In this paper,we get a necessary and sufficient condition such that a class of differential inequalities hold.Using this necessary and sufficient condition,we prove that a class of first order nonhomogeneous ordinary ...In this paper,we get a necessary and sufficient condition such that a class of differential inequalities hold.Using this necessary and sufficient condition,we prove that a class of first order nonhomogeneous ordinary differential equations have the Hyers-Ulam stability.And then,we prove that some first order nonhomogeneous ordinary differential equations and some second order nonhomogeneous ordinary differential equations do not have the Hyers-Ulam instability under some suitable conditions.展开更多
Channel prediction is critical to address the channel aging issue in mobile scenarios.Existing channel prediction techniques are mainly designed for discrete channel prediction,which can only predict the future channe...Channel prediction is critical to address the channel aging issue in mobile scenarios.Existing channel prediction techniques are mainly designed for discrete channel prediction,which can only predict the future channel in a fixed time slot per frame,while the other intra-frame channels are usually recovered by interpolation.However,these approaches suffer from a serious interpolation loss,especially for mobile millimeter-wave communications.To solve this challenging problem,we propose a tensor neural ordinary differential equation(TN-ODE)based continuous-time channel prediction scheme to realize the direct prediction of intra-frame channels.Specifically,inspired by the recently developed continuous mapping model named neural ODE in the field of machine learning,we first utilize the neural ODE model to predict future continuous-time channels.To improve the channel prediction accuracy and reduce computational complexity,we then propose the TN-ODE scheme to learn the structural characteristics of the high-dimensional channel by low-dimensional learnable transform.Simulation results show that the proposed scheme is able to achieve higher intra-frame channel prediction accuracy than existing schemes.展开更多
Existence of positive solution is established for boundary value problems of nonsingular for a class quasi-linear ordinary differential equation on the semi-infinite interval. The results are obtained by using the non...Existence of positive solution is established for boundary value problems of nonsingular for a class quasi-linear ordinary differential equation on the semi-infinite interval. The results are obtained by using the nonlinear alternative of Leray-Schauder method.展开更多
By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differenti...By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].展开更多
The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value...The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.展开更多
This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The gener...This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.展开更多
In this paper we study the singular perturbation of boundary value problems with perturbations both in the operator and in the interval ends. So as to prove the existence and uniqueness of solution of perturbed proble...In this paper we study the singular perturbation of boundary value problems with perturbations both in the operator and in the interval ends. So as to prove the existence and uniqueness of solution of perturbed problem, to establish the asymptotic expression involving three parameters. Thus, the iterative equation of finding the asymptotic solution is derived and the estimation of the remainder term is given out. We extend results of [l]-[5].展开更多
This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether me...This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.展开更多
In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the B...In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.展开更多
By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation abundant types of explicit and exact solutions for the double sinh-Gordon equation are derived in a simple...By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation abundant types of explicit and exact solutions for the double sinh-Gordon equation are derived in a simple manner.展开更多
基金Supported by Natural Science Research Projects of Liaoning Province Education Department(Grant No.LJ212410146024).
文摘In this paper,we get a necessary and sufficient condition such that a class of differential inequalities hold.Using this necessary and sufficient condition,we prove that a class of first order nonhomogeneous ordinary differential equations have the Hyers-Ulam stability.And then,we prove that some first order nonhomogeneous ordinary differential equations and some second order nonhomogeneous ordinary differential equations do not have the Hyers-Ulam instability under some suitable conditions.
基金supported in part by the National Key Research and Development Program of China(Grant No.2020YFB1805005)in part by the National Natural Science Foundation of China(Grant No.62031019)in part by the European Commission through the H2020-MSCA-ITN META WIRELESS Research Project under Grant 956256。
文摘Channel prediction is critical to address the channel aging issue in mobile scenarios.Existing channel prediction techniques are mainly designed for discrete channel prediction,which can only predict the future channel in a fixed time slot per frame,while the other intra-frame channels are usually recovered by interpolation.However,these approaches suffer from a serious interpolation loss,especially for mobile millimeter-wave communications.To solve this challenging problem,we propose a tensor neural ordinary differential equation(TN-ODE)based continuous-time channel prediction scheme to realize the direct prediction of intra-frame channels.Specifically,inspired by the recently developed continuous mapping model named neural ODE in the field of machine learning,we first utilize the neural ODE model to predict future continuous-time channels.To improve the channel prediction accuracy and reduce computational complexity,we then propose the TN-ODE scheme to learn the structural characteristics of the high-dimensional channel by low-dimensional learnable transform.Simulation results show that the proposed scheme is able to achieve higher intra-frame channel prediction accuracy than existing schemes.
文摘Existence of positive solution is established for boundary value problems of nonsingular for a class quasi-linear ordinary differential equation on the semi-infinite interval. The results are obtained by using the nonlinear alternative of Leray-Schauder method.
基金Project supported by the National Natural Science Foundation of China.
文摘By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].
基金Supported by the Scientific Research Foundation for Advisor Program of Higher Education of Gansu Province(1009-6)Supported by the Scientific Research Foundation for Youth Scholars of Hexi University(qn201015)
文摘The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.
基金supported by the National Natural Science Foundation of China (Grant No 10872037)the Natural Science Foundation of Anhui Province of China (Grant No 070416226)
文摘This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.
基金This research was supported by Fujian Science Foundation.
文摘In this paper we study the singular perturbation of boundary value problems with perturbations both in the operator and in the interval ends. So as to prove the existence and uniqueness of solution of perturbed problem, to establish the asymptotic expression involving three parameters. Thus, the iterative equation of finding the asymptotic solution is derived and the estimation of the remainder term is given out. We extend results of [l]-[5].
基金Project supported by the National Natural Science Foundation of China(Grant No10572021)Doctoral Programme Foundation of Institution of Higher Education of China(Grant No20040007022)
文摘This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.
文摘In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.
基金supported by the National Natural Science Foundation of China (Grant No 10672053) the Natural Science Foundation of Hunan Province of China (Grant No 05JJ30007)the Scientific Research Fund of Hunan Institute of Science and Technology of China (Grant No 2007Y047)
文摘By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation abundant types of explicit and exact solutions for the double sinh-Gordon equation are derived in a simple manner.
