期刊：
Applied and Computational Harmonic Analysis,2020年48(2):539-569 ISSN：1063-5203
通讯作者：
Li, Huiyuan
作者机构：
[Zhang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Jing] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Li, Huiyuan] Chinese Acad Sci, Inst Software, State Key Lab Comp Sci, Lab Parallel Comp, Beijing 100190, Peoples R China.;[Wang, Li-Lian] Nanyang Technol Univ, Sch Phys & Math Sci, Div Math Sci, Singapore 637371, Singapore.;[Zhang, Zhimin] Beijing Computat Sci Res Ctr, Beijing 100193, Peoples R China.
摘要：
In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order alpha > -1 on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both an integral operator, and a Sturm-Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalization of orthogonal ball polynomials in primitive variables with a tuning parameter c > 0, through a "perturbation" of the Sturm-Liouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutions. (C) 2018 Elsevier Inc. All rights reserved.
期刊：
Applied Mathematics and Computation,2020年370:124930 ISSN：0096-3003
通讯作者：
Zhang, Jing
作者机构：
[Ju, Lili; Tian, Hao] Ocean Univ China, Sch Math Sci, Qingdao 266100, Shandong, Peoples R China.;[Zhang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Zhang, Jing] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Ju, Lili] Univ South Carolina, Dept Math, Columbia, SC 29208 USA.
通讯机构：
[Zhang, Jing] 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.
摘要：
In this paper we study an SLIR epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero or the disease persists as time evolves. For the stochastic model, we prove the existence, uniqueness and positivity of the solution of the model. Then, we investigate the stability of the model and we prove that the infective tends asymptotically to zero exponentially almost surely as R-0 < 1. We also proved that the SLIR model has the ergodic property as the fluctuation is small, where the positive solution converges weakly to the unique stationary distribution.
期刊：
Journal of Scientific Computing,2017年70(2):451-477 ISSN：0885-7474
通讯作者：
Wang, Li-Lian
作者机构：
[Zhang, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Jing] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Wang, Li-Lian] Nanyang Technol Univ, Sch Phys & Math Sci, Div Math Sci, Singapore 637371, Singapore.;[Li, Huiyuan] Chinese Acad Sci, Inst Software, State Key Lab Comp Sci, Lab Parallel Comp, Beijing 100190, Peoples R China.;[Zhang, Zhimin] Beijing Computat Sci & Res Ctr, Beijing 100193, Peoples R China.
通讯机构：
[Wang, Li-Lian] N;Nanyang Technol Univ, Sch Phys & Math Sci, Div Math Sci, Singapore 637371, Singapore.
关键词：
Generalized prolate spheroidal wave functions of order \(-1\);Helmholtz equations;Optimal spectral schemes;Error analysis
摘要：
We introduce a family of generalized prolate spheroidal wave functions (PSWFs) of order
$$-1,$$
and develop new spectral schemes for second-order boundary value problems. Our technique differs from the differentiation approach based on PSWFs of order zero in Kong and Rokhlin (Appl Comput Harmon Anal 33(2):226–260,2012); in particular, our orthogonal basis can naturally include homogeneous boundary conditions without the re-orthogonalization of Kong and Rokhlin (2012). More notably, it leads to diagonal systems or direct “explicit” solutions to 1D Helmholtz problems in various situations. Using a rule optimally pairing the bandwidth parameter and the number of basis functions as in Kong and Rokhlin (2012), we demonstrate that the new method significantly outperforms the Legendre spectral method in approximating highly oscillatory solutions. We also conduct a rigorous error analysis of this new scheme. The idea and analysis can be extended to generalized PSWFs of negative integer order for higher-order boundary value and eigenvalue problems.
期刊：
Linear Algebra and its Applications,2015年483:293-308 ISSN：0024-3795
通讯作者：
Zhang, Jun
作者机构：
[Szokol, Patricia] Univ Debrecen, Inst Math, MTA DE Lendulet Funct Anal Res Grp, H-4010 Debrecen, Hungary.;[Tsai, Ming-Cheng] Natl Sun Yat Sen Univ, Dept Appl Math, Kaohsiung 80424, Taiwan.;[Zhang, Jun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Zhang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Chi-square test;Effective reproductive number;Goodness of fit;Parameter estimation
摘要：
This paper concentrates on the tuberculosis data of China from January 2005 to December 2012. We set up a mathematical model to fit those data with the goodness of fit and obtain the optimal parameter values of the model. By the Chi-square test of the statistical inference, the optimal parameter values of the model are reasonable. We get the effective reproductive number of the disease for each year, and also investigate the preventive measures to control the tuberculosis. (C) 2014 Elsevier Ltd. All rights reserved.
期刊：
Computational and Applied Mathematics,2015年34(3):1237-1249 ISSN：2238-3603
通讯作者：
Zhang, Jinhui
作者机构：
[Zhang, Jinhui] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Feng, Guangting] Hubei Univ Educ, Sch Math & Stat, Wuhan 430205, Peoples R China.
通讯机构：
[Zhang, Jinhui] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
期刊：
International Journal of Environmental Research and Public Health,2014年11(3):3108-3117 ISSN：1661-7827
通讯作者：
Zhang, Xinan
作者机构：
[Zhang, Xinan; Li, Yong; Zhang, Jinhui] Huazhong Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Xinan] Huazhong Normal Univ, Sch Math & Stat, 152 Luo Yu Rd, Wuhan 430079, Peoples R China.
通讯机构：
[Zhang, Xinan] H;Huazhong Normal Univ, Sch Math & Stat, 152 Luo Yu Rd, Wuhan 430079, Peoples R China.
关键词：
HFMD;the basic reproductive number;Chi-square test;goodness of fit
摘要：
This paper concentrates on the HFMD data of China from March 2009 to December 2012. We set up a mathematical model to fit those data with the goodness of fit and obtain the optimal parameter values of the model. By the Chi-square test of statistical inference, the optimal parameter values of the model are reasonable. We obtained the basic reproductive number of the disease for each year, and it is larger than 1. Thus, we conclude that HFMD will persist in China under the current conditions, so we investigate the preventive measures to control the HFMD. If the preventive measures proposed in our paper were implemented, HFMD would be controlled quickly and the number of infections would decline rapidly over a period of time.
期刊：
Journal of Computational Physics,2014年268:377-398 ISSN：0021-9991
通讯作者：
Zhang, Jing
作者机构：
[Wang, Li-Lian] Nanyang Technol Univ, Sch Math & Phys Sci, Div Math Sci, Singapore 637371, Singapore.;[Zhang, Jing] Huazhong Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Jing] Beijing Computat Sci Res Ctr, Beijing, Peoples R China.;[Zhang, Zhimin] Wayne State Univ, Detroit, MI 48202 USA.
通讯机构：
[Zhang, Jing] H;Huazhong Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
The first purpose of this paper is to provide further illustrations, from both theoretical and numerical perspectives, for the nonconvergence of h-refinement in hp-approximation by the prolate spheroidal wave functions (PSWFs), a surprising convergence property that was first discovered by Boyd et al. (2013) [3]. The second purpose is to offer a new basis that leads to prolate-collocation systems with condition numbers independent of (c, N), the intrinsic bandwidth parameter and the number of collocation points. We highlight that the collocation scheme together with a very practical rule for pairing up (c, N) significantly outperforms the Legendre polynomial-based method (and likewise other Jacobi polynomial-based methods) in approximating highly oscillatory bandlimited functions. (C) 2014 Elsevier Inc. All rights reserved.
作者：
Kuo, David Li-Wei;Tsai, Ming-Cheng;Wong, Ngai-Ching;Zhang, Jun*
期刊：
Abstract and Applied Analysis,2014年2014(SI52):1-5 ISSN：1085-3375
通讯作者：
Zhang, Jun
作者机构：
[Tsai, Ming-Cheng; Wong, Ngai-Ching; Kuo, David Li-Wei] Natl Sun Yat Sen Univ, Dept Appl Math, Kaohsiung 80424, Taiwan.;[Zhang, Jun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Zhang, Jun] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
We study maps ϕ of positive operators of the Schatten p -classes ( 1 < p < + ∞ );which preserve the p -norms of convex combinations;that is;∥ t � + ( 1 - t ) σ ∥ p = ∥ t ϕ ( � ) + ( 1 - t ) ϕ ( σ ) ∥ p;σ ∈ � p + ( H ) 1;t ∈ [ 0;1 ] . They are exactly those carrying the form ϕ ( � ) = U � U * for a unitary or antiunitary U . In the case p = 2;we have the same conclusion whenever it just holds ∥ � + σ ∥ 2 = ∥ ϕ ( � ) + ϕ ( σ ) ∥ 2 for all the positive Hilbert-Schmidt class operators �;σ of norm 1 . Some examples are demonstrated. Published: 2014 First available in Project Euclid: 26 March 2014 zbMATH: 07022546 MathSciNet: MR3166627 Digital Object Identifier: 10.1155/2014/520795
作者机构：
School of Mathematics, Yunnan Normal University, Kunming 650092, China;Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China;Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China;Department of Mathematics, Central China Normal University, Wuhan 430079, China
通讯机构：
Department of Mathematics, Shanghai Jiao Tong University, China
关键词：
Measure of non compactness;Mild solutions;Nonlinear evolution equations
摘要：
This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin (IPDG) formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Dirichlet-to-Neumann (DtN) boundary condition. An effective iterative approach is proposed to localize the global DtN boundary condition, which facilitates the implementation of multi-domain methods, and the treatment for complex geometry of the scatterers. Under a discontinuous Galerkin formulation, the proposed method allows to use polynomial basis functions of different degree on different subdomains, and more importantly, explicit wave number dependence estimates of the spectral scheme can be derived, which is somehow implausible for a multi-domain continuous Galerkin formulation.
期刊：
Abstract and Applied Analysis,2012年2012:1-15 ISSN：1085-3375
通讯作者：
Xiao, Ti-Jun
作者机构：
[Zhang, Jun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Xiao, Ti-Jun] Fudan Univ, Sch Math Sci, Shanghai Key Lab Contemporary Appl Math, Shanghai 200433, Peoples R China.;[Liang, Jin] Shanghai Jiao Tong Univ, Dept Math, Shanghai 200240, Peoples R China.
通讯机构：
[Xiao, Ti-Jun] F;Fudan Univ, Sch Math Sci, Shanghai Key Lab Contemporary Appl Math, Shanghai 200433, Peoples R China.
关键词：
We first give a solution to a key problem concerning the completeness of the space of weighted pseudo almost-periodic functions and then establish a new composition theorem with respect to these functions. Some important remarks with concrete examples are also presented. Moreover;we prove an existence theorem for the weighted pseudo almost-periodic mild solution to the semilinear evolution equation: x ′ ( t ) = A x ( t ) + f ( t;x ( t ) );t ∈ \Bbb R;where A is the infinitesimal generator of an exponentially stable C 0 -semigroup. An application is also given to illustrate the abstract existence theorem. Published: 2012 First available in Project Euclid: 14 December 2012 zbMATH: 1244.34083 MathSciNet: MR2903805 Digital Object Identifier: 10.1155/2012/179525
