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An accurate and efficient space-time Galerkin spectral method for the subdiffusion equation

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Science China Mathematics 2024年第10期67卷 2387-2408页

作者： Wei Zeng Chuanju Xu school of mathematical sciences and Fujian Provincial Key Laboratory of mathematical Modeling and High Performance Scientific computing Xiamen UniversityXiamen 361005China

In this paper, we design and analyze a space-time spectral method for the subdiffusion ***, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.

关键词： subdiffusion equations variable transformation Ψ-Sobolev spaces well-posedness space-time Galerkin spectral method error estimate fast algorithm

VARIATIONAL ANALYSIS FOR THE MAXIMAL TIME FUNCTION IN NORMED SPACES

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Acta Mathematica Scientia 2024年第5期44卷 1696-1706页

作者： Ziyi ZHOU Yi JIANG school of mathematical sciences Visual Computing and Virtual Reality Key LaboratorySichuan Normal UniversityChengdu610066China

For a general normed vector space,a special optimal value function called a maximal time function is *** covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball *** properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.

关键词： maximal time function subdifferential normal cone nonsmooth analysis

Flocking Effects of the Stochastic Cucker-Smale System with Noise

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Journal of Systems Science & Complexity 2024年第5期37卷 1978-1994页

作者： HAN Ning GE Bin LI Lin school of mathematical sciences Harbin Engineering University

In this paper, the flocking behavior of a Cucker-Smale model with a leader and noise is studied in a finite time. The authors present a Cucker-Smale system with two nonlinear controls for a complex network with stochastic synchronization in probability. Based on the finite-time stability theory of stochastic differential equations, the sufficient conditions for the flocking of stochastic systems in a finite time are obtained by using the Lyapunov function method. Finally, the numerical simulation of the particle system is carried out for the leader and noise, and the correctness of the results is verified.

关键词： Cucker-Smale model flocking behavior leader noise

Number of Solitons Emerged in the Initial Profile of Shallow Water Using Convolutional Neural Networks

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Journal of Systems Science & Complexity 2024年第2期37卷 463-479页

作者： WANG Zhen CUI Shikun school of mathematical sciences Dalian University of TechnologyDalian 116024China school of mathematical sciences Beihang UniversityBeijing 100191China

The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton ***,the problem of determining the number of solitons that form in a given initial profile remains unsolved,except for a few specific *** this paper,the authors use the deep learning method to predict the number of solitons in a given initial value of the Korteweg-de Vries(KdV)*** leveraging the analytical relationship between Asech^(2)(x)initial values and the number of solitons,the authors train a Convolutional Neural Network(CNN)that can accurately identify the soliton count from spatio-temporal *** trained neural network is capable of predicting the number of solitons with other given initial values without any additional *** extensive calculations,the authors demonstrate the effectiveness and high performance of the proposed method.

关键词： Convolutional neural network deep learning method KdV equation soliton

Rational Solutions of First Order Algebraic Ordinary Differential Equations

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Journal of Systems Science & Complexity 2024年第2期37卷 567-580页

作者： FENG Shuang SHEN Liyong school of Physical and mathematical sciences Nanjing Tech UniversityNanjing 211816China school of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

Let f(t,y,y')=∑ _(i=0)^(n )a_(i)(t,y)y'^(i)=0 be an irreducible first order ordinary differential equation with polynomial *** in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y')=0 is of degree not greater than *** show that this degree bound C depends not only on the degrees of f in t,y,y' but also on the coefficients of f viewed as the polynomial in t,y,y'.In this paper,the authors show that if f satisfies deg(f,y)0,then the degree bound C only depends on the degrees of f in t,y,y',and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y'.

关键词： Degree bound first order AODE height rational solution

Existence of Global Solutions to the Nonlocal mKdV Equation on the Line

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Chinese Annals of Mathematics,Series B 2024年第4期45卷 497-528页

作者： Anran LIU Engui FAN school of mathematical sciences Fudan University

In this paper, the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries（nonlocal mKdV for short） equation under the initial data u0 ∈ H3（R） ∩H1,1（R） with the L1（R） small-norm assumption. A Lipschitz L2-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem. The map from initial potential to reflection coefficient is obtained in direct scattering transform. The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator. The bijective relation naturally yields the existence of global solutions in a Sobolev space H3（R） ∩ H1,1（R） to the Cauchy problem.

关键词： Global solutions Lipschitz continuous Nonlocal mKdV equation Plemelj projection operator Riemann-Hilbert problem

A Deep Learning Method for computing Eigenvalues of the Fractional Schrödinger Operator

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Journal of Systems Science & Complexity 2024年第2期37卷 391-412页

作者： GUO Yixiao MING Pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China school of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger *** proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the *** improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded *** authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger *** an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

关键词： Eigenvalue problem deep learning fractional Schrödinger operator isospectral problem

The Strategic Analysis of Demand Forecast-Sharing in a Hybrid-Format Online Platform Supply Chain

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Journal of Systems Science and Systems Engineering 2024年第3期33卷 281-310页

作者： Jing Zhao Zijun Yin Guobiao Zhou school of mathematical sciences Tiangong UniversityTianjin300387China

This paper examines the optimal forecast-sharing strategy in a hybrid-format online platform supply chain where a supplier sells a product through agency format and reselling format provided by a platform retailer who possesses demand forecasts from two *** asymmetry and co-opetitive relationship arise between the platform retailer and the supplier,which affect their operational decisions and the supply chain’s *** improve supply chain efficiency,we compare different forecast-sharing strategies(i.e.,no forecast sharing,sharing a single forecast,and sharing two forecasts),and analyze the effects of co-opetitive parameters on the optimal forecast-sharing *** analysis shows that forecast sharing is always beneficial to the supplier,and sharing two forecasts is more beneficial than sharing a single *** for the platform retailer and the whole supply chain,forecast sharing is beneficial only under certain conditions,depending on the co-opetitive *** optimal forecast-sharing strategy is the result of a combination of the negative effect of double marginalization in reselling channel and the positive effect of responding pricing to demand uncertainty in agency *** illustrate the parameter regions of the platform retailer’s voluntary sharing,contract sharing,and no sharing,and also find that higher channel competition intensity,higher market share of agency channel,and higher commission rate can promote the platform retailer’s voluntary *** study extends the research scope of demand forecast-sharing and sheds light on the decision-making processes for managing a hybrid-format online platform supply chain.

关键词： Supply chain management hybrid-format online platform demand forecast-sharing co-opetitive relationship channel competition intensity

Global well-posedness for 2D inhomogeneous asymmetric fluids with large initial data

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Science China Mathematics 2024年第3期67卷 527-556页

作者： Chenyin Qian Beibei He Ting Zhang school of mathematical sciences Zhejiang Normal UniversityJinhua 321004China school of mathematical sciences Zhejiang UniversityHangzhou 310058China

In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids wi... 详细信息

关键词： inhomogeneous asymmetric fluids Littlewood-Paley theory Besov spaces global well-posedness