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Applied Analysis

The applied analysis group studies mathematical problems arising from physical, chemical, geophysical, biophysical, and materials sciences. These problems are often described by time-dependent partial, ordinary, or integral differential equations, together with sophisticated boundary conditions, interface conditions, and external forcing. Nonlinear dynamical systems offer a geometrical and topological framework for detecting, understanding, and quantifying complex phenomena of these time-dependent differential equations. Partial differential equation theory allows us to correctly formulate well-posed problems and to examine behaviors of solutions, and thus also allows us to set the stage for efficient numerical simulations. Nonlocal equations arise from macroscopic modeling of stochastic dynamical systems with Lévy noise and from modeling long-range interactions, and consequently give an understanding of anomalous diffusions.

Faculty with Primary Interests in Applied Analysis

» T. R. Bielecki » J. Duan » A.Lubin

Faculty with Secondary Interests in Applied Analysis

» I. Cialenco » F. Hickernell » S. Li » X. Li » F. Weening

Recent Publications

  • Z. Cheng, J. Duan, and L. Wang. Most Probable Dynamics of Some Non-Linear Systems under Noisy Fluctuations. Communications in Nonlinear Science and Numerical Simulation (2016), Vol. 30, Issue 1-3, p. 108-114.
  • T. Gao and J. Duan. Quantifying Model Uncertainty in Dynamical Systems Driven by Non-Gaussian Lévy Stable Noise with Observations on Mean Exit Time or Escape Probability. Communications in Nonlinear Science and Numerical Simulation (2016), Vol. 39, p. 1-6.
  • T. Gao, J. Duan, X. Kan, and Z. Cheng. Dynamical Inference for Transitions in Stochastic Systems with Alpha-Stable Lévy Noise (2016). Journal of Physics A: Mathematical and Theoretical, Vol. 49, No. 19, 294002.
  • T. Gao, J. Duan, and X. Li. Fokker-Planck Equations for Stochastic Dynamical Systems with Symmetric Lévy Motions. Applied Mathematics and Computation (2016), Vol. 278, p. 1-20.
  • G. Lv, J. Duan, H. Gao, and J.-L.Wu. On a Stochastic Nonlocal Conservation Law in a Bounded Domain (2016). Bulletin des Sciences Mathématiques, Vol. 140, Issue 6, p. 718 -746.
  • H. Qiao and J. Duan. Stationary Measures for Stochastic Differential Equations with Jumps (2016). Stochastics, Vol. 88, Issue 6, p. 864-883.
  • L. Serdukova, Y. Zheng, J. Duan, and J. Kurths. Stochastic Basins of Attraction for Metastable States (2016). Chaos 26, Article number 073117.
  • T. Wang, J. Duan, and T. Liu. Competition promotes the persistence of populations in ecosystems. Nature - Scientific Reports (2016) 6, Article number 30477.
  • Y. Zheng, J. Duan, L. Serdukova, and J. Kurths. Transitions in a Genetic Transcriptional Regulatory System under Lévy Motion (2016). Nature - Scientific Reports 6, Article Number 29274.
  • G. Lv and J. Duan. Impacts of Noise on a Class of Partial Differential Equations (2015). Journal of Differential Equations. Vol. 258, Issue: 6, p. 2196-2220.
  • H. Qiao and J. Duan, Nonlinear Filtering of Stochastic Dynamical Systems with Lévy Noises (2015). Advances in Applied Probability, Vol. 47, No. 3, p. 902-918.
  • J. Ren, J. Duan, and C. K. R. T. Jones. Approximation of Random Slow Manifolds and Settling of Inertial Particles under Uncertainty (2015). Journal of Dynamics and Differential Equations, Vol. 27, Issue 3, p. 961-979.
  • J. Ren, J. Duan, and X. Wang. A Parameter Estimation Method Based on Random Slow Manifolds. Applied Mathematical Modelling (2015), Vol. 39, Issue 13, p. 3721-3732.
  • W. Zou, D. V. Senthilkumar, R. Nagao, I. Z. Kiss, Y. Tang, A. Koseska, J. Duan, and J. Kurths. Restoration of Rhythmicity in Diffusively Coupled Dynamical Networks (2015). Nature - Communications 6, Article Number 7709.
  • G. Chen, J. Duan, and J. Zhang, Slow Foliation of a Slow-Fast Stochastic Evolutionary System (2014). Journal of Functional Analysis, Vol. 267, Issue 8, p. 2663-2697.
  • T. Gao, J. Duan, X. Li, and R. Song. Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noise. SIAM Journal on Scientific Computing (2014), Vol. 36, No. 3, p. A887-A906.
  • X. Sun, J. Duan, and X. Li. Modeling Nonlinear Oscillators under Excitation of Combined Gaussian and Poisson White Noise: A Viewpoint Based on Energy Conservation Law (2014). Nonlinear Dynamics, Vol. 84, Issue 3, p. 1311-1325.

Recent Research Grants

  • NSF DMS-1620449 (Co-PI J. Duan): Theoretical and numerical studies of nonlocal equations derived from stochastic differential equations with Levy noises, 2016-2020.
  • NSF DMS-1642545 (PI J. Duan), CBMS Conference: Nonlocal Dynamics Theory, Computation and Applications, 2016-2017.
  • NSF DMS-1025422 (Lead PI J. Duan). Collaborative Research: Mathematical Modeling by Bridging Primitive and Boussinesq Equations, 2010-2015.
  • NSF-0938235 (PI J. Duan): Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations, NSF-CBMS Regional Conference in the Mathematical Sciences, August 2010, Chicago.
  • NSF-0731201 (Co-PI J. Duan): Stochastic Agent-based Modeling of Angiogenesis and Tissue Growth, 2007-2010.