
2020
Sympnets: intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. Zbl 1475.68316
Jin, Pengzhan; Zhang, Zhen; Zhu, Aiqing; Tang, Yifa; Karniadakis, George Em
2020
A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation. Zbl 1442.65144
Li, Xianjuan; Mao, Zhiping; Wang, Nan; Song, Fangying; Wang, Hong; Karniadakis, George Em
2020
Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness. Zbl 1475.68315
Jin, Pengzhan; Lu, Lu; Tang, Yifa; Karniadakis, George Em
2020
Dying ReLU and initialization: theory and numerical examples. Zbl 07419148
Lu, Lu; Shin, Yeonjong; Su, Yanhui; Karniadakis, George Em
2020
Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Zbl 1415.68175
Raissi, M.; Perdikaris, P.; Karniadakis, G. E.
2019
FPINNs: fractional physics-informed neural networks. Zbl 1420.35459
Pang, Guofei; Lu, Lu; Karniadakis, George Em
2019
Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. Zbl 1454.65008
Zhang, Dongkun; Lu, Lu; Guo, Ling; Karniadakis, George Em
2019
Deep learning of vortex-induced vibrations. Zbl 1415.76177
Raissi, Maziar; Wang, Zhicheng; Triantafyllou, Michael S.; Karniadakis, George Em
2019
Efficient multistep methods for tempered fractional calculus: algorithms and simulations. Zbl 07099350
Guo, Ling; Zeng, Fanhai; Turner, Ian; Burrage, Kevin; Karniadakis, George Em
2019
Multi-domain spectral collocation method for variable-order nonlinear fractional differential equations. Zbl 1440.65258
Zhao, Tinggang; Mao, Zhiping; Karniadakis, George Em
2019
Machine learning of space-fractional differential equations. Zbl 1419.35209
Gulian, Mamikon; Raissi, Maziar; Perdikaris, Paris; Karniadakis, George
2019
Neural-net-induced Gaussian process regression for function approximation and PDE solution. Zbl 1451.68242
Pang, Guofei; Yang, Liu; Karniadakis, George Em
2019
Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network. Zbl 1434.76053
Mehta, Pavan Pranjivan; Pang, Guofei; Song, Fangying; Karniadakis, George Em
2019
An entropy-viscosity large eddy simulation study of turbulent flow in a flexible pipe. Zbl 1415.76369
Wang, Zhicheng; Triantafyllou, Michael S.; Constantinides, Yiannis; Karniadakis, George Em
2019
Fractional Gray-Scott model: well-posedness, discretization, and simulations. Zbl 1440.35344
Wang, Tingting; Song, Fangying; Wang, Hong; Karniadakis, George Em
2019
Fractional magneto-hydrodynamics: algorithms and applications. Zbl 1416.76127
2019
A spectral penalty method for two-sided fractional differential equations with General boundary conditions. Zbl 07099316
Wang, Nan; Mao, Zhiping; Huang, Chengming; Karniadakis, George Em
2019
Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: application to hydrodynamics. Zbl 1452.76172
Blumers, Ansel L.; Li, Zhen; Karniadakis, George Em
2019
Mapping the properties of the vortex-induced vibrations of flexible cylinders in uniform oncoming flow. Zbl 1430.76091
Fan, Dixia; Wang, Zhicheng; Triantafyllou, Michael S.; Karniadakis, George Em
2019
Nonlocal flocking dynamics: learning the fractional order of PDEs from particle simulations. Zbl 1449.35447
Mao, Zhiping; Li, Zhen; Karniadakis, George Em
2019
Numerical methods. Zbl 1410.65001
2019
A Computational Mechanics special issue on: Data-driven modeling and simulation – theory, methods, and applications. Zbl 1476.00073
2019
A stabilized phase-field method for two-phase flow at high Reynolds number and large density/viscosity ratio. Zbl 1453.76154
Wang, Zhicheng; Dong, Suchuan; Triantafyllou, Michael S.; Constantinides, Yiannis; Karniadakis, George Em
2019
One-dimensional modeling of fractional flow reserve in coronary artery disease: uncertainty quantification and Bayesian optimization. Zbl 1441.76150
Yin, Minglang; Yazdani, Alireza; Karniadakis, George Em
2019
Hidden physics models: machine learning of nonlinear partial differential equations. Zbl 1381.68248
2018
Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. Zbl 1386.65030
Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
2018
A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. Zbl 1422.65428
2018
A Riesz basis Galerkin method for the tempered fractional Laplacian. Zbl 1402.65120
Zhang, Zhijiang; Deng, Weihua; Karniadakis, George Em
2018
A partitioned coupling framework for peridynamics and classical theory: analysis and simulations. Zbl 1440.74045
Yu, Yue; Bargos, Fabiano F.; You, Huaiqian; Parks, Michael L.; Bittencourt, Marco L.; Karniadakis, George E.
2018
A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations. Zbl 1404.65105
Zeng, Fanhai; Turner, Ian; Burrage, Kevin; Karniadakis, George Em
2018
A dissipative particle dynamics method for arbitrarily complex geometries. Zbl 1380.76123
Li, Zhen; Bian, Xin; Tang, Yu-Hang; Karniadakis, George Em
2018
A computational stochastic methodology for the design of random meta-materials under geometric constraints. Zbl 1398.60073
Tsantili, Ivi C.; Cho, Min Hyung; Cai, Wei; Karniadakis, George Em
2018
Multi-fidelity optimization of super-cavitating hydrofoils. Zbl 1439.74256
Bonfiglio, L.; Perdikaris, P.; Brizzolara, S.; Karniadakis, G. E.
2018
Stochastic domain decomposition via moment minimization. Zbl 1391.60167
Zhang, Dongkun; Babaee, Hessam; Karniadakis, George Em
2018
A spectral-element/Fourier smoothed profile method for large-eddy simulations of complex VIV problems. Zbl 1410.76320
Wang, Zhicheng; Triantafyllou, Michael S.; Constantinides, Yiannis; Karniadakis, George Em
2018
Active learning of constitutive relation from mesoscopic dynamics for macroscopic modeling of non-Newtonian flows. Zbl 1392.76058
Zhao, Lifei; Li, Zhen; Caswell, Bruce; Ouyang, Jie; Karniadakis, George Em
2018
Analytical and computational studies of correlations of hydrodynamic fluctuations in shear flow. Zbl 07414373
Bian, Xin; Deng, Mingge; Karniadakis, George Em
2018
Machine learning of linear differential equations using Gaussian processes. Zbl 1380.68339
Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
2017
Inferring solutions of differential equations using noisy multi-fidelity data. Zbl 1382.65229
Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
2017
Petrov-Galerkin and spectral collocation methods for distributed order differential equations. Zbl 1367.65113
Kharazmi, Ehsan; Zayernouri, Mohsen; Karniadakis, George Em
2017
A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. Zbl 1431.65193
Zeng, Fanhai; Mao, Zhiping; Karniadakis, George Em
2017
Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling. Zbl 1407.62252
Perdikaris, P.; Raissi, M.; Damianou, A.; Lawrence, N. D.; Karniadakis, G. E.
2017
Numerical methods for stochastic partial differential equations with white noise. Zbl 1380.65021
2017
Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Zbl 1439.65081
Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2017
Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. Zbl 1380.65357
Song, Fangying; Xu, Chuanju; Karniadakis, George Em
2017
A Petrov-Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. Zbl 1367.65108
Lischke, Anna; Zayernouri, Mohsen; Karniadakis, George Em
2017
Multi-fidelity Gaussian process regression for prediction of random fields. Zbl 1419.62272
Parussini, L.; Venturi, D.; Perdikaris, P.; Karniadakis, G. E.
2017
Discovering variable fractional orders of advection-dispersion equations from field data using multi-fidelity Bayesian optimization. Zbl 1419.62499
Pang, Guofei; Perdikaris, Paris; Cai, Wei; Karniadakis, George Em
2017
A Petrov-Galerkin spectral element method for fractional elliptic problems. Zbl 1439.65205
Kharazmi, Ehsan; Zayernouri, Mohsen; Karniadakis, George Em
2017
Adaptive finite element method for fractional differential equations using hierarchical matrices. Zbl 1439.65091
Zhao, Xuan; Hu, Xiaozhe; Cai, Wei; Karniadakis, George Em
2017
A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems. Zbl 1380.76093
Babaee, Hessam; Choi, Minseok; Sapsis, Themistoklis P.; Karniadakis, George Em
2017
A tunable finite difference method for fractional differential equations with non-smooth solutions. Zbl 1439.65082
Chen, Xuejuan; Zeng, Fanhai; Karniadakis, George Em
2017
Fractional Burgers equation with nonlinear non-locality: spectral vanishing viscosity and local discontinuous Galerkin methods. Zbl 1380.65280
2017
Dissipative particle dynamics: foundation, evolution, implementation, and applications. Zbl 1387.35496
Li, Z.; Bian, X.; Li, X.; Deng, M.; Tang, Y.-H.; Caswell, B.; Karniadakis, George E.
2017
Numerical methods for high-dimensional kinetic equations. Zbl 1404.65117
Cho, Heyrim; Venturi, Daniele; Karniadakis, George Em
2017
Efficient two-dimensional simulations of the fractional Szabo equation with different time-stepping schemes. Zbl 1412.65091
Song, Fangying; Zeng, Fanhai; Cai, Wei; Chen, Wen; Karniadakis, George Em
2017
Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. Zbl 1355.65104
Cao, Wanrong; Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2016
Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. Zbl 1352.65278
Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2016
A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Zbl 1423.76102
Song, Fangying; Xu, Chuanju; Karniadakis, George Em
2016
Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms. Zbl 1415.74049
Yu, Yue; Perdikaris, Paris; Karniadakis, George Em
2016
Multifidelity information fusion algorithms for high-dimensional systems and massive data sets. Zbl 1342.62110
Perdikaris, Paris; Venturi, Daniele; Karniadakis, George Em
2016
Numerical methods for high-dimensional probability density function equations. Zbl 1349.65046
Cho, H.; Venturi, D.; Karniadakis, G. E.
2016
Strong and weak convergence order of finite element methods for stochastic PDEs with spatial white noise. Zbl 1357.65013
Zhang, Zhongqiang; Rozovskii, Boris; Karniadakis, George Em.
2016
Fractional-order uniaxial visco-elasto-plastic models for structural analysis. Zbl 1439.74077
Suzuki, J. L.; Zayernouri, M.; Bittencourt, M. L.; Karniadakis, G. E.
2016
Multi-fidelity modelling of mixed convection based on experimental correlations and numerical simulations. Zbl 1383.76438
Babaee, H.; Perdikaris, P.; Chryssostomidis, C.; Karniadakis, G. E.
2016
Flow in complex domains simulated by dissipative particle dynamics driven by geometry-specific body-forces. Zbl 1349.76069
Yazdani, Alireza; Deng, Mingge; Caswell, Bruce; Karniadakis, George Em
2016
Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. Zbl 1349.65531
2015
Second-order approximations for variable order fractional derivatives: algorithms and applications. Zbl 1349.65092
Zhao, Xuan; Sun, Zhi-zhong; Karniadakis, George Em
2015
A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. Zbl 1339.65197
Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2015
A unified Petrov-Galerkin spectral method for fractional PDEs. Zbl 1425.65127
Zayernouri, Mohsen; Ainsworth, Mark; Karniadakis, George Em.
2015
Tempered fractional Sturm-Liouville eigenproblems. Zbl 1323.34012
Zayernouri, Mohsen; Ainsworth, Mark; Karniadakis, George Em
2015
Optimal error estimates of spectral Petrov-Galerkin and collocation methods for initial value problems of fractional differential equations. Zbl 1326.65100
Zhang, Zhongqiang; Zeng, Fanhai; Karniadakis, George Em
2015
Time-splitting schemes for fractional differential equations I: Smooth solutions. Zbl 1320.65106
Cao, Wanrong; Zhang, Zhongqiang; Karniadakis, George Em
2015
Numerical methods for stochastic delay differential equations via the Wong-Zakai approximation. Zbl 1329.60233
Cao, Wanrong; Zhang, Zhongqiang; Karniadakis, George Em
2015
Multi-resolution flow simulations by smoothed particle hydrodynamics via domain decomposition. Zbl 1349.76663
Bian, Xin; Li, Zhen; Karniadakis, George Em
2015
Special issue on “Fractional PDEs: theory, numerics, and applications”. Zbl 1349.35004
2015
Multiscale universal interface: a concurrent framework for coupling heterogeneous solvers. Zbl 1349.76738
Tang, Yu-Hang; Kudo, Shuhei; Bian, Xin; Li, Zhen; Karniadakis, George Em
2015
Numerical methods for SPDEs with tempered stable processes. Zbl 1320.65020
2015
...and 176 more Documents
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### Cited by 7,186 Authors

 192 Karniadakis, George Em 70 Sherwin, Spencer J. 48 Lin, Guang 46 Wang, Hong 39 Xiu, Dongbin 32 Zheng, Xiangcheng 31 Thompson, Mark Christopher 29 Doostan, Alireza 28 Narayan, Akil C. 26 Shen, Jie 25 Dehghan Takht Fooladi, Mehdi 25 Le Maître, Olivier P. 23 Ghanem, Roger G. 23 Venturi, Daniele 22 Schwab, Christoph 22 Shu, Chi-Wang 22 Zhou, Tao 21 Hourigan, Kerry 21 Perdikaris, Paris G. 21 Zabaras, Nicholas J. 20 Blackburn, Hugh M. 20 Dumbser, Michael 20 Hesthaven, Jan S. 20 Kopriva, David Alan 20 Nobile, Fabio 20 Rozza, Gianluigi 19 Tartakovsky, Daniel M. 18 Lucor, Didier 18 Mao, Zhiping 18 Xu, Chuanju 18 Zaky, Mahmoud A. 18 Zayernouri, Mohsen 17 Abbaszadeh, Mostafa 17 Jin, Shi 17 Knio, Omar M. 17 Sagaut, Pierre 17 Wan, Xiaoliang 17 Wang, Lilian 16 Dong, Suchuan 16 Guo, Ben-Yu 16 Jakeman, John Davis 16 Najm, Habib N. 16 Noack, Bernd R. 16 Nordström, Jan 16 Quarteroni, Alfio M. 16 Zhang, Dongxiao 15 Giraldo, Francis X. 15 Iaccarino, Gianluca 15 Li, Zhen 15 Liu, Fawang 15 Warburton, Timothy 15 Zhang, Zhimin 14 Baccouch, Mahboub 14 Boscheri, Walter 14 Brunton, Steven L. 14 Kronbichler, Martin 14 Machado, José António Tenreiro 14 Mishra, Siddhartha 14 Pasquetti, Richard 14 Sheard, Gregory John 14 Tartakovsky, Alexandre M. 14 Xing, Yulong 14 Yang, Xiu 14 Zeng, Fanhai 14 Zhang, Zhongqiang 13 Ainsworth, Mark 13 Bourguet, Rémi 13 Elman, Howard C. 13 Ferrer, Esteban 13 Guo, Ling 13 Jiang, Xiaoyun 13 Luo, Hong 13 Pulch, Roland 13 Reddy, Junuthula Narasimha 13 Soize, Christian 13 Triantafyllou, Michael S. 12 Cheng, Liang 12 Congedo, Pietro Marco 12 Deng, Weihua 12 Gerritsma, Marc I. 12 Gunzburger, Max D. 12 Kutz, J. Nathan 12 Marzouk, Youssef M. 12 Matthies, Hermann Georg 12 Maxey, Martin R. 12 Nouy, Anthony 12 Trask, Nathaniel A. 12 Turner, Ian William 12 Xu, Yan 12 Yu, Yue 11 Abdelkawy, Mohamed A. 11 Băleanu, Dumitru I. 11 Cortés López, Juan Carlos 11 Dawson, Clint N. 11 Fischer, Paul F. 11 Gassner, Gregor J. 11 Hao, Zhaopeng 11 Leweke, Thomas 11 Li, Dongfang 11 Li, Hong ...and 7,086 more Authors
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### Cited in 327 Serials

 975 Journal of Computational Physics 429 Computer Methods in Applied Mechanics and Engineering 283 Journal of Fluid Mechanics 277 Journal of Scientific Computing 273 Computers and Fluids 149 SIAM Journal on Scientific Computing 127 Physics of Fluids 114 Applied Numerical Mathematics 108 Computers & Mathematics with Applications 100 Journal of Computational and Applied Mathematics 83 Applied Mathematics and Computation 76 International Journal for Numerical Methods in Engineering 69 SIAM/ASA Journal on Uncertainty Quantification 66 Applied Mathematical Modelling 65 International Journal for Numerical Methods in Fluids 52 Mathematics of Computation 49 Communications in Computational Physics 47 European Journal of Mechanics. B. Fluids 45 Computational Mechanics 43 Mathematics and Computers in Simulation 39 Numerical Algorithms 38 International Journal of Computer Mathematics 33 Numerische Mathematik 32 SIAM Journal on Numerical Analysis 32 Computational and Applied Mathematics 31 Communications in Nonlinear Science and Numerical Simulation 30 Computational Geosciences 29 Fractional Calculus & Applied Analysis 27 Physica D 27 Advances in Computational Mathematics 26 International Journal of Heat and Mass Transfer 25 Mathematical Problems in Engineering 25 International Journal of Computational Fluid Dynamics 23 Chaos, Solitons and Fractals 23 Engineering Analysis with Boundary Elements 22 Communications on Applied Mathematics and Computation 21 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 19 Journal of Engineering Mathematics 19 Journal of Mathematical Analysis and Applications 19 Nonlinear Dynamics 18 Advances in Difference Equations 17 Archives of Computational Methods in Engineering 17 Multiscale Modeling & Simulation 15 BIT 15 International Journal of Computational Methods 15 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 14 Applied Mathematics Letters 13 Journal of Statistical Physics 12 Numerical Methods for Partial Differential Equations 11 Acta Mechanica 11 Theoretical and Computational Fluid Dynamics 11 Mathematical and Computer Modelling 11 European Journal of Mechanics. A. Solids 10 Computer Physics Communications 10 Automatica 10 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 10 Mathematical Modelling of Natural Phenomena 10 East Asian Journal on Applied Mathematics 9 Computational Methods in Applied Mathematics 9 Discrete and Continuous Dynamical Systems. Series S 9 Advances in Applied Mathematics and Mechanics 9 Journal of Theoretical Biology 9 AMM. Applied Mathematics and Mechanics. (English Edition) 8 SIAM Review 8 Chaos 8 Journal of Turbulence 8 Journal of Applied Mathematics and Computing 8 Journal of Computational Acoustics 8 Acta Mechanica Sinica 8 Advances in Mathematical Physics 8 International Journal of Applied and Computational Mathematics 7 Journal of Mathematical Physics 7 Meccanica 7 International Journal of Numerical Methods for Heat & Fluid Flow 7 Combustion Theory and Modelling 7 Discrete and Continuous Dynamical Systems. Series B 7 SIAM Journal on Applied Dynamical Systems 7 International Journal of Numerical Analysis and Modeling 7 Statistics and Computing 6 ZAMP. Zeitschrift für angewandte Mathematik und Physik 6 SIAM Journal on Control and Optimization 6 Continuum Mechanics and Thermodynamics 6 Abstract and Applied Analysis 6 Journal of Numerical Mathematics 6 Acta Numerica 6 Communications in Applied and Industrial Mathematics 6 Stochastic and Partial Differential Equations. Analysis and Computations 6 Journal of Function Spaces 6 Results in Applied Mathematics 5 Journal of Mathematical Biology 5 Mathematical Methods in the Applied Sciences 5 Physics Letters. A 5 Wave Motion 5 Calcolo 5 Applied Mathematics and Mechanics. (English Edition) 5 European Journal of Applied Mathematics 5 Journal of Non-Newtonian Fluid Mechanics 5 Archive of Applied Mechanics 5 Fractals 5 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik ...and 227 more Serials
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### Cited in 46 Fields

 2,882 Numerical analysis (65-XX) 2,191 Fluid mechanics (76-XX) 1,406 Partial differential equations (35-XX) 493 Mechanics of deformable solids (74-XX) 465 Probability theory and stochastic processes (60-XX) 284 Computer science (68-XX) 261 Statistics (62-XX) 254 Ordinary differential equations (34-XX) 228 Biology and other natural sciences (92-XX) 203 Real functions (26-XX) 134 Statistical mechanics, structure of matter (82-XX) 121 Dynamical systems and ergodic theory (37-XX) 116 Classical thermodynamics, heat transfer (80-XX) 112 Approximations and expansions (41-XX) 104 Systems theory; control (93-XX) 100 Optics, electromagnetic theory (78-XX) 100 Geophysics (86-XX) 76 Calculus of variations and optimal control; optimization (49-XX) 57 Operations research, mathematical programming (90-XX) 56 Special functions (33-XX) 53 Integral equations (45-XX) 44 Mechanics of particles and systems (70-XX) 33 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 32 Harmonic analysis on Euclidean spaces (42-XX) 32 Information and communication theory, circuits (94-XX) 30 Linear and multilinear algebra; matrix theory (15-XX) 29 Operator theory (47-XX) 17 Functional analysis (46-XX) 16 Quantum theory (81-XX) 14 Global analysis, analysis on manifolds (58-XX) 11 Difference and functional equations (39-XX) 10 Integral transforms, operational calculus (44-XX) 8 General and overarching topics; collections (00-XX) 6 Functions of a complex variable (30-XX) 5 Combinatorics (05-XX) 5 Differential geometry (53-XX) 5 Astronomy and astrophysics (85-XX) 4 Number theory (11-XX) 3 Measure and integration (28-XX) 2 History and biography (01-XX) 2 Relativity and gravitational theory (83-XX) 2 Mathematics education (97-XX) 1 Mathematical logic and foundations (03-XX) 1 Potential theory (31-XX) 1 Convex and discrete geometry (52-XX) 1 General topology (54-XX)

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