摘要：
In this work, we consider the numerical integration of the nonlinear Dirac equation and the Dirac-Poisson system (NDEs) under rough initial data. We propose an ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in H-r for solutions in H-r, i.e., without requiring any additional regularity on the solution. In contrast to classical methods, a ULI overcomes the numerical loss of derivatives and is therefore more efficient and accurate for approximating low regular solutions. Convergence theorems and the extension of a ULI to second order are established. Numerical experiments confirm the theoretical results and underline the favourable error behaviour of the new method at low regularity compared to classical integration schemes.
期刊：
SIAM JOURNAL ON NUMERICAL ANALYSIS,2019年57(4):1602-1624 ISSN：0036-1429
通讯作者：
Wang, Yan
作者机构：
[Cai, Yongyong] Beijing Computat Sci Res Ctr, Beijing 100193, Peoples R China.;[Wang, Yan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Wang, Yan] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
摘要：
This paper is devoted to the construction and analysis of uniformly accurate nested Picard iterative integrators (NPI) for the Dirac equation in the nonrelativistic limit regime. In this regime, there is a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light and the equation admits propagating waves with O(1) wavelength in space and O(ε2) wavelength in time. To overcome the difficulty induced by the temporal ε dependent oscillation, we present the construction of several NPI methods which are uniformly first-, second-, and third-order convergent in time w.r.t. ε. The general idea is applying nested Picard iterations to the integral form of the Dirac equation and using exponential wave integrators to approximate the temporal integrals. Thanks to the nested Picard iterative idea, the NPI method can be extended to arbitrary higher-order in time with optimal and uniform accuracy. The implementation of the second-order in-time NPI method via Fourier pseudospectral discretization is clearly demonstrated, and the corresponding error bounds are rigorously established through the energy method as hm 0 + τ2, where h is the mesh size, τ is the time step, and m depends on the regularity of the solution. Numerical results are reported to confirm the error estimates for the second-order NPI method and show the uniform accurate properties (w.r.t. ε) for the first- and third-order NPI methods as well. 2019 Society for Industrial and Applied Mathematics
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2019年39(4):1745-1777 ISSN：1078-0947
通讯作者：
Yang, Jun
作者机构：
[Jiang, Ruiqi] Hunan Univ, Coll Math & Econometr, Changsha 410082, Hunan, Peoples R China.;[Wang, Youde] Guangzhou Univ, Coll Math & Informat Sci, Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Yang, Jun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
期刊：
IMA JOURNAL OF APPLIED MATHEMATICS,2018年83(2):283-301 ISSN：0272-4960
通讯作者：
Wang, Yan
作者机构：
[Wang, Yan] Hubei Univ Chinese Med, Informat Engn Coll, Wuhan 430065, Hubei, Peoples R China.;[Guo, Jun] South Cent Univ Nationalities, Sch Math & Stat, Wuhan 430074, Hubei, Peoples R China.;[Yan, Guozheng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Wang, Yan] H;Hubei Univ Chinese Med, Informat Engn Coll, Wuhan 430065, Hubei, Peoples R China.
关键词：
the factorization method;inverse scattering problem;oblique derivative boundary condition
摘要：
We consider a direct and inverse problem for the scattering of an obstacle with a generalized oblique derivative and impedance boundary condition due to the incident plane wave, which arises in some scattering phenomenons such as the scattering of tidal waves by islands under suitable assumptions. The solvability of the direct scattering problem is proven by using the boundary integral equation method. A special technique is proposed in order to show the formulated boundary integral operator is Fredholm with index 0. Then the factorization method is established to reconstruct the shape of the obstacle. We note that the tangential derivative on the obstacle boundary leads to the difficulties on mathematical analysis. Finally, we present some numerical examples in 2D to show the feasibility and effectiveness of the factorization method.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2017年37(6):3161-3182 ISSN：1078-0947
通讯作者：
Fan, Shilei
作者机构：
[Fan, Aihua; Fan, Shilei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Fan, Aihua] Univ Picardie Jules Verne, CNRS, LAMFA, UMR 7352, 33 Rue St Leu, F-80039 Amiens 1, France.;[Fan, Shilei] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Liao, Lingmin] Univ Paris Est Creteil, LAMA, UMR 8050, CNRS, 61 Ave Gen Gaulle, F-94010 Creteil, France.;[Wang, Yuefei] Chinese Acad Sci, AMSS, Inst Math, Beijing 100190, Peoples R China.
通讯机构：
[Fan, Shilei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
A rational map with good reduction in the field ℚp of p-adic numbers defines a 1-Lipschitz dynamical system on the projective line ℙ1(ℚp) over ℚp. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, ℙ1 (ℚp) is decomposed into three parts: Finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime p, a criterion of minimality for rational maps with good reduction is obtained. When p = 2, a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and 1-Lipschitz. It is also proved that a rational map having good reduction of degrees 2, 3 and 4 can never be minimal on the whole space ℙ1 (ℚ1).
期刊：
Journal of Theoretical Biology,2016年390:40-49 ISSN：0022-5193
通讯作者：
Zhou, Da;Zhang, Xingan
作者机构：
[Zhang, Xingan; Chen, Xiufang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhou, Da] Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China.;[Wang, Yue] Univ Washington, Dept Appl Math, Seattle, WA 98195 USA.;[Feng, Tianquan] Nanjing Normal Univ, Sch Teachers Educ, Nanjing 210023, Jiangsu, Peoples R China.;[Yi, Ming] Huazhong Agr Univ, Dept Phys, Coll Sci, Wuhan 430070, Hubei, Peoples R China.
通讯机构：
[Zhou, Da] X;[Zhang, Xingan] C;Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Cancer stem cell model;Cell plasticity;Multi-phenotypic model;Transient dynamics
摘要：
The paradigm of phenotypic plasticity indicates reversible relations of different cancer cell phenotypes, which extends the cellular hierarchy proposed by the classical cancer stem cell (CSC) theory. Since it is still questionable if the phenotypic plasticity is a crucial improvement to the hierarchical model or just a minor extension to it, it is worthwhile to explore the dynamic behavior characterizing the reversible phenotypic plasticity. In this study we compare the hierarchical model and the reversible model in predicting the cell-state dynamics observed in biological experiments. Our results show that the hierarchical model shows significant disadvantages over the reversible model in describing both long-term stability (phenotypic equilibrium) and short-term transient dynamics (overshoot) in cancer cell populations. In a very specific case in which the total growth of population due to each cell type is identical, the hierarchical model predicts neither phenotypic equilibrium nor overshoot, whereas the reversible model succeeds in predicting both of them. Even though the performance of the hierarchical model can be improved by relaxing the specific assumption, its prediction to the phenotypic equilibrium strongly depends on a precondition that may be unrealistic in biological experiments. Moreover, it still does not show as rich dynamics as the reversible model in capturing the overshoots of both CSCs and non-CSCs. By comparison, it is more likely for the reversible model to correctly predict the stability of the phenotypic mixture and various types of overshoot behavior.
期刊：
Advances in Mathematics,2015年281:857-885 ISSN：0001-8708
通讯作者：
Feng, De-Jun
作者机构：
[Feng, De-Jun] Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China.;[Rao, Hui] Cent China Normal Univ, Dept Math & Stat, Wuhan 430070, Hubei, Peoples R China.;[Wang, Yang] Hong Kong Univ Sci & Technol, Dept Math, Kowloon, Hong Kong, Peoples R China.;[Wang, Yang] Michigan State Univ, Dept Math, E Lansing, MI 48824 USA.
通讯机构：
[Feng, De-Jun] C;Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China.
关键词：
Middle-third Cantor set;Self-similar subsets;Ternary expansions;Set of uniqueness
期刊：
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A-OPTICS IMAGE SCIENCE AND VISION,2013年30(12):2473-2481 ISSN：1084-7529
通讯作者：
Li, Peijun
作者机构：
[Cheng, Ting] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Li, Peijun; Wang, Yuliang] Purdue Univ, Dept Math, W Lafayette, IN 47907 USA.
通讯机构：
[Li, Peijun] P;Purdue Univ, Dept Math, W Lafayette, IN 47907 USA.
摘要：
We investigate the zero dissipation limit problem of the one-dimensional compressible non-isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves. It is shown that the unique solution of the Navier-Stokes equations exists for all time, and converges to the Riemann solution of the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks, as both the viscosity and heat-conductivity tend to zero. In contrast to previous related works, where either shock waves are absent or the effects of initial layers are ignored, this gives the first mathematical justification of this limit for the compressible non-isentropic Navier-Stokes equations in the presence of both shocks and initial layers. Our method of proof consists of a scaling argument, the construction of the approximate solution, and delicate energy estimates.
