In the hyperbolic research community,there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability.In the first par...In the hyperbolic research community,there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability.In the first part of the series[6],the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly.By applying this technique,the authors demonstrate that a pure continu-ous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way.In this work,we extend this investigation to the nonlinear case and focus on entropy conservation.By switching to entropy variables,we provide an estimation of the boundary operators also for nonlinear problems,that guarantee conservation.In numerical simulations,we verify our theoretical analysis.展开更多
A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation ...A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration.Numerical results demonstrate the efficiency of the method and spectral accuracy.展开更多
By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s)....By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.展开更多
Using the Fokas unified method, we consider the initial boundary value problem for the Fokas-Lenells equation on the finite interval. We present that the Neumann boundary data can be explicitly expressed by Dirichlet ...Using the Fokas unified method, we consider the initial boundary value problem for the Fokas-Lenells equation on the finite interval. We present that the Neumann boundary data can be explicitly expressed by Dirichlet boundary conditions prescribed, and extend the idea of the linearizable boundary conditions for equations on the half line to Fokas-Lenells equation on the finite interval.展开更多
The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line.We show that the solution of the coupled Hirota equation can be expressed in terms of the solut...The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line.We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3×3 matrix Riemann-Hilbert problem formulated in the complex k-plane.The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k)and S(k)that depend on the initial data and boundary values,respectively.Then,applying nonlinear steepest descent techniques to the associated 3×3 matrix-valued Riemann-Hilbert problem,we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation.展开更多
The Gelfand-Levitan-Marchenko representation is used to analyze the initialboundary value problem of two-component nonlinear Schr¨odinger equation on the half-line.It has shown that the global relation can be eff...The Gelfand-Levitan-Marchenko representation is used to analyze the initialboundary value problem of two-component nonlinear Schr¨odinger equation on the half-line.It has shown that the global relation can be effectively analyzed by the Gelfand-LevitanMarchenko representation. we also derive expressions for the Dirichlet-to-Neumann map to characterize the unknown boundary values.展开更多
In this paper we present a necessary and sufficient condition to guarantee that the zeroextended function of the solution for the heat equation in a smaller cylinder is still the solution of the corresponding extensio...In this paper we present a necessary and sufficient condition to guarantee that the zeroextended function of the solution for the heat equation in a smaller cylinder is still the solution of the corresponding extension problem in a larger cylinder.We prove the results under the frameworks of classical solutions,strong solutions and weak solutions.Moreover,we generalize these results to uniformly parabolic equations of divergence form.展开更多
In this work,we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schrödinger(DNLS)equation.By establishing a matrix Riemann-Hilbert pr...In this work,we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schrödinger(DNLS)equation.By establishing a matrix Riemann-Hilbert problem and reconstructing potential function q(x,t)from eigenfunctions{Gj(x,t,η)}3/1 in the inverse problem,the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed.Moreover,we also obtain that the spectral functions f(η),s(η),F(η),S(η)are not independent of each other,but meet an important global relation.As applications,the generalized DNLS equation can be reduced to the Kaup-Newell equation and Chen-Lee-Liu equation on the half-line.展开更多
In this paper,we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate....In this paper,we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate.Utilizing energy methods,we show that under time-dependent Dirichlet boundary conditions,long-time dynamics of solutions are driven by their boundary data,and there is no restriction on the magnitude of initial energy.Moreover,the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions,which has not been observed in previous studies on related models.展开更多
We investigate the thermal instability of a three-dimensional Rayleigh–Bénard(RB for short)problem without thermal diffusion in a bounded domain.First we construct unstable solutions in exponential growth modes ...We investigate the thermal instability of a three-dimensional Rayleigh–Bénard(RB for short)problem without thermal diffusion in a bounded domain.First we construct unstable solutions in exponential growth modes for the linear RB problem.Then we derive energy estimates for the nonlinear solutions by a method of a prior energy estimates,and establish a Gronwall-type energy inequality for the nonlinear solutions.Finally,we estimate for the error of L^(1)-norm between the both solutions of the linear and nonlinear problems,and prove the existence of escape times of nonlinear solutions.Thus we get the instability of nonlinear solutions under L^(1)-norm.展开更多
The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and a...The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and another is a linear equation coupling the reactant concentration for the resultant concentration. This paper is concerned with the initial-boundary value problem for the above model. By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial initial-boundary value problem for Riemann type of initial-boundary data. Moreover, as examples, we apply the obtained results to the cases of head-on and wide pulse injections and give the expression of the global weak entropy solution.展开更多
In this paper, the Fokas unified method is used to analyze the initial-boundary value problem of a complex Sharma–Tasso–Olver(c STO) equation on the half line. We show that the solution can be expressed in terms of ...In this paper, the Fokas unified method is used to analyze the initial-boundary value problem of a complex Sharma–Tasso–Olver(c STO) equation on the half line. We show that the solution can be expressed in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly given in terms of the matrix-value spectral functions spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which depending on initial data u_0(x) = u(x, 0) and boundary data g_0(y) = u(0, y), g_1(y) = ux(0, y), g_2(y) = u_(xx)(0, y). These spectral functions are not independent, they satisfy a global relation.展开更多
A new penalty-free neural network method,PFNN-2,is presented for solving partial differential equations,which is a subsequent improvement of our previously proposed PFNN method[1].PFNN-2 inherits all advantages of PFN...A new penalty-free neural network method,PFNN-2,is presented for solving partial differential equations,which is a subsequent improvement of our previously proposed PFNN method[1].PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries,and extends the application to a broader range of non-self-adjoint time-dependent differential equations.In addition,PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy.Experiments results on a series of partial differential equations are reported,which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy,convergence speed,and parallel scalability.展开更多
We address the well-posedness of the 2D(Euler)–Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich’s type data,which gives a positive answer to part of the q...We address the well-posedness of the 2D(Euler)–Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich’s type data,which gives a positive answer to part of the questions raised in Lai(Arch Ration Mech Anal 199(3):739–760,2011).Our analysis on the the polygonallike domains essentially relies on the recent elliptic regularity results for such domains proved in Bardos et al.(J Math Anal Appl 407(1):69–89,2013)and Di Plinio(SIAM J Math Anal 47(1):159–178,2015).展开更多
The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous b...The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous boundary condition.We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces.Moreover,we show that the nonlinear part of the solution on the half-line is smoother than the initial data.展开更多
The admissibility of the initial-boundary data,which characterizes the existence of solution for the initial-boundary value problem,is important.Based on the Fokas method and the inverse scattering transformation,an a...The admissibility of the initial-boundary data,which characterizes the existence of solution for the initial-boundary value problem,is important.Based on the Fokas method and the inverse scattering transformation,an approach is developed to solve the initial-boundary value problem of the nonlinear Schrodinger equation on a finite interval.A necessary and sufficient condition for the admissibility of the initial-boundary data is given,and the reconstruction of the potential is obtained.展开更多
We propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime.The typical example we dis...We propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime.The typical example we discuss is the derivation of the Euler system from the BGK equation.The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flux might be far from the thermodynamic equilibrium.展开更多
基金funded by the SNF Grant(Number 200021175784)the UZH Postdoc grant+1 种基金funded by an SNF Grant 200021_153604The Los Alamos unlimited release number is LA-UR-19-32411.
文摘In the hyperbolic research community,there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability.In the first part of the series[6],the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly.By applying this technique,the authors demonstrate that a pure continu-ous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way.In this work,we extend this investigation to the nonlinear case and focus on entropy conservation.By switching to entropy variables,we provide an estimation of the boundary operators also for nonlinear problems,that guarantee conservation.In numerical simulations,we verify our theoretical analysis.
基金Supported by National Natural Science Foundation of China(Grant Nos.11771299,11371123)Natural Science Foundation of Henan Province(Grant No.202300410156).
文摘A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration.Numerical results demonstrate the efficiency of the method and spectral accuracy.
文摘By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.
基金supported by grants from the National Natural Science Foundation of China(11271079,11626090)
文摘Using the Fokas unified method, we consider the initial boundary value problem for the Fokas-Lenells equation on the finite interval. We present that the Neumann boundary data can be explicitly expressed by Dirichlet boundary conditions prescribed, and extend the idea of the linearizable boundary conditions for equations on the half line to Fokas-Lenells equation on the finite interval.
基金supported by the China Postdoctoral Science Foundation(Grant No.2019TQ0041)。
文摘The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line.We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3×3 matrix Riemann-Hilbert problem formulated in the complex k-plane.The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k)and S(k)that depend on the initial data and boundary values,respectively.Then,applying nonlinear steepest descent techniques to the associated 3×3 matrix-valued Riemann-Hilbert problem,we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation.
基金support by NSFC(11671095,11501365)Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant NO.15YF1408100+1 种基金Shanghai Youth Teacher Assistance program NO.ZZslg15056the Hujiang Foundation of China(B14005)
文摘The Gelfand-Levitan-Marchenko representation is used to analyze the initialboundary value problem of two-component nonlinear Schr¨odinger equation on the half-line.It has shown that the global relation can be effectively analyzed by the Gelfand-LevitanMarchenko representation. we also derive expressions for the Dirichlet-to-Neumann map to characterize the unknown boundary values.
基金Supported by NSFC(Grant No.12071009)the Fundamental Research Funds for the Central Universities(Grant No.lzujbky-2019-21)。
文摘In this paper we present a necessary and sufficient condition to guarantee that the zeroextended function of the solution for the heat equation in a smaller cylinder is still the solution of the corresponding extension problem in a larger cylinder.We prove the results under the frameworks of classical solutions,strong solutions and weak solutions.Moreover,we generalize these results to uniformly parabolic equations of divergence form.
基金This work is supported by the Natural Science Foundation of China(Nos.11601055,11805114 and 11975145)the Natural Science Research Projects of Anhui Province(No.KJ2019A0637)University Excellent Talent Fund of Anhui Province(No.gxyq2019096).
文摘In this work,we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schrödinger(DNLS)equation.By establishing a matrix Riemann-Hilbert problem and reconstructing potential function q(x,t)from eigenfunctions{Gj(x,t,η)}3/1 in the inverse problem,the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed.Moreover,we also obtain that the spectral functions f(η),s(η),F(η),S(η)are not independent of each other,but meet an important global relation.As applications,the generalized DNLS equation can be reduced to the Kaup-Newell equation and Chen-Lee-Liu equation on the half-line.
基金partially supported by China Scholarship Council(No.201906150159)partially supported by China Scholarship Council(No.201906150101)+2 种基金National Natural Science Foundation of China(No.11971176,No.11871226)partially supported by Fundamental Research Funds for the Central Universities of China(No.3072020CFT2402)partially supported by Simons Foundation Collaboration Grant for Mathematicians(No.413028)。
文摘In this paper,we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate.Utilizing energy methods,we show that under time-dependent Dirichlet boundary conditions,long-time dynamics of solutions are driven by their boundary data,and there is no restriction on the magnitude of initial energy.Moreover,the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions,which has not been observed in previous studies on related models.
基金supported by the NSF of China(Grant No.11901100)the Natural Science Foundation of Fujian Province of China(Grant No.2020J02001)Funds of Education Department of Fujian Province(Grant No.510881/GXRC-20046)。
文摘We investigate the thermal instability of a three-dimensional Rayleigh–Bénard(RB for short)problem without thermal diffusion in a bounded domain.First we construct unstable solutions in exponential growth modes for the linear RB problem.Then we derive energy estimates for the nonlinear solutions by a method of a prior energy estimates,and establish a Gronwall-type energy inequality for the nonlinear solutions.Finally,we estimate for the error of L^(1)-norm between the both solutions of the linear and nonlinear problems,and prove the existence of escape times of nonlinear solutions.Thus we get the instability of nonlinear solutions under L^(1)-norm.
基金supported by the State Key Program of National Natural Science Foundation of China(Grants No.11731008)the National Natural Science Foundation of China(Grants No.10771087)。
文摘The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and another is a linear equation coupling the reactant concentration for the resultant concentration. This paper is concerned with the initial-boundary value problem for the above model. By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial initial-boundary value problem for Riemann type of initial-boundary data. Moreover, as examples, we apply the obtained results to the cases of head-on and wide pulse injections and give the expression of the global weak entropy solution.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271008 and 61072147
文摘In this paper, the Fokas unified method is used to analyze the initial-boundary value problem of a complex Sharma–Tasso–Olver(c STO) equation on the half line. We show that the solution can be expressed in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly given in terms of the matrix-value spectral functions spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which depending on initial data u_0(x) = u(x, 0) and boundary data g_0(y) = u(0, y), g_1(y) = ux(0, y), g_2(y) = u_(xx)(0, y). These spectral functions are not independent, they satisfy a global relation.
基金supported in part by National Natural Science Foundation of China(grants#12131002 and#12288101)Huawei Technologies Co.,Ltd.
文摘A new penalty-free neural network method,PFNN-2,is presented for solving partial differential equations,which is a subsequent improvement of our previously proposed PFNN method[1].PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries,and extends the application to a broader range of non-self-adjoint time-dependent differential equations.In addition,PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy.Experiments results on a series of partial differential equations are reported,which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy,convergence speed,and parallel scalability.
文摘We address the well-posedness of the 2D(Euler)–Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich’s type data,which gives a positive answer to part of the questions raised in Lai(Arch Ration Mech Anal 199(3):739–760,2011).Our analysis on the the polygonallike domains essentially relies on the recent elliptic regularity results for such domains proved in Bardos et al.(J Math Anal Appl 407(1):69–89,2013)and Di Plinio(SIAM J Math Anal 47(1):159–178,2015).
文摘The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous boundary condition.We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces.Moreover,we show that the nonlinear part of the solution on the half-line is smoother than the initial data.
基金supported by the National Natural Science Foundation of China(Grant Nos.11931017 and 11871440)by the Henan Youth Talent Support Project(Grant No.2020HYTP001)。
文摘The admissibility of the initial-boundary data,which characterizes the existence of solution for the initial-boundary value problem,is important.Based on the Fokas method and the inverse scattering transformation,an approach is developed to solve the initial-boundary value problem of the nonlinear Schrodinger equation on a finite interval.A necessary and sufficient condition for the admissibility of the initial-boundary data is given,and the reconstruction of the potential is obtained.
基金This work is supported by Thales Alenia Space.We are gratefully indebted to J.-F.Coulombel,F.GolseK.Aoki for many useful advices concerning this work and for their kind encouragements。
文摘We propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime.The typical example we discuss is the derivation of the Euler system from the BGK equation.The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flux might be far from the thermodynamic equilibrium.
