By defining fuzzy valued simple functions and giving L1(μ) approximations of fuzzy valued integrably bounded functions by such simple functions, the paper analyses by L1(μ)-norm the approximation capability of four-...By defining fuzzy valued simple functions and giving L1(μ) approximations of fuzzy valued integrably bounded functions by such simple functions, the paper analyses by L1(μ)-norm the approximation capability of four-layer feedforward regular fuzzy neural networks to the fuzzy valued integrably bounded function F : Rn → FcO(R). That is, if the transfer functionσ: R→R is non-polynomial and integrable function on each finite interval, F may be innorm approximated by fuzzy valued functions defined as to anydegree of accuracy. Finally some real examples demonstrate the conclusions.展开更多
The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based...The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented. To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1 functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance.展开更多
Study of the SISO mixed H2/l1 problem for discrete time systems showed that there exists a unique optimal solution which can be approximated within any prescribed missing error bound in l2 norm with solvable suboptima...Study of the SISO mixed H2/l1 problem for discrete time systems showed that there exists a unique optimal solution which can be approximated within any prescribed missing error bound in l2 norm with solvable suboptimal solutions and solvable superoptimal solutions.展开更多
In recent years,variable-order fractional partial differential equations have attracted growing interest due to their enhanced ability tomodel complex physical phenomena withmemory and spatial heterogeneity.However,ex...In recent years,variable-order fractional partial differential equations have attracted growing interest due to their enhanced ability tomodel complex physical phenomena withmemory and spatial heterogeneity.However,existing numerical methods often struggle with the computational challenges posed by such equations,especially in nonlinear,multi-term formulations.This study introduces two hybrid numerical methods—the Linear-Sine and Cosine(L1-CAS)and fast-CAS schemes—for solving linear and nonlinear multi-term Caputo variable-order(CVO)fractional partial differential equations.These methods combine CAS wavelet-based spatial discretization with L1 and fast algorithms in the time domain.A key feature of the approach is its ability to efficiently handle fully coupled spacetime variable-order derivatives and nonlinearities through a second-order interpolation technique.In addition,we derive CAS wavelet operational matrices for variable-order integration and for boundary value problems,forming the foundation of the spatial discretization.Numerical experiments confirm the accuracy,stability,and computational efficiency of the proposed methods.展开更多
By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the m...By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation,the mixed case contains a special time-space coupled derivative,which leads to many difficulties in numerical analysis.Firstly,a fully discrete scheme is established by using nonconforming finite element method(FEM)in spatial direction and L1 approximation coupled with Crank-Nicolson(L1-CN)scheme in temporal direction.Furthermore,the fully discrete scheme is proved to be unconditional stable.Besides,convergence and superclose results are derived by using the properties of EQ^(ROT)_(1) nonconforming finite element.What's more,the global superconvergence is obtained via the interpolation postprocessing technique.Finally,several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.展开更多
In this paper,we develop a two-grid method(TGM)based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations.A two-grid algorithm is proposed for solving the nonlinear sys...In this paper,we develop a two-grid method(TGM)based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations.A two-grid algorithm is proposed for solving the nonlinear system,which consists of two steps:a nonlinear FE system is solved on a coarse grid,then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution.The fully discrete numerical approximation is analyzed,where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with orderα∈(1,2)andα1∈(0,1).Numerical stability and optimal error estimate O(h^(r+1)+H^(2r+2)+τ^(min{3−α,2−α1}))in L^(2)-norm are presented for two-grid scheme,where t,H and h are the time step size,coarse grid mesh size and fine grid mesh size,respectively.Finally,numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.展开更多
为解决测距仪脉冲信号干扰L频段数字航空通信系统1(L-band digital aeronautical communications system 1,L-DACS1)正交频分复用接收机的问题,提出一种基于特征矩阵联合对角化(joint approximate diagonalization of eigen-matrices,JA...为解决测距仪脉冲信号干扰L频段数字航空通信系统1(L-band digital aeronautical communications system 1,L-DACS1)正交频分复用接收机的问题,提出一种基于特征矩阵联合对角化(joint approximate diagonalization of eigen-matrices,JADE)的测距仪脉冲干扰抑制方法。首先将干扰抑制问题转化为盲源分离问题,在接收端建立频域盲分离模型,利用JADE算法将接收到有用信号与测距仪干扰信号分离;然后根据干扰信号的功率特性进行分离后信号的识别;最后通过训练序列解决盲源分离固有的幅度模糊性问题,最终恢复出有用接收信号。仿真结果表明:所提出的基于JADE的干扰抑制方法可有效消除测距仪脉冲信号干扰,改善系统的误比特性能,增加传输可靠性。展开更多
Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, et...Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.展开更多
In this article,we study the energy dissipation property of time-fractional Allen–Cahn equation.On the continuous level,we propose an upper bound of energy that decreases with respect to time and coincides with the o...In this article,we study the energy dissipation property of time-fractional Allen–Cahn equation.On the continuous level,we propose an upper bound of energy that decreases with respect to time and coincides with the original energy at t=0 and as t tends to∞.This upper bound can also be viewed as a nonlocal-in-time modified energy which is the summation of the original energy and an accumulation term due to the memory effect of time-fractional derivative.In particular,the decrease of the modified energy indicates that the original energy indeed decays w.r.t.time in a small neighborhood at t=0.We illustrate the theory mainly with the time-fractional Allen-Cahn equation but it could also be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation.On the discrete level,the decreasing upper bound of energy is useful for proving energy dissipation of numerical schemes.First-order L1 and second-order L2 schemes for the time-fractional Allen-Cahn equation have similar decreasing modified energies,so that stability can be established.Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.展开更多
The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which...The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which contains a time-space coupled derivative.The nonconforming EQ^(rot)_(1)element and Raviart-Thomas element are employed for spatial discretization,and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization.Firstly,based on some significant lemmas,the unconditional stability analysis of the fully-discrete scheme is acquired.With the assistance of the interpolation operator I_(h)and projection operator Rh,superclose and convergence results of the variable u in H^(1)-norm and the flux~p=k_(5)(x)ru(x,t)in L^(2)-norm are obtained,respectively.Furthermore,the global superconvergence results are derived by applying the interpolation postprocessing technique.Finally,the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.展开更多
基金Supported by the National Natural Science Foundation of China(No:69872039)
文摘By defining fuzzy valued simple functions and giving L1(μ) approximations of fuzzy valued integrably bounded functions by such simple functions, the paper analyses by L1(μ)-norm the approximation capability of four-layer feedforward regular fuzzy neural networks to the fuzzy valued integrably bounded function F : Rn → FcO(R). That is, if the transfer functionσ: R→R is non-polynomial and integrable function on each finite interval, F may be innorm approximated by fuzzy valued functions defined as to anydegree of accuracy. Finally some real examples demonstrate the conclusions.
基金Supported by 973-Project of China(2006cb303102)the National Science Foundation of China(11461161006,11201079)
文摘The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented. To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1 functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance.
文摘Study of the SISO mixed H2/l1 problem for discrete time systems showed that there exists a unique optimal solution which can be approximated within any prescribed missing error bound in l2 norm with solvable suboptimal solutions and solvable superoptimal solutions.
基金supported by the National Research Foundation of Korea(NRF)grant funded by the Korean government(MSIT)(NRF-2021R1A2C1011817)the BK21 Program(Next Generation Education Program for Mathematical Sciences,4299990414089)funded by the Ministry of Education(MOE,Republic of Korea).
文摘In recent years,variable-order fractional partial differential equations have attracted growing interest due to their enhanced ability tomodel complex physical phenomena withmemory and spatial heterogeneity.However,existing numerical methods often struggle with the computational challenges posed by such equations,especially in nonlinear,multi-term formulations.This study introduces two hybrid numerical methods—the Linear-Sine and Cosine(L1-CAS)and fast-CAS schemes—for solving linear and nonlinear multi-term Caputo variable-order(CVO)fractional partial differential equations.These methods combine CAS wavelet-based spatial discretization with L1 and fast algorithms in the time domain.A key feature of the approach is its ability to efficiently handle fully coupled spacetime variable-order derivatives and nonlinearities through a second-order interpolation technique.In addition,we derive CAS wavelet operational matrices for variable-order integration and for boundary value problems,forming the foundation of the spatial discretization.Numerical experiments confirm the accuracy,stability,and computational efficiency of the proposed methods.
基金National Natural Science Foundation of China(No.11971416)Scientific Research Innovation Team of Xuchang University(No.2022CXTD002)+3 种基金Foundation for University Key Young Teacher of Henan Province(No.2019GGJS214)Key Scientific Research Projects in Universities of Henan Province(Nos.21B110007,22A110022)National Natural Science Foundation of China(International cooperation key project:No.12120101001)Australian Research Council via the Discovery Project(DP190101889).
文摘By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation,the mixed case contains a special time-space coupled derivative,which leads to many difficulties in numerical analysis.Firstly,a fully discrete scheme is established by using nonconforming finite element method(FEM)in spatial direction and L1 approximation coupled with Crank-Nicolson(L1-CN)scheme in temporal direction.Furthermore,the fully discrete scheme is proved to be unconditional stable.Besides,convergence and superclose results are derived by using the properties of EQ^(ROT)_(1) nonconforming finite element.What's more,the global superconvergence is obtained via the interpolation postprocessing technique.Finally,several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.
基金This work is supported by the State Key Program of National Natural Science Foundation of China(11931003)National Natural Science Foundation of China(41974133,11971410)+2 种基金Project for Hunan National Applied Mathematics Center of Hunan Provincial Science and Technology Department(2020ZYT003)Hunan Provincial Innovation Foundation for Postgraduate,China(XDCX2020B082,XDCX2021B098)Postgraduate Scientific Research Innovation Project of Hunan Province(CX20210597).
文摘In this paper,we develop a two-grid method(TGM)based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations.A two-grid algorithm is proposed for solving the nonlinear system,which consists of two steps:a nonlinear FE system is solved on a coarse grid,then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution.The fully discrete numerical approximation is analyzed,where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with orderα∈(1,2)andα1∈(0,1).Numerical stability and optimal error estimate O(h^(r+1)+H^(2r+2)+τ^(min{3−α,2−α1}))in L^(2)-norm are presented for two-grid scheme,where t,H and h are the time step size,coarse grid mesh size and fine grid mesh size,respectively.Finally,numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.
文摘为解决测距仪脉冲信号干扰L频段数字航空通信系统1(L-band digital aeronautical communications system 1,L-DACS1)正交频分复用接收机的问题,提出一种基于特征矩阵联合对角化(joint approximate diagonalization of eigen-matrices,JADE)的测距仪脉冲干扰抑制方法。首先将干扰抑制问题转化为盲源分离问题,在接收端建立频域盲分离模型,利用JADE算法将接收到有用信号与测距仪干扰信号分离;然后根据干扰信号的功率特性进行分离后信号的识别;最后通过训练序列解决盲源分离固有的幅度模糊性问题,最终恢复出有用接收信号。仿真结果表明:所提出的基于JADE的干扰抑制方法可有效消除测距仪脉冲信号干扰,改善系统的误比特性能,增加传输可靠性。
文摘Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.
基金partially supported by the National Natural Science Foundation of China/Hong Kong RGC Joint Research Scheme(NSFC/RGC 11961160718)the fund of the Guangdong Provincial Key Laboratory of Computational Science And Material Design(No.2019B030301001)+4 种基金supported in part by the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science under UIC 2022B1212010006supported by the National Science Foundation of China(NSFC)Grant No.12271240supported by NSFC Grant 12271241Guangdong Basic and Applied Basic Research Foundation(No.2023B1515020030)Shenzhen Science and Technology Program(Grant No.RCYX20210609104358076).
文摘In this article,we study the energy dissipation property of time-fractional Allen–Cahn equation.On the continuous level,we propose an upper bound of energy that decreases with respect to time and coincides with the original energy at t=0 and as t tends to∞.This upper bound can also be viewed as a nonlocal-in-time modified energy which is the summation of the original energy and an accumulation term due to the memory effect of time-fractional derivative.In particular,the decrease of the modified energy indicates that the original energy indeed decays w.r.t.time in a small neighborhood at t=0.We illustrate the theory mainly with the time-fractional Allen-Cahn equation but it could also be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation.On the discrete level,the decreasing upper bound of energy is useful for proving energy dissipation of numerical schemes.First-order L1 and second-order L2 schemes for the time-fractional Allen-Cahn equation have similar decreasing modified energies,so that stability can be established.Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.
基金The work is supported by the National Natural Science Foundation of China(Nos.11971416 and 11871441)the Scientific Research Innovation Team of Xuchang University(No.2022CXTD002)the Foundation for University Key Young Teacher of Henan Province(No.2019GGJS214).
文摘The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which contains a time-space coupled derivative.The nonconforming EQ^(rot)_(1)element and Raviart-Thomas element are employed for spatial discretization,and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization.Firstly,based on some significant lemmas,the unconditional stability analysis of the fully-discrete scheme is acquired.With the assistance of the interpolation operator I_(h)and projection operator Rh,superclose and convergence results of the variable u in H^(1)-norm and the flux~p=k_(5)(x)ru(x,t)in L^(2)-norm are obtained,respectively.Furthermore,the global superconvergence results are derived by applying the interpolation postprocessing technique.Finally,the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.
