To obtain the approximate solution of the nonlinear ordinary differential equations requires the solution to systems of nonlinear equations. The authors study the conditions for the existence and uniqueness of the sol...To obtain the approximate solution of the nonlinear ordinary differential equations requires the solution to systems of nonlinear equations. The authors study the conditions for the existence and uniqueness of the solutions to the algebraic equations in multiderivative block methods.展开更多
In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt ...In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.展开更多
In this work,we study the error estimates of the fully discrete Fourier pseudospectral numerical scheme for solving the nonlocal volume-conserved Allen-Cahn(AC)equation.The time marching method of the numerical scheme...In this work,we study the error estimates of the fully discrete Fourier pseudospectral numerical scheme for solving the nonlocal volume-conserved Allen-Cahn(AC)equation.The time marching method of the numerical scheme is based on the well-known Invariant Energy Quadratization(IEQ)method.We demonstrate that the proposed fully discrete numerical method is uniquely solvable,unconditionally energy stable,and obtain the optimal error estimate of the scheme for both space and time.Additionally,we conduct several numerical tests to verify the theoretical results.展开更多
In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We ...In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.展开更多
The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Nav...The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. The paper also describes the time blowup of classical solutions for the Navier-Stokes equations by the smoothness assumption.展开更多
The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational pri...The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.展开更多
This paper deals with the numerical computation and analysis for Caputo fractional differential equations(CFDEs).By combining the p-order boundary value methods(B-VMs)and the m-th Lagrange interpolation,a type of exte...This paper deals with the numerical computation and analysis for Caputo fractional differential equations(CFDEs).By combining the p-order boundary value methods(B-VMs)and the m-th Lagrange interpolation,a type of extended BVMs for the CFDEs with y-order(0<r<1)Caputo derivatives are derived.The local stability,unique solvability and convergence of the methods are studied.It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at min{p,m-γ+1}.In the end,by performing several numerical examples,the computational efficiency,accuracy and comparability of the methods are further ilustrated.展开更多
Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBV...Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP.Besides,whenever the classic Lipschitz conditions are satisfied,the extended BBVMs are preconsistent and pth order consistent.Moreover,through some numerical examples,the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.展开更多
文摘To obtain the approximate solution of the nonlinear ordinary differential equations requires the solution to systems of nonlinear equations. The authors study the conditions for the existence and uniqueness of the solutions to the algebraic equations in multiderivative block methods.
文摘In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12261017,62062018)the Foundation of Science and Technology of Guizhou Province(Grant No.ZK[2022]031)the Scientific Research Foundation of Guizhou University of Finance and Economics(Grant Nos.2022KYYB08,2022ZCZX077)。
文摘In this work,we study the error estimates of the fully discrete Fourier pseudospectral numerical scheme for solving the nonlocal volume-conserved Allen-Cahn(AC)equation.The time marching method of the numerical scheme is based on the well-known Invariant Energy Quadratization(IEQ)method.We demonstrate that the proposed fully discrete numerical method is uniquely solvable,unconditionally energy stable,and obtain the optimal error estimate of the scheme for both space and time.Additionally,we conduct several numerical tests to verify the theoretical results.
基金the Ministry of Education and Science of the Republic of Kazakhstan for a grant
文摘In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.
基金the Ministry of Education and Science of the Republic of Kazakhstan for a grant
文摘The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. The paper also describes the time blowup of classical solutions for the Navier-Stokes equations by the smoothness assumption.
基金This work was partly supported by the 973 projectthe National Natural Science Foundation of China(Grant No.10371076)+1 种基金E-Institutes of Shanghai Municipal Education Commission(Grant No.E03004)The Science Foundation of Shanghai(Grant No.04JC14062).
文摘The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.
基金The authors would like to thank the anonymous referees for their valuable comments and helpful suggestions.The second author Chengian Zhang(corresponding author)is supported by NSFC(Grant No.11971010).
文摘This paper deals with the numerical computation and analysis for Caputo fractional differential equations(CFDEs).By combining the p-order boundary value methods(B-VMs)and the m-th Lagrange interpolation,a type of extended BVMs for the CFDEs with y-order(0<r<1)Caputo derivatives are derived.The local stability,unique solvability and convergence of the methods are studied.It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at min{p,m-γ+1}.In the end,by performing several numerical examples,the computational efficiency,accuracy and comparability of the methods are further ilustrated.
基金supported by the National Key R&D Program of China(2020YFA0709800)the National Natural Science Foundation of China(Nos.11901577,11971481,12071481,12001539)+4 种基金the Natural Science Foundation of Hunan(No.S2017JJQNJJ-0764)the fund from Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering(No.2018MMAEZD004)the Basic Research Foundation of National Numerical Wind Tunnel Project(No.NNW2018-ZT4A08)the Research Fund of National University of Defense Technology(No.ZK19-37)The science and technology innovation Program of Hunan Province(No.2020RC2039).
文摘Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP.Besides,whenever the classic Lipschitz conditions are satisfied,the extended BBVMs are preconsistent and pth order consistent.Moreover,through some numerical examples,the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.
