In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 rep...In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.展开更多
By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε...By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.展开更多
The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic sol...The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic solutions,we transform such equations into boundary value problems with discontinuous coefficients by the properties of Fourier analysis.In view of the analytical Riemann-Hilbert method,the generalized Liouville theorem and Sokhotski-Plemelj formula,we get the uniqueness and existence of solutions for such problems,and study the asymptotic property of solutions at nodes.Therefore,this paper improves the theory of singular integral equations and boundary value problems.展开更多
In this paper,a class of quaternion-valued cellular neural networks(QVCNNS)with time-varying delays are considered.Combining graph theory with the continuation theorem of Mawhin’s coincidence degree theory as well as...In this paper,a class of quaternion-valued cellular neural networks(QVCNNS)with time-varying delays are considered.Combining graph theory with the continuation theorem of Mawhin’s coincidence degree theory as well as Lyapunov functional method,we establish new criteria on the existence and exponential stability of periodic solutions for QVCNNS by removing the assumptions for the boundedness on the activation functions and the assumptions that the values of the activation functions are zero at origin.Hence,our results are less conservative and new.展开更多
We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave pr...We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiffness matrix can be used repeatedly in multi-query scenarios.展开更多
The reliability assessment of unit-system near two levels is the mostimportant content in the reliability multi-level synthesis of complex systems. Introducing theinformation theory into system reliability assessment,...The reliability assessment of unit-system near two levels is the mostimportant content in the reliability multi-level synthesis of complex systems. Introducing theinformation theory into system reliability assessment, using the addible characteristic ofinformation quantity and the principle of equivalence of information quantity, an entropy method ofdata information conversion is presented for the system consisted of identical exponential units.The basic conversion formulae of entropy method of unit test data are derived based on the principleof information quantity equivalence. The general models of entropy method synthesis assessment forsystem reliability approximate lower limits are established according to the fundamental principleof the unit reliability assessment. The applications of the entropy method are discussed by way ofpractical examples. Compared with the traditional methods, the entropy method is found to be validand practicable and the assessment results are very satisfactory.展开更多
Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear ...Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.展开更多
We perform benchmark calculations of the p-wave resonances in the exponentially cosine screened Coulomb potential using the uniform complex-scaling generalized pseudo-spectral method.The present results show significa...We perform benchmark calculations of the p-wave resonances in the exponentially cosine screened Coulomb potential using the uniform complex-scaling generalized pseudo-spectral method.The present results show significant improvement in calculation accuracy compared to previous predictions and correct the misidentification of resonance electron configuration in previous works.It is found that the resonance states approximately follow an n^(2)-scaling law which is similar to the bound counterparts.The birth of a new resonance would distort the trajectory of an adjacent higher-lying resonance.展开更多
This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,...This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,and Applications”.With respect to classical Finite Elements Methods,Trefftz methods are unconventional methods because of the way the basis functions are generated.Trefftz discontinuous Galerkin(TDG)methods have recently shown potential for numerical approximation of transport equations[6,26]with vectorial exponential modes.This paper focuses on a proof of the approximation properties of these exponential solutions.We show that vectorial exponential functions can achieve high order convergence.The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.展开更多
We present the application of differential quadrature（DQ） method for the buckling analysis of nanobeams with exponentially varying stiffness based on four different beam theories of Euler-Bernoulli, Timoshenko, Redd...We present the application of differential quadrature（DQ） method for the buckling analysis of nanobeams with exponentially varying stiffness based on four different beam theories of Euler-Bernoulli, Timoshenko, Reddy, and Levison.The formulation is based on the nonlocal elasticity theory of Eringen. New results are presented for the guided and simply supported guided boundary conditions. Numerical results are obtained to investigate the effects of the nonlocal parameter,length-to-height ratio, boundary condition, and nonuniform parameter on the critical buckling load parameter. It is observed that the critical buckling load decreases with increase in the nonlocal parameter while the critical buckling load parameter increases with increase in the length-to-height ratio.展开更多
A non-orthogonal predefined exponential basis set is used to handle half-bounded domains in multi domain spectral method (MDSM). This approach works extremely well for real-valued semi-infinite differential problems. ...A non-orthogonal predefined exponential basis set is used to handle half-bounded domains in multi domain spectral method (MDSM). This approach works extremely well for real-valued semi-infinite differential problems. It spans simultaneously wide range of exponential decay rates with multi scaling and does not suffer from zero crossing. These two conditions are necessary for many physical problems. For comparison, the method is used to solve different problems and compared with analytical and published results. The comparison exhibits the strengths and accuracy of the presented basis set.展开更多
We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponent...We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams–Bashforth–Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.展开更多
In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathema...In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.展开更多
The effects of plasma screening on the ^(1)P^(o) resonance states of H-and He below the n=3 and n=4 thresholds of the respective subsystemsare investigated using the stabilization method and correlated exponential wav...The effects of plasma screening on the ^(1)P^(o) resonance states of H-and He below the n=3 and n=4 thresholds of the respective subsystemsare investigated using the stabilization method and correlated exponential wave functions.Two plasma mediums,namely,the Debye plasma and quantum plasma environments are considered.The screened Coulomb potential(SCP)obtained from Debye-Hückel model is used to represent Debye plasma environments and the exponential cosine screened Coulomb potential(ECSCP)obtained from a modified Debye-Hückel model is used to represent quantum plasma environments.The resonance parameters(resonance positions and widths)are presented in terms of the screening parameters.展开更多
In this paper, an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions. Regularized long wave equation （RLW） is integrated fully by using an exponentia...In this paper, an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions. Regularized long wave equation （RLW） is integrated fully by using an exponential B-spline Galerkin method in space together with Crank-Nicolson method in time. Three numerical examples related to propagation of sin- gle solitary wave, interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method. Obtained results are compared with some early studies.展开更多
By the discussion of the formula and properties of (4,4) parametric form rational approximation to function exp(q), the fourth order derivative one_step exponentially fitted method and the third order derivative hybri...By the discussion of the formula and properties of (4,4) parametric form rational approximation to function exp(q), the fourth order derivative one_step exponentially fitted method and the third order derivative hybrid one_step exponentially fitted method are presented, their order p satisfying 6≤p≤8. The necessary and sufficient conditions for the two methods to be A_ stable are given. Finally, for the fourth order derivative method, the error bound and the necessary and sufficient conditions for it to be median are discussed.展开更多
The Filon-type quadrature is efficient for highly oscillatory functions - Fourier transforms. Based on Cox and Matthews' ETD schemes, the higher order single step exponential time differencing schemes are presente...The Filon-type quadrature is efficient for highly oscillatory functions - Fourier transforms. Based on Cox and Matthews' ETD schemes, the higher order single step exponential time differencing schemes are presented based on the Filon-type integration and the A-stability of the two-order Adams-Bashforth exponential time differencing scheme is considered. The effectiveness and accuracy of the schemes is tested.展开更多
Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Th...Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Though, this paper shows that recent advance methods can be more favored. In this work, we have incorporated, throughout numerical comparison experiments, spectral methods, for the space discretization, in conjunction with second and fourth-order time integrating methods for approximating the solution of the reaction-diffusion differential equations. The results have revealed that these methods have advantages over the conventional methods, some of which to mention are: the ease of implementation, accuracy and CPU time.展开更多
We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponent...We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11671403 and 11671236)Henan Provincial General Natural Science Foundation Project(Grant No.232300420113)。
文摘In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.
基金supported by National Natural Science Foundation of China(No.12171152)。
文摘By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.
基金Supported by National Natural Science Foundation of China(Grant No.11971015).
文摘The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic solutions,we transform such equations into boundary value problems with discontinuous coefficients by the properties of Fourier analysis.In view of the analytical Riemann-Hilbert method,the generalized Liouville theorem and Sokhotski-Plemelj formula,we get the uniqueness and existence of solutions for such problems,and study the asymptotic property of solutions at nodes.Therefore,this paper improves the theory of singular integral equations and boundary value problems.
基金Supported by the Innovation Platform Open Fund in Hunan Province Colleges and Universities of China(201485).
文摘In this paper,a class of quaternion-valued cellular neural networks(QVCNNS)with time-varying delays are considered.Combining graph theory with the continuation theorem of Mawhin’s coincidence degree theory as well as Lyapunov functional method,we establish new criteria on the existence and exponential stability of periodic solutions for QVCNNS by removing the assumptions for the boundedness on the activation functions and the assumptions that the values of the activation functions are zero at origin.Hence,our results are less conservative and new.
基金part supported by the NSF Grants DMS-1912654 and DMS 2205590。
文摘We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiffness matrix can be used repeatedly in multi-query scenarios.
文摘The reliability assessment of unit-system near two levels is the mostimportant content in the reliability multi-level synthesis of complex systems. Introducing theinformation theory into system reliability assessment, using the addible characteristic ofinformation quantity and the principle of equivalence of information quantity, an entropy method ofdata information conversion is presented for the system consisted of identical exponential units.The basic conversion formulae of entropy method of unit test data are derived based on the principleof information quantity equivalence. The general models of entropy method synthesis assessment forsystem reliability approximate lower limits are established according to the fundamental principleof the unit reliability assessment. The applications of the entropy method are discussed by way ofpractical examples. Compared with the traditional methods, the entropy method is found to be validand practicable and the assessment results are very satisfactory.
基金supported by the National Natural Science Foundation of China(No.10632040)
文摘Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.
基金supported by the National Natural Science Foundation of China(Grant No.12174147)the Chinese Scholarship Council(Grant Nos.202108210152 and 202006175016).
文摘We perform benchmark calculations of the p-wave resonances in the exponentially cosine screened Coulomb potential using the uniform complex-scaling generalized pseudo-spectral method.The present results show significant improvement in calculation accuracy compared to previous predictions and correct the misidentification of resonance electron configuration in previous works.It is found that the resonance states approximately follow an n^(2)-scaling law which is similar to the bound counterparts.The birth of a new resonance would distort the trajectory of an adjacent higher-lying resonance.
文摘This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,and Applications”.With respect to classical Finite Elements Methods,Trefftz methods are unconventional methods because of the way the basis functions are generated.Trefftz discontinuous Galerkin(TDG)methods have recently shown potential for numerical approximation of transport equations[6,26]with vectorial exponential modes.This paper focuses on a proof of the approximation properties of these exponential solutions.We show that vectorial exponential functions can achieve high order convergence.The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.
文摘We present the application of differential quadrature（DQ） method for the buckling analysis of nanobeams with exponentially varying stiffness based on four different beam theories of Euler-Bernoulli, Timoshenko, Reddy, and Levison.The formulation is based on the nonlocal elasticity theory of Eringen. New results are presented for the guided and simply supported guided boundary conditions. Numerical results are obtained to investigate the effects of the nonlocal parameter,length-to-height ratio, boundary condition, and nonuniform parameter on the critical buckling load parameter. It is observed that the critical buckling load decreases with increase in the nonlocal parameter while the critical buckling load parameter increases with increase in the length-to-height ratio.
文摘A non-orthogonal predefined exponential basis set is used to handle half-bounded domains in multi domain spectral method (MDSM). This approach works extremely well for real-valued semi-infinite differential problems. It spans simultaneously wide range of exponential decay rates with multi scaling and does not suffer from zero crossing. These two conditions are necessary for many physical problems. For comparison, the method is used to solve different problems and compared with analytical and published results. The comparison exhibits the strengths and accuracy of the presented basis set.
基金The project supported by National Natural Science Foundation of China under Grant No.19902002
文摘We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams–Bashforth–Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.
文摘In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.
基金Supported by the Natural Science Foundation of Heilongjiang Province(LH2024A025)。
文摘The effects of plasma screening on the ^(1)P^(o) resonance states of H-and He below the n=3 and n=4 thresholds of the respective subsystemsare investigated using the stabilization method and correlated exponential wave functions.Two plasma mediums,namely,the Debye plasma and quantum plasma environments are considered.The screened Coulomb potential(SCP)obtained from Debye-Hückel model is used to represent Debye plasma environments and the exponential cosine screened Coulomb potential(ECSCP)obtained from a modified Debye-Hückel model is used to represent quantum plasma environments.The resonance parameters(resonance positions and widths)are presented in terms of the screening parameters.
基金supported by the Scientific and Technological Research Council of Turkey(Grant No.113F394)
文摘In this paper, an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions. Regularized long wave equation （RLW） is integrated fully by using an exponential B-spline Galerkin method in space together with Crank-Nicolson method in time. Three numerical examples related to propagation of sin- gle solitary wave, interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method. Obtained results are compared with some early studies.
基金the Science Technology Foundation of Ministry of Machine_ Buildin
文摘By the discussion of the formula and properties of (4,4) parametric form rational approximation to function exp(q), the fourth order derivative one_step exponentially fitted method and the third order derivative hybrid one_step exponentially fitted method are presented, their order p satisfying 6≤p≤8. The necessary and sufficient conditions for the two methods to be A_ stable are given. Finally, for the fourth order derivative method, the error bound and the necessary and sufficient conditions for it to be median are discussed.
基金Supported by National Natural Science Foundation of China(10571036)the Key Discipline Development Program of Beijing Municipal Commission (XK100080537)
基金Projects(02JJY2006, 03JJY2001) supported by Natural Science Foundation of Hunan Province project supported by JSPS Fellowship Research Program
文摘The Filon-type quadrature is efficient for highly oscillatory functions - Fourier transforms. Based on Cox and Matthews' ETD schemes, the higher order single step exponential time differencing schemes are presented based on the Filon-type integration and the A-stability of the two-order Adams-Bashforth exponential time differencing scheme is considered. The effectiveness and accuracy of the schemes is tested.
文摘Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Though, this paper shows that recent advance methods can be more favored. In this work, we have incorporated, throughout numerical comparison experiments, spectral methods, for the space discretization, in conjunction with second and fourth-order time integrating methods for approximating the solution of the reaction-diffusion differential equations. The results have revealed that these methods have advantages over the conventional methods, some of which to mention are: the ease of implementation, accuracy and CPU time.
基金The project supported by Liu Hui Applied Mathematics Center of Nankai University and 985 Education Development Plan of Tianjin University
文摘We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.
