Using the Nevanlinna theory of the value distribution of meromorphic functions and theory of differential algebra, we investigate the problem of the forms of meromorphic solutions of some specific systems of generaliz...Using the Nevanlinna theory of the value distribution of meromorphic functions and theory of differential algebra, we investigate the problem of the forms of meromorphic solutions of some specific systems of generalized higher order algebraic differential equations with exponential coefficients and obtain some results.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
The oscillatory behavior of neutral differential equation with positive and negative coefficients is investigated by mathematics analysis technique and the fixed point principle. Some sufficient conditions for oscilla...The oscillatory behavior of neutral differential equation with positive and negative coefficients is investigated by mathematics analysis technique and the fixed point principle. Some sufficient conditions for oscillation of neutral differential equation with positive and negative coefficients are obtained.展开更多
A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpos...A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpose is to prove that the semi-implicit Euler solutions converge to the true solutions in the mean-square sense. An example is given for illustration.展开更多
Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential e...Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential equations with variable coefficients. On most occasions and due to the nonuniformity nature, nonlinearity property can cause the equations of the kinds. Using the model, the satisfactory valuable results with only a few units can be obtained.展开更多
In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted,...In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1 - 4] are improved and extended by means of the modified method of multiple scales.展开更多
A class of stochastic differential equations(SDEs) driven by semimartingale with non-Lipschitz coefficients was studied.By using Gronwall inequality,the non-confluence of solutions is proved under the general conditions.
How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducin...How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.展开更多
This article investigates the property of linearly dependence of solutions f（z） and f（z ＋ 2πi） for higher order linear differential equations with entire periodic coefficients.
In this paper, the second order nonlinear elliptic differential equations (E) (n)Sigma (i,j=1) partial derivative/partial derivativex(j)[a(i,j)(x,y) partial derivative/partial derivativex(j)y] + q(x)f(y) = e(x) are co...In this paper, the second order nonlinear elliptic differential equations (E) (n)Sigma (i,j=1) partial derivative/partial derivativex(j)[a(i,j)(x,y) partial derivative/partial derivativex(j)y] + q(x)f(y) = e(x) are considered in an exterior Omega subset of R-n, where q(x) is allowed to change sign. Some sufficient conditions for any solutions y(x) of (E) to be satisfied liminf\\x\--> infinity \y(x)\ = 0 are obtained. Particularly, these results improve the previous results for second order ordinary differential equations.展开更多
A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable ...A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.展开更多
In this paper,the polymer chain of rotator(PCOR) equation of state(EOS) was used together with an EOS/G^E mixing rule(MHV1) and the Wilson's equation as an excess-Gibbs-energy model in the proposed approach to ext...In this paper,the polymer chain of rotator(PCOR) equation of state(EOS) was used together with an EOS/G^E mixing rule(MHV1) and the Wilson's equation as an excess-Gibbs-energy model in the proposed approach to extend the capability and improve the accuracy of the PCOR EOS for predicting the Henry's constant of solutions containing polymers.The results of the proposed method compared with two equation of state(van der Waals and GC-Flory) and three activity coefficient models(UNIFAC,UNIFAC-FV and Entropic-FV) indicated that the PCOR EOS/Wilson's equation provided more accurate results.The interaction parameters of Wilson's equation were fitted with Henry's constant experimental data and the property parameters of PCOR,a and b,were fitted with experimental volume data(Tait equation).As a result,the present work provided a simple and useful model for prediction of Henry's constant for polymer solutions.展开更多
This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect...This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.展开更多
In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part,...In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.展开更多
Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian ...Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.展开更多
The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to impl...The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to implement and mathematically simple.In this paper,the commonly⁃used multiquadric RBF,conical RBF,and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients.Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems.It is shown that the conical RBF numerical results were more stable than the other two radial basis functions.From the comparison of three commonly⁃used RBFs,one may obtain the best numerical solutions for boundary value problems.展开更多
In this paper, we give sufficient conditions for the existence and uniqueness of asymptotically w-antiperiodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space when w ...In this paper, we give sufficient conditions for the existence and uniqueness of asymptotically w-antiperiodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space when w is an integer. This is done using the Banach fixed point theorem. An example involving the heat operator is discussed as an illustration of the theory.展开更多
In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-u...In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-uh = O(hn+2), n ≥ 2, at (n + 1)-order Lobatto points in each element respectively is proved. Finally the theoretical results are tested by a numerical example.展开更多
This paper studies for ward-back ward differential equations with Poisson jumps and with stopping time as termination. Under some weak monotonicity conditions and for non-Lipschitzian coefficients, the existence and u...This paper studies for ward-back ward differential equations with Poisson jumps and with stopping time as termination. Under some weak monotonicity conditions and for non-Lipschitzian coefficients, the existence and uniqueness of solutions are proved via a purely probabilistic approach, while a priori estimate is given. Here, we allow the forward equation to be degenerate.展开更多
A formula has been deduced, and named as 2ESKV, relating the concentration of two consecutive substrates of a metabolic cycle with the kinetic constants of the two enzymes involved in their synthesis and degradation. ...A formula has been deduced, and named as 2ESKV, relating the concentration of two consecutive substrates of a metabolic cycle with the kinetic constants of the two enzymes involved in their synthesis and degradation. After application of formula 2ESKV to consecutive pairs of substrates and enzymes, a system of interrelated equations was obtained allowing a great variety of theoretical postulates to calculate, back and forth: bunches of unknown enzyme kinetic constants and substrate concentrations, from complementary sets of known data. This vision of a metabolic cycle is of partial application to irreversible pathways and can be useful for modeling and understanding of metabolomics data. To our knowledge, the formula 2ESKV is here described for the first time.展开更多
基金Project Supported by the Natural Science Foundation of China (10471065)the Natural Science Foundation of Guangdong Province (04010474)
文摘Using the Nevanlinna theory of the value distribution of meromorphic functions and theory of differential algebra, we investigate the problem of the forms of meromorphic solutions of some specific systems of generalized higher order algebraic differential equations with exponential coefficients and obtain some results.
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.
文摘The oscillatory behavior of neutral differential equation with positive and negative coefficients is investigated by mathematics analysis technique and the fixed point principle. Some sufficient conditions for oscillation of neutral differential equation with positive and negative coefficients are obtained.
基金National Natural Science Foundations of China(Nos.11401261,11471071)Qing Lan Project of Jiangsu Province,China(No.2012)+2 种基金Natural Science Foundation of Higher Education Institutions of Jiangsu Province(No.13KJB110005)the Grant of Jiangsu Second Normal University(No.JSNU-ZY-02)the Jiangsu Government Overseas Study Scholarship,China
文摘A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpose is to prove that the semi-implicit Euler solutions converge to the true solutions in the mean-square sense. An example is given for illustration.
文摘Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential equations with variable coefficients. On most occasions and due to the nonuniformity nature, nonlinearity property can cause the equations of the kinds. Using the model, the satisfactory valuable results with only a few units can be obtained.
基金The Project Supported by the National Natural Science Foundation of China
文摘In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1 - 4] are improved and extended by means of the modified method of multiple scales.
基金National Natural Science Foundation of China(No.71171003)Natural Science Foundation of Anhui Province of China(No.090416225)Natural Science Foundation of Universities of Anhui Province of China(No.KJ2010A037)
文摘A class of stochastic differential equations(SDEs) driven by semimartingale with non-Lipschitz coefficients was studied.By using Gronwall inequality,the non-confluence of solutions is proved under the general conditions.
基金Project supported by the National Natural Science Foundation of China(No.10461005)
文摘How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.
基金Supported by the Brain Pool Program of Korea Federation of Science and Technology Societies(072-1-3-0164)the Natural Science Foundation of Guangdong Province in China(06025059)
文摘This article investigates the property of linearly dependence of solutions f（z） and f（z ＋ 2πi） for higher order linear differential equations with entire periodic coefficients.
基金Project supported by the Natural Science Foundation of Guangdong Province
文摘In this paper, the second order nonlinear elliptic differential equations (E) (n)Sigma (i,j=1) partial derivative/partial derivativex(j)[a(i,j)(x,y) partial derivative/partial derivativex(j)y] + q(x)f(y) = e(x) are considered in an exterior Omega subset of R-n, where q(x) is allowed to change sign. Some sufficient conditions for any solutions y(x) of (E) to be satisfied liminf\\x\--> infinity \y(x)\ = 0 are obtained. Particularly, these results improve the previous results for second order ordinary differential equations.
文摘A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.
基金financial support provided by Islamic Azad University of Mahshahr Branch,Iran
文摘In this paper,the polymer chain of rotator(PCOR) equation of state(EOS) was used together with an EOS/G^E mixing rule(MHV1) and the Wilson's equation as an excess-Gibbs-energy model in the proposed approach to extend the capability and improve the accuracy of the PCOR EOS for predicting the Henry's constant of solutions containing polymers.The results of the proposed method compared with two equation of state(van der Waals and GC-Flory) and three activity coefficient models(UNIFAC,UNIFAC-FV and Entropic-FV) indicated that the PCOR EOS/Wilson's equation provided more accurate results.The interaction parameters of Wilson's equation were fitted with Henry's constant experimental data and the property parameters of PCOR,a and b,were fitted with experimental volume data(Tait equation).As a result,the present work provided a simple and useful model for prediction of Henry's constant for polymer solutions.
基金supported by the National Natural Science Foundation of China (11101096)
文摘This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.
基金Provincial Science and Technology Foundation of Guizhou
文摘In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.
基金The supports of the National Natural Science Foundation of China(Grant Nos.51725804 and U1711264)the Research Fund for State Key Laboratories of Ministry of Science and Technology of China(SLDRCE19-B-23)the Shanghai Post-Doctoral Excellence Program(2022558)。
文摘Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.
基金the Natural Science Foundation of Anhui Province(Grant No.1908085QA09)the University Natural Science Research Project of Anhui Province(KJ2019A0591).
文摘The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to implement and mathematically simple.In this paper,the commonly⁃used multiquadric RBF,conical RBF,and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients.Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems.It is shown that the conical RBF numerical results were more stable than the other two radial basis functions.From the comparison of three commonly⁃used RBFs,one may obtain the best numerical solutions for boundary value problems.
文摘In this paper, we give sufficient conditions for the existence and uniqueness of asymptotically w-antiperiodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space when w is an integer. This is done using the Banach fixed point theorem. An example involving the heat operator is discussed as an illustration of the theory.
基金The work was supported in part by the Special Funds of State Major Basic Research Projects (Grant No.1999032804) by scientific Research Fund of Hunan Provincial Education Department (03C508).
文摘In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-uh = O(hn+2), n ≥ 2, at (n + 1)-order Lobatto points in each element respectively is proved. Finally the theoretical results are tested by a numerical example.
文摘This paper studies for ward-back ward differential equations with Poisson jumps and with stopping time as termination. Under some weak monotonicity conditions and for non-Lipschitzian coefficients, the existence and uniqueness of solutions are proved via a purely probabilistic approach, while a priori estimate is given. Here, we allow the forward equation to be degenerate.
文摘A formula has been deduced, and named as 2ESKV, relating the concentration of two consecutive substrates of a metabolic cycle with the kinetic constants of the two enzymes involved in their synthesis and degradation. After application of formula 2ESKV to consecutive pairs of substrates and enzymes, a system of interrelated equations was obtained allowing a great variety of theoretical postulates to calculate, back and forth: bunches of unknown enzyme kinetic constants and substrate concentrations, from complementary sets of known data. This vision of a metabolic cycle is of partial application to irreversible pathways and can be useful for modeling and understanding of metabolomics data. To our knowledge, the formula 2ESKV is here described for the first time.
