To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’...To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.展开更多
The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple ba...The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.展开更多
By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn...By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.展开更多
By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its...By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations.展开更多
In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational c...In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.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
This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which ...This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary展开更多
The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of ...The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.展开更多
In this paper, we are concerned with the existence of solutions to a class of Atangana-Baleanu-Caputo impulsive fractional differential equation. The existence and uniqueness of the solution of the fractional differen...In this paper, we are concerned with the existence of solutions to a class of Atangana-Baleanu-Caputo impulsive fractional differential equation. The existence and uniqueness of the solution of the fractional differential equation are obtained by Banach and Krasnoselakii fixed point theorems, and sufficient conditions for the existence and uniqueness of the solution are also developed. In addition, the Hyers-Ulam stability of the solution is considered. At last, an example is given to illustrate the main results.展开更多
Using Nevanlinna theory of the value distribution of meromorphic functions, the author investigates the problem of the growth of solutions of two types of algebraic differential equation and obtains some results.
A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive...A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.展开更多
In this paper, a new approach for solving the second order nonlinear ordinary differential equation y’’ + p(x;y)y’ = G(x;y) is considered. The results obtained by this approach are illustrated by examples and show ...In this paper, a new approach for solving the second order nonlinear ordinary differential equation y’’ + p(x;y)y’ = G(x;y) is considered. The results obtained by this approach are illustrated by examples and show that this method is powerful for this type of equations.展开更多
In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and t...In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.展开更多
In this paper, we first employ the complex method to deritive all meromorphic solutions of an auxiliary ordinary differential equation, and then find all meromorphic exact solutions of the modified ZK equation, modifi...In this paper, we first employ the complex method to deritive all meromorphic solutions of an auxiliary ordinary differential equation, and then find all meromorphic exact solutions of the modified ZK equation, modified KdV equation, nonlinear Klein-Gordon equation and modified BBM equation. Our work shows that there exist some classes of rational solutions wr,2 (z) and simple periodic solutions ws,1 (z) which are new and are not degenerated successively to by the elliptic function solutions.展开更多
In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s p...In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.展开更多
In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the nth-order homogeneou...In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the nth-order homogeneous linear differential equation with constant coefficients. In fact, we give the general formula for calculating the elements of the matrix exp[At] . We also give the results when the characteristic equation has the repeated roots. The present method is simpler and better than the other methods.展开更多
The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theore...The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.展开更多
A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations u' = nu, epsilon nu' + f(x, u, u')nu' - g(x, ...A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations u' = nu, epsilon nu' + f(x, u, u')nu' - g(x, u, u') nu = 0 (0 < epsilon much less than 1). The asymptotic expansions of solutions are constructed, the remainders are estimated. The former works are improved and generalized.展开更多
In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative a...In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software;the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.展开更多
This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtain...This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtained. Simultaneously the figures of the novel one-soliton solution and two-soliton solution were given and the singularity of the novel multisoliton solutions was discussed. Finally it was pointed out that the multisoliton solutions with sigularity can only be called soliton-like solutions. Key words differential-difference KdV equation - Hirota method - multisoliton-like solutions MSC 2000 35Q51 Project supported by the National Natural Science Foundation of China(Grant No. 19571052)展开更多
文摘To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.
文摘The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.
基金supported partly by the National Natural Science Foundation of China(12171050,11871260)National Science Foundation of Guangdong Province(2018A030313508)。
文摘By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.
文摘By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations.
文摘In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.
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This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary
文摘The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.
文摘In this paper, we are concerned with the existence of solutions to a class of Atangana-Baleanu-Caputo impulsive fractional differential equation. The existence and uniqueness of the solution of the fractional differential equation are obtained by Banach and Krasnoselakii fixed point theorems, and sufficient conditions for the existence and uniqueness of the solution are also developed. In addition, the Hyers-Ulam stability of the solution is considered. At last, an example is given to illustrate the main results.
基金The project Supported by NNSF of China(19971052)
文摘Using Nevanlinna theory of the value distribution of meromorphic functions, the author investigates the problem of the growth of solutions of two types of algebraic differential equation and obtains some results.
文摘A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.
文摘In this paper, a new approach for solving the second order nonlinear ordinary differential equation y’’ + p(x;y)y’ = G(x;y) is considered. The results obtained by this approach are illustrated by examples and show that this method is powerful for this type of equations.
基金Project Supported by the Science Fund of the Chinese Academy of Sciences
文摘In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.
基金supported by the NSFC(11271090)NSF of Guangdong(S2012010010121)
文摘In this paper, we first employ the complex method to deritive all meromorphic solutions of an auxiliary ordinary differential equation, and then find all meromorphic exact solutions of the modified ZK equation, modified KdV equation, nonlinear Klein-Gordon equation and modified BBM equation. Our work shows that there exist some classes of rational solutions wr,2 (z) and simple periodic solutions ws,1 (z) which are new and are not degenerated successively to by the elliptic function solutions.
文摘In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.
文摘In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the nth-order homogeneous linear differential equation with constant coefficients. In fact, we give the general formula for calculating the elements of the matrix exp[At] . We also give the results when the characteristic equation has the repeated roots. The present method is simpler and better than the other methods.
基金Supported by Guangdong Natural Science Foundation(2015A030313628,S2012010010376)Training plan for Distinguished Young Teachers in Higher Education of Guangdong(Yqgdufe1405)+1 种基金Guangdong Education Science Planning Project(2014GXJK091,GDJG20142304)the National Natural Science Foundation of China(11301140,11101096)
文摘The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.
文摘A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations u' = nu, epsilon nu' + f(x, u, u')nu' - g(x, u, u') nu = 0 (0 < epsilon much less than 1). The asymptotic expansions of solutions are constructed, the remainders are estimated. The former works are improved and generalized.
文摘In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software;the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.
文摘This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtained. Simultaneously the figures of the novel one-soliton solution and two-soliton solution were given and the singularity of the novel multisoliton solutions was discussed. Finally it was pointed out that the multisoliton solutions with sigularity can only be called soliton-like solutions. Key words differential-difference KdV equation - Hirota method - multisoliton-like solutions MSC 2000 35Q51 Project supported by the National Natural Science Foundation of China(Grant No. 19571052)