Linear differential-algebraic equations (DAEs) with time-varying coefficients A(t)x(1)(t) + B(t)x(t) = q(t), which are tractable with a higher index. are discussed. Their essential properties are investigated. Some eq...Linear differential-algebraic equations (DAEs) with time-varying coefficients A(t)x(1)(t) + B(t)x(t) = q(t), which are tractable with a higher index. are discussed. Their essential properties are investigated. Some equivalent system,,; are given. Using them the paper shows how to state properly initial and boundary conditions for these DAEs. The existence and uniqueness theory of the solution of the initial and boundary value problems for higher index DAEs are proposed.展开更多
In this article, linear regular index 2 DAEs A（t）[D（t）x（t）]＇ ＋ B（t）x（t） = q（t） are considered. Using a decoupling technique, initial condition and boundary condition are properly formulated. Regular inde...In this article, linear regular index 2 DAEs A（t）[D（t）x（t）]＇ ＋ B（t）x（t） = q（t） are considered. Using a decoupling technique, initial condition and boundary condition are properly formulated. Regular index 1 DAEs are obtained by a regularization method. We study the behavior of the solution of the regularization system via asymptotic expansions. The error analysis between the solutions of the DAEs and its regularization system is given.展开更多
Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As...Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.展开更多
In this paper,we propose a novel Legendre neural network combined with the extreme learning machine algorithm to solve variable coefficients linear delay differential-algebraic equations with weak discontinuities.Firs...In this paper,we propose a novel Legendre neural network combined with the extreme learning machine algorithm to solve variable coefficients linear delay differential-algebraic equations with weak discontinuities.First,the solution interval is divided into multiple subintervals by weak discontinuity points.Then,Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials on each subinterval.Finally,the parameters of the neural network are obtained by training with the extreme learning machine.The numerical examples show that the proposed method can effectively deal with the difficulty of numerical simulation caused by the discontinuities.展开更多
Linear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay.The corresponding convergence results are obtained and successfully confi...Linear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay.The corresponding convergence results are obtained and successfully confirmed by some numerical examples.The results obtained in this work extend the corresponding ones in literature.展开更多
This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-co...This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.展开更多
This paper deals with quasilinear elliptic equations of singular growth like-Δu-uΔ(u^(2))=a(x)u^(-1).We establish the existence of positive solutions for general a(x)∈L^(p)(Ω),p>2,whereΩis a bounded domain inℝ...This paper deals with quasilinear elliptic equations of singular growth like-Δu-uΔ(u^(2))=a(x)u^(-1).We establish the existence of positive solutions for general a(x)∈L^(p)(Ω),p>2,whereΩis a bounded domain inℝ^(N)with N≥1.展开更多
In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with ...In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods.展开更多
In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,t...In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.展开更多
Biomass is among the most important state variables used to characterize ecosystems. Estimation of tree biomass involves the development of species-specific “allometric equations” that describe the relationship betw...Biomass is among the most important state variables used to characterize ecosystems. Estimation of tree biomass involves the development of species-specific “allometric equations” that describe the relationship between tree biomass and tree diameter and/or height. While many allometric equations were developed for northern hemisphere and tropical species, rarely have they been developed for trees in arid ecosystems, limiting, amongst other things, our ability to estimate carbon stocks in arid regions. Acacia raddiana and A. tortilis are major components of savannas and arid regions in the Middle East and Africa, where they are considered keystone species. Using the opportunity that trees were being uprooted for land development, we measured height (H), north-south (C1) and east-west (C2) canopy diameters, stem diameter at 1.3 meters of the largest stem (D1.3 or DBH), and aboveground fresh and dry weight (FW and DW, respectively) of nine trees (n = 9) from each species. For A. tortilis only, we recorded the number of trunks, and measured the diameter of the largest trunk at ground level (D0). While the average crown (canopy) size (C1 + C2) was very similar among the two species, Acacia raddiana trees were found to be significantly taller than their Acacia tortilis counterparts. Results show that in the arid Arava (southern Israel), an average adult acacia tree has ~200 kg of aboveground dry biomass and that a typical healthy acacia ecosystem in this region, may include ~41 tons of tree biomass per km2. The coefficients of DBH (tree diameter at breast height) to biomass and wood volume, could be used by researchers studying acacia trees throughout the Middle East and Africa, enabling them to estimate biomass of acacia trees and to evaluate their importance for carbon stocks in their arid regions. Highlights: 1) Estimations of tree biomass in arid regions are rare. 2) Biomass allometric equations were developed for A. raddiana and A. tortilis trees. 3) Equations contribute to the estimation of carbon stocks in arid regions.展开更多
Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay di...Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data.展开更多
In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use ...In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use the Z1transformation to get the general solutions of some nonlinear partial differential equations for the first time, and use the general solutions to obtain the exact solutions of some typical definite solution problems.展开更多
ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lew...ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.展开更多
In this paper, we get the N-fold Darboux transformation with multi-parameters for the coupled mKdV equations with the help of a guage transformation of the spectral problem. As an application, some new multi-soliton s...In this paper, we get the N-fold Darboux transformation with multi-parameters for the coupled mKdV equations with the help of a guage transformation of the spectral problem. As an application, some new multi-soliton solutions and complexiton solutions are obtained from choosing the appropriate seed solution. All obtained solutions and N-fold Darboux transformations are expressed using the Vandermonde-like determinants.展开更多
In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with ...In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with N≥1,m>0,p>1,such that m(p-1)>1.We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)^(-α/β)((βt)^(-1/β)|x|)withα∈R andβ=α[m(p-1)-1]+p,regular or singular at the origin point.The existence and uniqueness of some solutions are established by the phase plane analysis method,and the asymptotic properties of the solutions near the origin and the infinity are also described.This paper extends the classical results of self-similar solutions for degeneratep-Laplace heat equation by Bidaut-Véron[Proc Royal Soc Edinburgh,2009,139:1-43]to the doubly nonlinear degenerate diffusion equations.展开更多
In this paper,we consider the p-Laplacian Schrödinger-Poisson equation with L^(2)-norm constraint-Δ_(p)u+|u|^(p-2)u+λu+(1/4π|x|*|u|^(2))u=|u|^(q-2)u,x∈R^(3),where 2≤p<3,5p/3<q<p*=3p/3-p,λ>0 is a...In this paper,we consider the p-Laplacian Schrödinger-Poisson equation with L^(2)-norm constraint-Δ_(p)u+|u|^(p-2)u+λu+(1/4π|x|*|u|^(2))u=|u|^(q-2)u,x∈R^(3),where 2≤p<3,5p/3<q<p*=3p/3-p,λ>0 is a Lagrange multiplier.We obtain the critical point of the corresponding functional of the problem on mass constraint by the variational method and the Mountain pass lemma,and then find a normalized solution to this equation.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
We implemented a two-step Asymmetric Perfectly Matched Layer (APML) model in High-Order Finite Difference Time Domain (FDTD) algorithm for solving two-dimensional Maxwell’s equations. Initially, we applied the APML m...We implemented a two-step Asymmetric Perfectly Matched Layer (APML) model in High-Order Finite Difference Time Domain (FDTD) algorithm for solving two-dimensional Maxwell’s equations. Initially, we applied the APML method to the standard second-order FDTD algorithm to derive a two-step time-staggered APML (APML-2SS) and a two-step time-centered APML (APML-2SC) formulation for these equations, afterwards, we extended these formulations in high-order FDTD algorithm in order to derive a APML high-order FDTD (APML-HOFDTD) formulation for our Maxwell’s equations. To examine the performance and check out the accuracy of APML model, we conducted a numerical study using a 2D fluid where the three derived formulations were to analyze selected phenomena in terahertz radiation production by the filamentation of two femtosecond laser beams in air plasma. Numerical results illustrated that the two-step APML model is sufficiently accurate for solving our 2D Maxwell’s equations in high-order FDTD discretization and it demonstrated a great performance in studying the THz radiation production.展开更多
We investigate the radial symmetry of minimizers on the Pohozaev-Nehari manifold to the Schrodinger Poisson equation with a general nonlinearity f(u).Particularly,we allow that f is L^(2) supercritical.The main result...We investigate the radial symmetry of minimizers on the Pohozaev-Nehari manifold to the Schrodinger Poisson equation with a general nonlinearity f(u).Particularly,we allow that f is L^(2) supercritical.The main result shows that minimizers are radially symmetric modulo suitable translations.展开更多
In the study of the extremal for Sobolev inequality on the Heisenberg group and the Cauchy-Riemann(CR)Yamabe problem,Jerison-Lee found a three-dimensional family of differential identities for critical exponent subell...In the study of the extremal for Sobolev inequality on the Heisenberg group and the Cauchy-Riemann(CR)Yamabe problem,Jerison-Lee found a three-dimensional family of differential identities for critical exponent subelliptic equation on Heisenberg groupℍn by using the computer in[5].They wanted to know whether there is a theoretical framework that would predict the existence and the structure of such formulae.With the help of dimension conservation and invariant tensors,we can answer the above question.展开更多
基金Project supported by the National Natural Science Foundation of China by Jiangsu Provincial Natural Science Foundation
文摘Linear differential-algebraic equations (DAEs) with time-varying coefficients A(t)x(1)(t) + B(t)x(t) = q(t), which are tractable with a higher index. are discussed. Their essential properties are investigated. Some equivalent system,,; are given. Using them the paper shows how to state properly initial and boundary conditions for these DAEs. The existence and uniqueness theory of the solution of the initial and boundary value problems for higher index DAEs are proposed.
基金Project supported by the Foundation for the Authors of the National Excellent Doctoral Thesis Award of China (200720)
文摘In this article, linear regular index 2 DAEs A（t）[D（t）x（t）]＇ ＋ B（t）x（t） = q（t） are considered. Using a decoupling technique, initial condition and boundary condition are properly formulated. Regular index 1 DAEs are obtained by a regularization method. We study the behavior of the solution of the regularization system via asymptotic expansions. The error analysis between the solutions of the DAEs and its regularization system is given.
基金supported by the Natural Science Foundation of China(NSFC)under grant 11501436Young Talent fund of University Association for Science and Technology in Shaanxi,China(20170701)
文摘Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.
基金supported by the National Natural Science Foundation of China(No.11971412)the Natural Science Foundation of Hunan Province of China(No.2018JJ2378)Scientific Research Fund of Hunan Provincial Science and Technology Department(No.2018WK4006).
文摘In this paper,we propose a novel Legendre neural network combined with the extreme learning machine algorithm to solve variable coefficients linear delay differential-algebraic equations with weak discontinuities.First,the solution interval is divided into multiple subintervals by weak discontinuity points.Then,Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials on each subinterval.Finally,the parameters of the neural network are obtained by training with the extreme learning machine.The numerical examples show that the proposed method can effectively deal with the difficulty of numerical simulation caused by the discontinuities.
基金This work is supported by NSF of China(10971175)Specialized Research Fund for the Doctoral Program of Higher Education of China(20094301110001)+2 种基金Program for Changjiang Scholars and Innovative Research Team in University of China(IRT1179)NSF of Hunan Province(10JJ7001)the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province,and Fund Project of Hunan Province Education Office(11C1220).
文摘Linear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay.The corresponding convergence results are obtained and successfully confirmed by some numerical examples.The results obtained in this work extend the corresponding ones in literature.
文摘This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.
基金Supported by National Science Foundation of China(11971027,12171497)。
文摘This paper deals with quasilinear elliptic equations of singular growth like-Δu-uΔ(u^(2))=a(x)u^(-1).We establish the existence of positive solutions for general a(x)∈L^(p)(Ω),p>2,whereΩis a bounded domain inℝ^(N)with N≥1.
基金Supported by the National Natural Science Foundation of China(11671403,11671236,12101192)Henan Provincial General Natural Science Foundation Project(232300420113)。
文摘In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods.
基金Supported by Research Project Supported by Shanxi Scholarship Council of China(2021-029)International Cooperation Base and Platform Project of Shanxi Province(202104041101019)+2 种基金Basic Research Plan of Shanxi Province(202203021211129)Shanxi Province Natural Science Research(202203021212249)Special/Youth Foundation of Taiyuan University of Technology(2022QN101)。
文摘In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.
文摘Biomass is among the most important state variables used to characterize ecosystems. Estimation of tree biomass involves the development of species-specific “allometric equations” that describe the relationship between tree biomass and tree diameter and/or height. While many allometric equations were developed for northern hemisphere and tropical species, rarely have they been developed for trees in arid ecosystems, limiting, amongst other things, our ability to estimate carbon stocks in arid regions. Acacia raddiana and A. tortilis are major components of savannas and arid regions in the Middle East and Africa, where they are considered keystone species. Using the opportunity that trees were being uprooted for land development, we measured height (H), north-south (C1) and east-west (C2) canopy diameters, stem diameter at 1.3 meters of the largest stem (D1.3 or DBH), and aboveground fresh and dry weight (FW and DW, respectively) of nine trees (n = 9) from each species. For A. tortilis only, we recorded the number of trunks, and measured the diameter of the largest trunk at ground level (D0). While the average crown (canopy) size (C1 + C2) was very similar among the two species, Acacia raddiana trees were found to be significantly taller than their Acacia tortilis counterparts. Results show that in the arid Arava (southern Israel), an average adult acacia tree has ~200 kg of aboveground dry biomass and that a typical healthy acacia ecosystem in this region, may include ~41 tons of tree biomass per km2. The coefficients of DBH (tree diameter at breast height) to biomass and wood volume, could be used by researchers studying acacia trees throughout the Middle East and Africa, enabling them to estimate biomass of acacia trees and to evaluate their importance for carbon stocks in their arid regions. Highlights: 1) Estimations of tree biomass in arid regions are rare. 2) Biomass allometric equations were developed for A. raddiana and A. tortilis trees. 3) Equations contribute to the estimation of carbon stocks in arid regions.
文摘Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data.
文摘In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use the Z1transformation to get the general solutions of some nonlinear partial differential equations for the first time, and use the general solutions to obtain the exact solutions of some typical definite solution problems.
文摘ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.
文摘In this paper, we get the N-fold Darboux transformation with multi-parameters for the coupled mKdV equations with the help of a guage transformation of the spectral problem. As an application, some new multi-soliton solutions and complexiton solutions are obtained from choosing the appropriate seed solution. All obtained solutions and N-fold Darboux transformations are expressed using the Vandermonde-like determinants.
基金supported by the NSFC(12271178,12171166)the Guangzhou Basic and Applied Basic Research Foundation(2024A04J2022)the TCL Young Scholar(2024-2027).
文摘In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with N≥1,m>0,p>1,such that m(p-1)>1.We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)^(-α/β)((βt)^(-1/β)|x|)withα∈R andβ=α[m(p-1)-1]+p,regular or singular at the origin point.The existence and uniqueness of some solutions are established by the phase plane analysis method,and the asymptotic properties of the solutions near the origin and the infinity are also described.This paper extends the classical results of self-similar solutions for degeneratep-Laplace heat equation by Bidaut-Véron[Proc Royal Soc Edinburgh,2009,139:1-43]to the doubly nonlinear degenerate diffusion equations.
基金supported by the National Natural Science Foundation of China(No.12461024)the Natural Science Research Project of Department of Education of Guizhou Province(Nos.QJJ2023012,QJJ2023061,QJJ2023062)the Natural Science Research Project of Guizhou Minzu University(No.GZMUZK[2022]YB06)。
文摘In this paper,we consider the p-Laplacian Schrödinger-Poisson equation with L^(2)-norm constraint-Δ_(p)u+|u|^(p-2)u+λu+(1/4π|x|*|u|^(2))u=|u|^(q-2)u,x∈R^(3),where 2≤p<3,5p/3<q<p*=3p/3-p,λ>0 is a Lagrange multiplier.We obtain the critical point of the corresponding functional of the problem on mass constraint by the variational method and the Mountain pass lemma,and then find a normalized solution to this equation.
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.
文摘We implemented a two-step Asymmetric Perfectly Matched Layer (APML) model in High-Order Finite Difference Time Domain (FDTD) algorithm for solving two-dimensional Maxwell’s equations. Initially, we applied the APML method to the standard second-order FDTD algorithm to derive a two-step time-staggered APML (APML-2SS) and a two-step time-centered APML (APML-2SC) formulation for these equations, afterwards, we extended these formulations in high-order FDTD algorithm in order to derive a APML high-order FDTD (APML-HOFDTD) formulation for our Maxwell’s equations. To examine the performance and check out the accuracy of APML model, we conducted a numerical study using a 2D fluid where the three derived formulations were to analyze selected phenomena in terahertz radiation production by the filamentation of two femtosecond laser beams in air plasma. Numerical results illustrated that the two-step APML model is sufficiently accurate for solving our 2D Maxwell’s equations in high-order FDTD discretization and it demonstrated a great performance in studying the THz radiation production.
基金supported by the NSFC(12031015)Song's research was supported by the Shuimu Tsinghua Scholar Program,the National Funded Postdoctoral Research Program(GZB20230368)the China Postdoctoral Science Foundation(2024T170452)。
文摘We investigate the radial symmetry of minimizers on the Pohozaev-Nehari manifold to the Schrodinger Poisson equation with a general nonlinearity f(u).Particularly,we allow that f is L^(2) supercritical.The main result shows that minimizers are radially symmetric modulo suitable translations.
基金supported by the National Natural Science Foundation of China(12141105,12471194)the first author’s research also was supported by the National Key Research and Development Project(SQ2020YFA070080).
文摘In the study of the extremal for Sobolev inequality on the Heisenberg group and the Cauchy-Riemann(CR)Yamabe problem,Jerison-Lee found a three-dimensional family of differential identities for critical exponent subelliptic equation on Heisenberg groupℍn by using the computer in[5].They wanted to know whether there is a theoretical framework that would predict the existence and the structure of such formulae.With the help of dimension conservation and invariant tensors,we can answer the above question.
